Tiffin Girls Stage One - 48 Hour Maths Master Pack

1. What Tiffin Girls officially says about Stage One

For September 2027 entry, the school states that Stage One and Stage Two both test English and Maths. Stage One English is multiple choice. Stage One Maths uses mixed question types: some may be multiple choice and others require a numerical answer. All answers are entered on a computer-marked answer sheet. The school does not publish past papers.

The official admissions page currently lists Stage One on 1 and 2 October 2026. If your individual invitation gives a different date or session, follow the invitation.

Important: this pack uses the 19-topic 11+ Maths syllabus you uploaded as the practice framework. Tiffin Girls does not publish an official topic-by-topic Stage One syllabus, so no external pack can guarantee the exact question mix.

Official source: https://www.tiffingirls.org/admissions/year-7/

2. Best use of the next 48 hours

Day 1 morning - Core accuracy

Learn the formula wall below.
Do Topics 1-6.
Mark immediately and record every error type.
Redo only the incorrect varieties once.

Day 1 afternoon - Algebra and geometry

Do Topics 7-13.
Prioritise fractions, percentages, ratio, algebra, angles, area and units.
Finish with a 20-minute mixed set under time pressure.

Day 2 morning - Data and applied maths

Do Topics 14-19.
Practise graphs, mean/median, probability, Venn diagrams and multi-step word problems.
Redo yesterday's error list.

Day 2 afternoon/evening - Exam simulation

Do two short mixed timed papers rather than learning new topics.
After each, classify errors: knowledge, arithmetic, reading, units or time pressure.
Stop heavy work early enough to sleep properly.

Exam morning

10 minutes of fractions/percentages.
5 minutes of angle and area formulas.
No new material.
Eat normally, hydrate and arrive with enough time not to rush.

With only two days, accuracy on high-frequency foundations is more valuable than trying to master obscure material from scratch.

3. Formula and fact wall

Number and arithmetic

Divisibility: by 2 if last digit even; by 3 if digit sum divisible by 3; by 4 if last two digits divisible by 4; by 5 if last digit 0 or 5; by 9 if digit sum divisible by 9; by 10 if last digit 0.
BODMAS/BIDMAS: Brackets, Orders/Indices, Division and Multiplication from left to right, Addition and Subtraction from left to right.
Useful fraction-decimal-percentage facts: 1/2=0.5=50%, 1/4=0.25=25%, 3/4=0.75=75%, 1/5=0.2=20%, 1/10=0.1=10%, 1/8=0.125=12.5%, 3/8=0.375=37.5%, 5/8=0.625=62.5%, 7/8=0.875=87.5%, 1/3≈33.3%, 2/3≈66.7%.

Fractions, decimals and percentages

Add/subtract fractions: make denominators the same first.
Multiply fractions: numerator × numerator and denominator × denominator. Cancel before multiplying where possible.
Divide fractions: keep the first fraction, change ÷ to ×, flip the second fraction.
Percentage of amount = percentage/100 × amount.
Percentage change = change ÷ original × 100%.
Reverse percentage: if final amount is p% of original, original = final ÷ (p/100).

Ratio and proportion

Share total T in ratio a:b: one part = T ÷ (a+b).
Direct proportion: find the value of one unit first, then scale.
Scale factor = new length ÷ original length.
Map scale 1:n means 1 map unit represents n of the same real-world units.

Algebra and sequences

For a linear sequence with common difference d, nth term begins dn, then adjust by a constant.
Equation rule: whatever you do to one side, do to the other.
Substitution: replace the letter with its value before calculating.
Consecutive integers can be written n, n+1, n+2.

Geometry

Angles on a straight line = 180°.
Angles around a point = 360°.
Angles in a triangle = 180°.
Angles in a quadrilateral = 360°.
Vertically opposite angles are equal.
For parallel lines: corresponding angles are equal; alternate interior angles are equal.
Rectangle perimeter = 2(l+w).
Rectangle area = l×w.
Triangle area = 1/2 × base × perpendicular height.
Parallelogram area = base × perpendicular height.
Trapezium area = 1/2 × (sum of parallel sides) × perpendicular height.
Cube volume = side³. Cuboid volume = l×w×h.
Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh).

Measures and data

Speed = distance ÷ time; distance = speed × time; time = distance ÷ speed.
1 km=1000 m; 1 m=100 cm=1000 mm; 1 kg=1000 g; 1 L=1000 ml.
1 m²=10,000 cm². Area conversions square the linear conversion.
Mean = total ÷ number of values.
Range = highest - lowest.
Probability = favourable outcomes ÷ total equally likely outcomes.
Probabilities range from 0 to 1 and all mutually exclusive outcomes total 1.
Pie chart angle = fraction × 360°.

Finance

Profit = selling price - cost price.
Loss = cost price - selling price.
Profit percentage = profit ÷ cost price × 100%.
Simple interest = principal × rate × time.
Best buy: compare prices using the same quantity, usually cost per 100 g, per kg or per item.

4. High-value exam tactics

Read the final instruction twice. Many 11+ errors come from solving the calculation correctly but answering the wrong thing, such as giving the number of full coaches instead of the number needed.
Estimate before calculating. If an exact answer is far from the estimate, check it.
For multiple choice, eliminate impossible answers first. This is especially effective with parity, size, units and estimation.
Do not leave a numerical-answer box blank just because the arithmetic looks long. Look for factors, cancellation, complements to 10/100 and equivalent forms.
Write units beside intermediate results in measurement questions. This prevents km/m, kg/g and hour/minute mistakes.
On fractions, simplify at the end unless simplifying earlier makes the arithmetic easier.
On percentages, build from 10%, 5%, 1%, 25%, 50% and 75% rather than using long multiplication whenever possible.
On ratio questions, always calculate the total number of parts first.
On geometry diagrams, do not trust how an angle or length looks. Use only the stated information and known angle facts.
On data questions, inspect the scale before reading a bar or point. Check whether each minor interval is 1, 2, 5, 10 or something else.
On sequence questions, check first differences, then second differences, then multiplication/division, then alternating rules.
At the end, use remaining time only on flagged questions. Do not change an answer unless you can identify a specific error.

Fast mental shortcuts worth memorising

×25 = ×100 ÷4 ×50 = ×100 ÷2 ×5 = ×10 ÷2 ÷0.5 = ×2 ÷0.25 = ×4 10% = ÷10 5% = half of 10% 1% = ÷100 25% = quarter 75% = three quarters 12.5% = one eighth

5. 620 worked examples: original syllabus + Tiffin-focused extensions

For every question type, attempt Question A first, then the matched Question B. The new Tiffin-focused sections add special-number reasoning, digit puzzles, mixed fractions/decimals/percentages, ratio chains, timetables and time zones, speed and rates, angle reasoning, reverse geometry, transformations, statistics, algebra/function machines and multi-step answer-entry traps. Open the solution only after answering.

Tiffin-specific expansion: what is verified and what is inferred

The Tiffin Girls' School officially confirms that Stage One tests English and Maths and does not publish full past papers. No external source can therefore guarantee the exact topic mix. The additional practice below is deliberately broader than the original 19-topic syllabus.

Independent preparation sources commonly group Tiffin Stage One Maths around arithmetic/number, fractions-decimals-percentages, ratio/proportion, measurement, angles/geometry, statistics, probability, position/direction, sequences/algebra, speed-distance-time and multi-step problem solving. These extra sections target those recurring categories and unfamiliar presentations.

1. Number Concepts

#	Question type	Example	Worked solution
1A	Place value	What is the value of 7 in 5,274,816?	Show answer + formula + trick 70,000. The 7 is in the ten-thousands column. Formula / rule Digit value = digit × its place value. Applied to 1A: the worked result is 70,000. Why it works here The 7 is in the ten-thousands column. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Place value. Checkpoint for 1A: 70,000.
1B	Place value Similar practice	What is the value of 4 in 8,643,219?	Show answer + formula + trick 40,000. The 4 is in the ten-thousands column. Formula / rule Digit value = digit × its place value. Applied to 1B: the worked result is 40,000. Why it works here The 4 is in the ten-thousands column. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Place value. Checkpoint for 1B: 40,000.
2A	Rounding	Round 4,567,891 to the nearest 10,000.	Show answer + formula + trick 4,570,000. Look at the thousands digit, 7. Since it is 5 or more, round the ten-thousands digit up. Formula / rule To round, inspect the digit immediately to the right of the required place: 5–9 round up, 0–4 keep. Applied to 2A: the worked result is 4,570,000. Why it works here Look at the thousands digit, 7. Since it is 5 or more, round the ten-thousands digit up. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 2A, re-check the exact numbers in this row before entering the answer. Remember Look one digit to the right. Checkpoint for 2A: 4,570,000.
2B	Rounding Similar practice	Round 7,842,615 to the nearest 100,000.	Show answer + formula + trick 7,800,000. Look at the ten-thousands digit, 4. Since it is less than 5, keep the hundred-thousands digit unchanged. Formula / rule To round, inspect the digit immediately to the right of the required place: 5–9 round up, 0–4 keep. Applied to 2B: the worked result is 7,800,000. Why it works here Look at the ten-thousands digit, 4. Since it is less than 5, keep the hundred-thousands digit unchanged. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 2B, re-check the exact numbers in this row before entering the answer. Remember Look one digit to the right. Checkpoint for 2B: 7,800,000.
3A	Highest common factor	Find the HCF of 36 and 48.	Show answer + formula + trick 12. Factors common to both include 1, 2, 3, 4, 6 and 12. The greatest is 12. Formula / rule HCF = the greatest factor that divides every given number exactly. Applied to 3A: the worked result is 12. Why it works here Factors common to both include 1, 2, 3, 4, 6 and 12. The greatest is 12. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 3A, re-check the exact numbers in this row before entering the answer. Remember Factors divide exactly. Checkpoint for 3A: 12.
3B	Highest common factor Similar practice	Find the HCF of 54 and 72.	Show answer + formula + trick 18. Common factors include 1, 2, 3, 6, 9 and 18. The greatest is 18. Formula / rule HCF = the greatest factor that divides every given number exactly. Applied to 3B: the worked result is 18. Why it works here Common factors include 1, 2, 3, 6, 9 and 18. The greatest is 18. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 3B, re-check the exact numbers in this row before entering the answer. Remember Factors divide exactly. Checkpoint for 3B: 18.
4A	Lowest common multiple	Find the LCM of 12 and 18.	Show answer + formula + trick 36. Multiples of 12 are 12, 24, 36... and multiples of 18 are 18, 36... Formula / rule LCM = the smallest positive multiple shared by all given numbers. Applied to 4A: the worked result is 36. Why it works here Multiples of 12 are 12, 24, 36... and multiples of 18 are 18, 36... Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 4A, re-check the exact numbers in this row before entering the answer. Remember Multiples come from times tables. Checkpoint for 4A: 36.
4B	Lowest common multiple Similar practice	Find the LCM of 15 and 20.	Show answer + formula + trick 60. Multiples of 15 are 15, 30, 45, 60... and multiples of 20 are 20, 40, 60... Formula / rule LCM = the smallest positive multiple shared by all given numbers. Applied to 4B: the worked result is 60. Why it works here Multiples of 15 are 15, 30, 45, 60... and multiples of 20 are 20, 40, 60... Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 4B, re-check the exact numbers in this row before entering the answer. Remember Multiples come from times tables. Checkpoint for 4B: 60.
5A	Prime numbers	List all prime numbers between 40 and 50.	Show answer + formula + trick 41, 43, 47. A prime has exactly two factors: 1 and itself. Formula / rule A prime number has exactly two positive factors: 1 and itself. Applied to 5A: the worked result is 41, 43, 47. Why it works here A prime has exactly two factors: 1 and itself. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 5A, re-check the exact numbers in this row before entering the answer. Remember Prime = exactly two factors. Checkpoint for 5A: 41, 43, 47.
5B	Prime numbers Similar practice	List all prime numbers between 50 and 60.	Show answer + formula + trick 53, 59. 53 and 59 have exactly two factors: 1 and themselves. Formula / rule A prime number has exactly two positive factors: 1 and itself. Applied to 5B: the worked result is 53, 59. Why it works here 53 and 59 have exactly two factors: 1 and themselves. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 5B, re-check the exact numbers in this row before entering the answer. Remember Prime = exactly two factors. Checkpoint for 5B: 53, 59.
6A	Odd and even	Is 135 × 27 odd or even?	Show answer + formula + trick Odd. Odd × odd = odd. You do not need to calculate the product. Formula / rule Parity rules: odd×odd=odd; any product containing an even factor is even; odd±odd=even; odd±even=odd. Applied to 6A: the worked result is Odd. Why it works here Odd × odd = odd. You do not need to calculate the product. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Odd and even. Checkpoint for 6A: Odd.
6B	Odd and even Similar practice	Is 246 × 19 odd or even?	Show answer + formula + trick Even. Even × odd = even. You do not need to calculate the product. Formula / rule Parity rules: odd×odd=odd; any product containing an even factor is even; odd±odd=even; odd±even=odd. Applied to 6B: the worked result is Even. Why it works here Even × odd = even. You do not need to calculate the product. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Odd and even. Checkpoint for 6B: Even.
7A	Negative numbers	Calculate -7 + 12 - 9.	Show answer + formula + trick -4. -7 + 12 = 5, then 5 - 9 = -4. Formula / rule Use the number line/sign rules: adding a positive moves right; subtracting a positive moves left. Applied to 7A: the worked result is -4. Why it works here -7 + 12 = 5, then 5 - 9 = -4. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Negative numbers. Checkpoint for 7A: -4.
7B	Negative numbers Similar practice	Calculate -12 + 7 + 9.	Show answer + formula + trick 4. -12 + 7 = -5, then -5 + 9 = 4. Formula / rule Use the number line/sign rules: adding a positive moves right; subtracting a positive moves left. Applied to 7B: the worked result is 4. Why it works here -12 + 7 = -5, then -5 + 9 = 4. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Negative numbers. Checkpoint for 7B: 4.
8A	Roman numerals	Write 49 in Roman numerals.	Show answer + formula + trick XLIX. 40 = XL and 9 = IX, so 49 = XLIX. Formula / rule Build Roman numerals from place values: I=1, V=5, X=10, L=50, C=100; subtractive pairs include IV, IX, XL, XC. Applied to 8A: the worked result is XLIX. Why it works here 40 = XL and 9 = IX, so 49 = XLIX. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Roman numerals. Checkpoint for 8A: XLIX.
8B	Roman numerals Similar practice	Write 74 in Roman numerals.	Show answer + formula + trick LXXIV. 70 = LXX and 4 = IV, so 74 = LXXIV. Formula / rule Build Roman numerals from place values: I=1, V=5, X=10, L=50, C=100; subtractive pairs include IV, IX, XL, XC. Applied to 8B: the worked result is LXXIV. Why it works here 70 = LXX and 4 = IV, so 74 = LXXIV. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Roman numerals. Checkpoint for 8B: LXXIV.
9A	Divisibility	Without dividing, decide whether 7,362 is divisible by 9.	Show answer + formula + trick Yes. Add the digits: 7 + 3 + 6 + 2 = 18. Since 18 is divisible by 9, so is 7,362. Formula / rule Use divisibility tests before long division: 2 even last digit; 3/9 digit sum; 4 last two digits; 5 last digit 0 or 5. Applied to 9A: the worked result is Yes. Why it works here Add the digits: 7 + 3 + 6 + 2 = 18. Since 18 is divisible by 9, so is 7,362. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 9A, re-check the exact numbers in this row before entering the answer. Remember Use the divisibility test, not long division. Checkpoint for 9A: Yes.
9B	Divisibility Similar practice	Without dividing, decide whether 8,127 is divisible by 3.	Show answer + formula + trick Yes. Add the digits: 8 + 1 + 2 + 7 = 18. Since 18 is divisible by 3, so is 8,127. Formula / rule Use divisibility tests before long division: 2 even last digit; 3/9 digit sum; 4 last two digits; 5 last digit 0 or 5. Applied to 9B: the worked result is Yes. Why it works here Add the digits: 8 + 1 + 2 + 7 = 18. Since 18 is divisible by 3, so is 8,127. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 9B, re-check the exact numbers in this row before entering the answer. Remember Use the divisibility test, not long division. Checkpoint for 9B: Yes.
10A	Square numbers	144 is the square of which whole number?	Show answer + formula + trick 12. 12 × 12 = 144. Formula / rule A square number has the form n² = n×n. Applied to 10A: the worked result is 12. Why it works here 12 × 12 = 144. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Square numbers. Checkpoint for 10A: 12.
10B	Square numbers Similar practice	169 is the square of which whole number?	Show answer + formula + trick 13. 13 × 13 = 169. Formula / rule A square number has the form n² = n×n. Applied to 10B: the worked result is 13. Why it works here 13 × 13 = 169. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Square numbers. Checkpoint for 10B: 13.

2. Arithmetic Operations

#	Question type	Example	Worked solution
1A	Mental multiplication	Calculate 48 × 25 mentally.	Show answer + formula + trick 1,200. 25 is one quarter of 100. So 48 × 25 = 48 × 100 ÷ 4 = 1,200. Formula / rule Use factor-friendly equivalences, e.g. ×25 = ×100÷4, ×50 = ×100÷2, ×5 = ×10÷2. Applied to 1A: the worked result is 1,200. Why it works here 25 is one quarter of 100. So 48 × 25 = 48 × 100 ÷ 4 = 1,200. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Mental multiplication. Checkpoint for 1A: 1,200.
1B	Mental multiplication Similar practice	Calculate 64 × 25 mentally.	Show answer + formula + trick 1,600. 25 is one quarter of 100. So 64 × 25 = 64 × 100 ÷ 4 = 1,600. Formula / rule Use factor-friendly equivalences, e.g. ×25 = ×100÷4, ×50 = ×100÷2, ×5 = ×10÷2. Applied to 1B: the worked result is 1,600. Why it works here 25 is one quarter of 100. So 64 × 25 = 64 × 100 ÷ 4 = 1,600. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Mental multiplication. Checkpoint for 1B: 1,600.
2A	Long multiplication	Calculate 307 × 46.	Show answer + formula + trick 14,122. 307 × 40 = 12,280 and 307 × 6 = 1,842. Total = 14,122. Formula / rule Partition one factor by place value, multiply each part, then add the partial products. Applied to 2A: the worked result is 14,122. Why it works here 307 × 40 = 12,280 and 307 × 6 = 1,842. Total = 14,122. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Long multiplication. Checkpoint for 2A: 14,122.
2B	Long multiplication Similar practice	Calculate 428 × 37.	Show answer + formula + trick 15,836. 428 × 30 = 12,840 and 428 × 7 = 2,996. Total = 15,836. Formula / rule Partition one factor by place value, multiply each part, then add the partial products. Applied to 2B: the worked result is 15,836. Why it works here 428 × 30 = 12,840 and 428 × 7 = 2,996. Total = 15,836. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Long multiplication. Checkpoint for 2B: 15,836.
3A	Division with remainder	Calculate 987 ÷ 23.	Show answer + formula + trick 42 remainder 21. 23 × 42 = 966, and 987 - 966 = 21. Formula / rule Dividend = divisor × whole-number quotient + remainder. Applied to 3A: the worked result is 42 remainder 21. Why it works here 23 × 42 = 966, and 987 - 966 = 21. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Division with remainder. Checkpoint for 3A: 42 remainder 21.
3B	Division with remainder Similar practice	Calculate 1,075 ÷ 24.	Show answer + formula + trick 44 remainder 19. 24 × 44 = 1,056, and 1,075 - 1,056 = 19. Formula / rule Dividend = divisor × whole-number quotient + remainder. Applied to 3B: the worked result is 44 remainder 19. Why it works here 24 × 44 = 1,056, and 1,075 - 1,056 = 19. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Division with remainder. Checkpoint for 3B: 44 remainder 19.
4A	BODMAS	Calculate 18 - 3 × (4 + 2) + 24 ÷ 6.	Show answer + formula + trick 4. Brackets first: 6. Then 3 × 6 = 18 and 24 ÷ 6 = 4. So 18 - 18 + 4 = 4. Formula / rule BODMAS/BIDMAS: Brackets, Orders/Indices, Division and Multiplication left-to-right, Addition and Subtraction left-to-right. Applied to 4A: the worked result is 4. Why it works here Brackets first: 6. Then 3 × 6 = 18 and 24 ÷ 6 = 4. So 18 - 18 + 4 = 4. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: BODMAS. Checkpoint for 4A: 4.
4B	BODMAS Similar practice	Calculate 30 - 4 × (5 + 1) + 18 ÷ 3.	Show answer + formula + trick 12. Brackets first: 6. Then 4 × 6 = 24 and 18 ÷ 3 = 6. So 30 - 24 + 6 = 12. Formula / rule BODMAS/BIDMAS: Brackets, Orders/Indices, Division and Multiplication left-to-right, Addition and Subtraction left-to-right. Applied to 4B: the worked result is 12. Why it works here Brackets first: 6. Then 4 × 6 = 24 and 18 ÷ 3 = 6. So 30 - 24 + 6 = 12. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: BODMAS. Checkpoint for 4B: 12.
5A	Estimation	Estimate 398 × 51.	Show answer + formula + trick About 20,000. Round to 400 × 50 = 20,000. Formula / rule Estimate by rounding to friendly numbers, then compare the exact or proposed answer with the expected size. Applied to 5A: the worked result is About 20,000. Why it works here Round to 400 × 50 = 20,000. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Estimation. Checkpoint for 5A: About 20,000.
5B	Estimation Similar practice	Estimate 612 × 39.	Show answer + formula + trick About 24,000. Round to 600 × 40 = 24,000. Formula / rule Estimate by rounding to friendly numbers, then compare the exact or proposed answer with the expected size. Applied to 5B: the worked result is About 24,000. Why it works here Round to 600 × 40 = 24,000. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Estimation. Checkpoint for 5B: About 24,000.
6A	Inverse operations	A number divided by 8 gives 37. What is the number?	Show answer + formula + trick 296. Reverse the division: 37 × 8 = 296. Formula / rule Inverse pairs: + ↔ − and × ↔ ÷. Undo operations in reverse order. Applied to 6A: the worked result is 296. Why it works here Reverse the division: 37 × 8 = 296. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 6A: 296.
6B	Inverse operations Similar practice	A number multiplied by 7 gives 364. What is the number?	Show answer + formula + trick 52. Reverse the multiplication: 364 ÷ 7 = 52. Formula / rule Inverse pairs: + ↔ − and × ↔ ÷. Undo operations in reverse order. Applied to 6B: the worked result is 52. Why it works here Reverse the multiplication: 364 ÷ 7 = 52. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 6B: 52.
7A	Missing number	□ × 16 = 1,248. Find □.	Show answer + formula + trick 78. 1,248 ÷ 16 = 78. Formula / rule Use the inverse operation to isolate the missing number. Applied to 7A: the worked result is 78. Why it works here 1,248 ÷ 16 = 78. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing number. Checkpoint for 7A: 78.
7B	Missing number Similar practice	□ × 18 = 1,674. Find □.	Show answer + formula + trick 93. 1,674 ÷ 18 = 93. Formula / rule Use the inverse operation to isolate the missing number. Applied to 7B: the worked result is 93. Why it works here 1,674 ÷ 18 = 93. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing number. Checkpoint for 7B: 93.
8A	Efficient subtraction	Calculate 10,000 - 3,998.	Show answer + formula + trick 6,002. Subtract 4,000 to get 6,000, then add back 2. Formula / rule Compensate around a friendly number: subtract the nearby round number, then correct the difference. Applied to 8A: the worked result is 6,002. Why it works here Subtract 4,000 to get 6,000, then add back 2. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Efficient subtraction. Checkpoint for 8A: 6,002.
8B	Efficient subtraction Similar practice	Calculate 8,000 - 2,997.	Show answer + formula + trick 5,003. Subtract 3,000 to get 5,000, then add back 3. Formula / rule Compensate around a friendly number: subtract the nearby round number, then correct the difference. Applied to 8B: the worked result is 5,003. Why it works here Subtract 3,000 to get 5,000, then add back 3. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Efficient subtraction. Checkpoint for 8B: 5,003.
9A	Remainder interpretation	253 pupils travel in coaches holding 48 each. What is the minimum number of coaches?	Show answer + formula + trick 6. 253 ÷ 48 = 5 remainder 13. Five coaches are not enough, so 6 are needed. Formula / rule If every item/person must fit and division leaves a remainder, required containers = quotient + 1. Applied to 9A: the worked result is 6. Why it works here 253 ÷ 48 = 5 remainder 13. Five coaches are not enough, so 6 are needed. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Remainder interpretation. Checkpoint for 9A: 6.
9B	Remainder interpretation Similar practice	327 pupils travel in minibuses holding 15 each. What is the minimum number of minibuses?	Show answer + formula + trick 22. 327 ÷ 15 = 21 remainder 12. Twenty-one are not enough, so 22 are needed. Formula / rule If every item/person must fit and division leaves a remainder, required containers = quotient + 1. Applied to 9B: the worked result is 22. Why it works here 327 ÷ 15 = 21 remainder 12. Twenty-one are not enough, so 22 are needed. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Remainder interpretation. Checkpoint for 9B: 22.
10A	Multi-step arithmetic	A shop has 24 boxes of 36 pencils. It sells 275 pencils. How many remain?	Show answer + formula + trick 589. 24 × 36 = 864. Then 864 - 275 = 589. Formula / rule Carry out the operations in the order demanded by the story, keeping an intermediate total after each step. Applied to 10A: the worked result is 589. Why it works here 24 × 36 = 864. Then 864 - 275 = 589. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Multi-step arithmetic. Checkpoint for 10A: 589.
10B	Multi-step arithmetic Similar practice	A warehouse has 18 crates of 42 bottles. It ships 289 bottles. How many remain?	Show answer + formula + trick 467. 18 × 42 = 756. Then 756 - 289 = 467. Formula / rule Carry out the operations in the order demanded by the story, keeping an intermediate total after each step. Applied to 10B: the worked result is 467. Why it works here 18 × 42 = 756. Then 756 - 289 = 467. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Multi-step arithmetic. Checkpoint for 10B: 467.

3. Fractions

#	Question type	Example	Worked solution
1A	Equivalent fractions	Complete: 3/5 = ?/40.	Show answer + formula + trick 24/40. Multiply numerator and denominator by 8. Formula / rule Equivalent fractions multiply or divide numerator and denominator by the same non-zero number. Applied to 1A: the worked result is 24/40. Why it works here Multiply numerator and denominator by 8. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 1A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 1A: 24/40.
1B	Equivalent fractions Similar practice	Complete: 7/9 = ?/63.	Show answer + formula + trick 49/63. Multiply numerator and denominator by 7. Formula / rule Equivalent fractions multiply or divide numerator and denominator by the same non-zero number. Applied to 1B: the worked result is 49/63. Why it works here Multiply numerator and denominator by 7. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 1B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 1B: 49/63.
2A	Simplifying	Simplify 42/56.	Show answer + formula + trick 3/4. Divide numerator and denominator by 14. Formula / rule Simplify a fraction by dividing numerator and denominator by their highest common factor. Applied to 2A: the worked result is 3/4. Why it works here Divide numerator and denominator by 14. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Simplifying. Checkpoint for 2A: 3/4.
2B	Simplifying Similar practice	Simplify 63/84.	Show answer + formula + trick 3/4. Divide numerator and denominator by 21. Formula / rule Simplify a fraction by dividing numerator and denominator by their highest common factor. Applied to 2B: the worked result is 3/4. Why it works here Divide numerator and denominator by 21. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Simplifying. Checkpoint for 2B: 3/4.
3A	Comparing	Which is larger: 5/8 or 7/12?	Show answer + formula + trick 5/8. Cross-multiply: 5 × 12 = 60 and 7 × 8 = 56. Since 60 > 56, 5/8 is larger. Formula / rule Compare/subtract fractions using a common denominator or cross-multiplication. Applied to 3A: the worked result is 5/8. Why it works here Cross-multiply: 5 × 12 = 60 and 7 × 8 = 56. Since 60 > 56, 5/8 is larger. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Comparing. Checkpoint for 3A: 5/8.
3B	Comparing Similar practice	Which is larger: 7/10 or 5/8?	Show answer + formula + trick 7/10. Cross-multiply: 7 × 8 = 56 and 5 × 10 = 50. Since 56 > 50, 7/10 is larger. Formula / rule Compare/subtract fractions using a common denominator or cross-multiplication. Applied to 3B: the worked result is 7/10. Why it works here Cross-multiply: 7 × 8 = 56 and 5 × 10 = 50. Since 56 > 50, 7/10 is larger. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Comparing. Checkpoint for 3B: 7/10.
4A	Mixed to improper	Convert 3 2/5 to an improper fraction.	Show answer + formula + trick 17/5. 3 × 5 + 2 = 17. Formula / rule Improper numerator = whole number × denominator + numerator. Applied to 4A: the worked result is 17/5. Why it works here 3 × 5 + 2 = 17. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Mixed to improper. Checkpoint for 4A: 17/5.
4B	Mixed to improper Similar practice	Convert 4 3/7 to an improper fraction.	Show answer + formula + trick 31/7. 4 × 7 + 3 = 31. Formula / rule Improper numerator = whole number × denominator + numerator. Applied to 4B: the worked result is 31/7. Why it works here 4 × 7 + 3 = 31. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Mixed to improper. Checkpoint for 4B: 31/7.
5A	Adding unlike fractions	Calculate 3/4 + 5/6.	Show answer + formula + trick 1 7/12. Common denominator 12: 9/12 + 10/12 = 19/12 = 1 7/12. Formula / rule For fraction addition/subtraction, first convert to a common denominator. Applied to 5A: the worked result is 1 7/12. Why it works here Common denominator 12: 9/12 + 10/12 = 19/12 = 1 7/12. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 5A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 5A: 1 7/12.
5B	Adding unlike fractions Similar practice	Calculate 5/6 + 3/8.	Show answer + formula + trick 1 5/24. Common denominator 24: 20/24 + 9/24 = 29/24 = 1 5/24. Formula / rule For fraction addition/subtraction, first convert to a common denominator. Applied to 5B: the worked result is 1 5/24. Why it works here Common denominator 24: 20/24 + 9/24 = 29/24 = 1 5/24. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 5B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 5B: 1 5/24.
6A	Subtracting fractions	Calculate 7/8 - 1/3.	Show answer + formula + trick 13/24. Common denominator 24: 21/24 - 8/24 = 13/24. Formula / rule For fraction addition/subtraction, first convert to a common denominator. Applied to 6A: the worked result is 13/24. Why it works here Common denominator 24: 21/24 - 8/24 = 13/24. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 6A: 13/24.
6B	Subtracting fractions Similar practice	Calculate 11/12 - 2/5.	Show answer + formula + trick 31/60. Common denominator 60: 55/60 - 24/60 = 31/60. Formula / rule For fraction addition/subtraction, first convert to a common denominator. Applied to 6B: the worked result is 31/60. Why it works here Common denominator 60: 55/60 - 24/60 = 31/60. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 6B: 31/60.
7A	Multiplying fractions	Calculate 3/5 × 10/21.	Show answer + formula + trick 2/7. Cancel 10 with 5 to get 2; cancel 3 with 21 to get 1/7. Answer 2/7. Formula / rule Multiply fractions straight across after cancelling common factors; convert mixed numbers to improper fractions first. Applied to 7A: the worked result is 2/7. Why it works here Cancel 10 with 5 to get 2; cancel 3 with 21 to get 1/7. Answer 2/7. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 7A: 2/7.
7B	Multiplying fractions Similar practice	Calculate 4/7 × 21/10.	Show answer + formula + trick 6/5 or 1 1/5. Cancel 21 with 7 to get 3; cancel 4 with 10 to get 2/5. Then 2 × 3 / 5 = 6/5. Formula / rule Multiply fractions straight across after cancelling common factors; convert mixed numbers to improper fractions first. Applied to 7B: the worked result is 6/5 or 1 1/5. Why it works here Cancel 21 with 7 to get 3; cancel 4 with 10 to get 2/5. Then 2 × 3 / 5 = 6/5. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 7B: 6/5 or 1 1/5.
8A	Dividing fractions	Calculate 4/9 ÷ 2/3.	Show answer + formula + trick 2/3. Keep 4/9, change ÷ to ×, flip 2/3 to 3/2: 4/9 × 3/2 = 2/3. Formula / rule Divide fractions using keep-change-flip: a/b ÷ c/d = a/b × d/c. Applied to 8A: the worked result is 2/3. Why it works here Keep 4/9, change ÷ to ×, flip 2/3 to 3/2: 4/9 × 3/2 = 2/3. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 8A: 2/3.
8B	Dividing fractions Similar practice	Calculate 5/8 ÷ 15/16.	Show answer + formula + trick 2/3. Keep 5/8, change ÷ to ×, flip 15/16 to 16/15: 5/8 × 16/15 = 2/3. Formula / rule Divide fractions using keep-change-flip: a/b ÷ c/d = a/b × d/c. Applied to 8B: the worked result is 2/3. Why it works here Keep 5/8, change ÷ to ×, flip 15/16 to 16/15: 5/8 × 16/15 = 2/3. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 8B: 2/3.
9A	Fraction of an amount	Find 7/12 of 360.	Show answer + formula + trick 210. 360 ÷ 12 = 30, then 30 × 7 = 210. Formula / rule a/b of N = N÷b×a. Applied to 9A: the worked result is 210. Why it works here 360 ÷ 12 = 30, then 30 × 7 = 210. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 9A: 210.
9B	Fraction of an amount Similar practice	Find 5/9 of 324.	Show answer + formula + trick 180. 324 ÷ 9 = 36, then 36 × 5 = 180. Formula / rule a/b of N = N÷b×a. Applied to 9B: the worked result is 180. Why it works here 324 ÷ 9 = 36, then 36 × 5 = 180. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 9B: 180.
10A	Missing fraction	2/3 + ? = 11/12. Find ?.	Show answer + formula + trick 1/4. 2/3 = 8/12. So ? = 11/12 - 8/12 = 3/12 = 1/4. Formula / rule For fraction addition/subtraction, first convert to a common denominator. Applied to 10A: the worked result is 1/4. Why it works here 2/3 = 8/12. So ? = 11/12 - 8/12 = 3/12 = 1/4. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 10A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 10A: 1/4.
10B	Missing fraction Similar practice	3/4 + ? = 11/12. Find ?.	Show answer + formula + trick 1/6. 3/4 = 9/12. So ? = 2/12 = 1/6. Formula / rule For fraction addition/subtraction, first convert to a common denominator. Applied to 10B: the worked result is 1/6. Why it works here 3/4 = 9/12. So ? = 2/12 = 1/6. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 10B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 10B: 1/6.

4. Decimals

#	Question type	Example	Worked solution
1A	Place value	What is the value of 6 in 14.063?	Show answer + formula + trick 0.06. It is in the hundredths column. Formula / rule Digit value = digit × its place value. Applied to 1A: the worked result is 0.06. Why it works here It is in the hundredths column. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Place value. Checkpoint for 1A: 0.06.
1B	Place value Similar practice	What is the value of 8 in 72.184?	Show answer + formula + trick 0.08. It is in the hundredths column. Formula / rule Digit value = digit × its place value. Applied to 1B: the worked result is 0.08. Why it works here It is in the hundredths column. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Place value. Checkpoint for 1B: 0.08.
2A	Comparing decimals	Which is greater: 4.507 or 4.57?	Show answer + formula + trick 4.57. Write 4.57 as 4.570. Then 4.570 > 4.507. Formula / rule Align place values and append trailing zeros if useful before comparing. Applied to 2A: the worked result is 4.57. Why it works here Write 4.57 as 4.570. Then 4.570 > 4.507. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Comparing decimals. Checkpoint for 2A: 4.57.
2B	Comparing decimals Similar practice	Which is greater: 6.305 or 6.35?	Show answer + formula + trick 6.35. Write 6.35 as 6.350. Then 6.350 > 6.305. Formula / rule Align place values and append trailing zeros if useful before comparing. Applied to 2B: the worked result is 6.35. Why it works here Write 6.35 as 6.350. Then 6.350 > 6.305. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Comparing decimals. Checkpoint for 2B: 6.35.
3A	Addition	Calculate 18.75 + 6.089.	Show answer + formula + trick 24.839. Align decimal points: 18.750 + 6.089 = 24.839. Formula / rule Align decimal points vertically, then add by place value. Applied to 3A: the worked result is 24.839. Why it works here Align decimal points: 18.750 + 6.089 = 24.839. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Addition. Checkpoint for 3A: 24.839.
3B	Addition Similar practice	Calculate 27.406 + 8.95.	Show answer + formula + trick 36.356. Align decimal points: 27.406 + 8.950 = 36.356. Formula / rule Align decimal points vertically, then add by place value. Applied to 3B: the worked result is 36.356. Why it works here Align decimal points: 27.406 + 8.950 = 36.356. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Addition. Checkpoint for 3B: 36.356.
4A	Subtraction	Calculate 20 - 7.386.	Show answer + formula + trick 12.614. Write 20 as 20.000, then subtract. Formula / rule Align decimal points vertically, append placeholder zeros, then subtract. Applied to 4A: the worked result is 12.614. Why it works here Write 20 as 20.000, then subtract. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Subtraction. Checkpoint for 4A: 12.614.
4B	Subtraction Similar practice	Calculate 50 - 18.725.	Show answer + formula + trick 31.275. Write 50 as 50.000, then subtract. Formula / rule Align decimal points vertically, append placeholder zeros, then subtract. Applied to 4B: the worked result is 31.275. Why it works here Write 50 as 50.000, then subtract. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Subtraction. Checkpoint for 4B: 31.275.
5A	Multiply by whole number	Calculate 3.75 × 24.	Show answer + formula + trick 90. 3.75 × 6 = 22.5, then ×4 gives 90, or use 3.75 × (20 + 4). Formula / rule Multiply as whole numbers while keeping the decimal place value of the decimal factor. Applied to 5A: the worked result is 90. Why it works here 3.75 × 6 = 22.5, then ×4 gives 90, or use 3.75 × (20 + 4). Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Multiply by whole number. Checkpoint for 5A: 90.
5B	Multiply by whole number Similar practice	Calculate 4.25 × 16.	Show answer + formula + trick 68. 4.25 × 8 = 34, then double to get 68. Formula / rule Multiply as whole numbers while keeping the decimal place value of the decimal factor. Applied to 5B: the worked result is 68. Why it works here 4.25 × 8 = 34, then double to get 68. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Multiply by whole number. Checkpoint for 5B: 68.
6A	Multiply decimals	Calculate 0.6 × 0.35.	Show answer + formula + trick 0.21. 6 × 35 = 210. There are three decimal places in total, so 0.210 = 0.21. Formula / rule Multiply as integers, then restore the total number of decimal places from both factors. Applied to 6A: the worked result is 0.21. Why it works here 6 × 35 = 210. There are three decimal places in total, so 0.210 = 0.21. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Multiply decimals. Checkpoint for 6A: 0.21.
6B	Multiply decimals Similar practice	Calculate 0.8 × 0.45.	Show answer + formula + trick 0.36. 8 × 45 = 360. There are three decimal places in total, so 0.360 = 0.36. Formula / rule Multiply as integers, then restore the total number of decimal places from both factors. Applied to 6B: the worked result is 0.36. Why it works here 8 × 45 = 360. There are three decimal places in total, so 0.360 = 0.36. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Multiply decimals. Checkpoint for 6B: 0.36.
7A	Divide decimals	Calculate 7.2 ÷ 0.3.	Show answer + formula + trick 24. Multiply both numbers by 10: 72 ÷ 3 = 24. Formula / rule Scale both dividend and divisor by the same power of 10 until the divisor is a whole number. Applied to 7A: the worked result is 24. Why it works here Multiply both numbers by 10: 72 ÷ 3 = 24. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Divide decimals. Checkpoint for 7A: 24.
7B	Divide decimals Similar practice	Calculate 8.4 ÷ 0.7.	Show answer + formula + trick 12. Multiply both numbers by 10: 84 ÷ 7 = 12. Formula / rule Scale both dividend and divisor by the same power of 10 until the divisor is a whole number. Applied to 7B: the worked result is 12. Why it works here Multiply both numbers by 10: 84 ÷ 7 = 12. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Divide decimals. Checkpoint for 7B: 12.
8A	Fraction to decimal	Convert 7/20 to a decimal.	Show answer + formula + trick 0.35. Multiply by 5 to get 35/100 = 0.35. Formula / rule Fraction to decimal = numerator ÷ denominator. Applied to 8A: the worked result is 0.35. Why it works here Multiply by 5 to get 35/100 = 0.35. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 8A: 0.35.
8B	Fraction to decimal Similar practice	Convert 9/25 to a decimal.	Show answer + formula + trick 0.36. Multiply numerator and denominator by 4 to get 36/100 = 0.36. Formula / rule Fraction to decimal = numerator ÷ denominator. Applied to 8B: the worked result is 0.36. Why it works here Multiply numerator and denominator by 4 to get 36/100 = 0.36. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 8B: 0.36.
9A	Decimal to fraction	Convert 0.375 to a fraction in simplest form.	Show answer + formula + trick 3/8. 0.375 = 375/1000. Divide by 125 to get 3/8. Formula / rule Write the decimal over 10, 100, 1000, ... according to place value, then simplify. Applied to 9A: the worked result is 3/8. Why it works here 0.375 = 375/1000. Divide by 125 to get 3/8. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 9A: 3/8.
9B	Decimal to fraction Similar practice	Convert 0.625 to a fraction in simplest form.	Show answer + formula + trick 5/8. 0.625 = 625/1000. Divide by 125 to get 5/8. Formula / rule Write the decimal over 10, 100, 1000, ... according to place value, then simplify. Applied to 9B: the worked result is 5/8. Why it works here 0.625 = 625/1000. Divide by 125 to get 5/8. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 9B: 5/8.
10A	Rounding decimals	Round 8.746 to 2 decimal places.	Show answer + formula + trick 8.75. Look at the third decimal digit, 6, so 8.74 rounds up to 8.75. Formula / rule To round, inspect the digit immediately to the right of the required place: 5–9 round up, 0–4 keep. Applied to 10A: the worked result is 8.75. Why it works here Look at the third decimal digit, 6, so 8.74 rounds up to 8.75. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 10A, re-check the exact numbers in this row before entering the answer. Remember Look one digit to the right. Checkpoint for 10A: 8.75.
10B	Rounding decimals Similar practice	Round 13.485 to 2 decimal places.	Show answer + formula + trick 13.49. Look at the third decimal digit, 5, so 13.48 rounds up to 13.49. Formula / rule To round, inspect the digit immediately to the right of the required place: 5–9 round up, 0–4 keep. Applied to 10B: the worked result is 13.49. Why it works here Look at the third decimal digit, 5, so 13.48 rounds up to 13.49. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 10B, re-check the exact numbers in this row before entering the answer. Remember Look one digit to the right. Checkpoint for 10B: 13.49.

5. Percentages

#	Question type	Example	Worked solution
1A	Fraction to percentage	Convert 3/5 to a percentage.	Show answer + formula + trick 60%. 3 ÷ 5 = 0.6 = 60%. Formula / rule Fraction to percentage = numerator ÷ denominator ×100%. Applied to 1A: the worked result is 60%. Why it works here 3 ÷ 5 = 0.6 = 60%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 1A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 1A: 60%.
1B	Fraction to percentage Similar practice	Convert 7/20 to a percentage.	Show answer + formula + trick 35%. 7 ÷ 20 = 0.35 = 35%. Formula / rule Fraction to percentage = numerator ÷ denominator ×100%. Applied to 1B: the worked result is 35%. Why it works here 7 ÷ 20 = 0.35 = 35%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 1B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 1B: 35%.
2A	Decimal to percentage	Convert 0.275 to a percentage.	Show answer + formula + trick 27.5%. Multiply by 100. Formula / rule Decimal to percentage = decimal ×100%. Applied to 2A: the worked result is 27.5%. Why it works here Multiply by 100. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 2A: 27.5%.
2B	Decimal to percentage Similar practice	Convert 0.462 to a percentage.	Show answer + formula + trick 46.2%. Multiply by 100. Formula / rule Decimal to percentage = decimal ×100%. Applied to 2B: the worked result is 46.2%. Why it works here Multiply by 100. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 2B: 46.2%.
3A	Percentage of amount	Find 35% of £240.	Show answer + formula + trick £84. 10% = £24, 30% = £72, 5% = £12. Total £84. Formula / rule Percentage of amount = percentage ÷100 × amount. Applied to 3A: the worked result is £84. Why it works here 10% = £24, 30% = £72, 5% = £12. Total £84. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 3A: £84.
3B	Percentage of amount Similar practice	Find 45% of £320.	Show answer + formula + trick £144. 10% = £32, 40% = £128, 5% = £16. Total £144. Formula / rule Percentage of amount = percentage ÷100 × amount. Applied to 3B: the worked result is £144. Why it works here 10% = £32, 40% = £128, 5% = £16. Total £144. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 3B: £144.
4A	Quick 15%	Find 15% of 360.	Show answer + formula + trick 54. 10% = 36 and 5% = 18, total 54. Formula / rule Percentage of amount = percentage ÷100 × amount. Applied to 4A: the worked result is 54. Why it works here 10% = 36 and 5% = 18, total 54. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Quick 15%. Checkpoint for 4A: 54.
4B	Quick 15% Similar practice	Find 15% of 480.	Show answer + formula + trick 72. 10% = 48 and 5% = 24, total 72. Formula / rule Percentage of amount = percentage ÷100 × amount. Applied to 4B: the worked result is 72. Why it works here 10% = 48 and 5% = 24, total 72. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Quick 15%. Checkpoint for 4B: 72.
5A	Percentage increase	Increase £80 by 25%.	Show answer + formula + trick £100. 25% of 80 = 20. Add: 80 + 20 = 100. Formula / rule New amount = original × (1 + increase rate). Applied to 5A: the worked result is £100. Why it works here 25% of 80 = 20. Add: 80 + 20 = 100. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 5A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 5A: £100.
5B	Percentage increase Similar practice	Increase £120 by 15%.	Show answer + formula + trick £138. 15% of 120 = 18. Add: 120 + 18 = 138. Formula / rule New amount = original × (1 + increase rate). Applied to 5B: the worked result is £138. Why it works here 15% of 120 = 18. Add: 120 + 18 = 138. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 5B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 5B: £138.
6A	Percentage decrease	Decrease 350 by 12%.	Show answer + formula + trick 308. 10% = 35, 2% = 7, so 12% = 42. Then 350 - 42 = 308. Formula / rule New amount = original × (1 − decrease rate). Applied to 6A: the worked result is 308. Why it works here 10% = 35, 2% = 7, so 12% = 42. Then 350 - 42 = 308. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 6A: 308.
6B	Percentage decrease Similar practice	Decrease 500 by 18%.	Show answer + formula + trick 410. 10% = 50, 5% = 25, 3% = 15, so 18% = 90. Then 500 - 90 = 410. Formula / rule New amount = original × (1 − decrease rate). Applied to 6B: the worked result is 410. Why it works here 10% = 50, 5% = 25, 3% = 15, so 18% = 90. Then 500 - 90 = 410. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 6B: 410.
7A	Reverse percentage	After a 20% discount, a coat costs £72. What was the original price?	Show answer + formula + trick £90. £72 is 80% of the original. 1% = 72 ÷ 80 = 0.9, so 100% = £90. Formula / rule Original amount = final amount ÷ remaining multiplier (for example 80% → ÷0.80). Applied to 7A: the worked result is £90. Why it works here £72 is 80% of the original. 1% = 72 ÷ 80 = 0.9, so 100% = £90. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 7A: £90.
7B	Reverse percentage Similar practice	After a 25% discount, a bag costs £60. What was the original price?	Show answer + formula + trick £80. £60 is 75% of the original. 1% = 60 ÷ 75 = 0.8, so 100% = £80. Formula / rule Original amount = final amount ÷ remaining multiplier (for example 80% → ÷0.80). Applied to 7B: the worked result is £80. Why it works here £60 is 75% of the original. 1% = 60 ÷ 75 = 0.8, so 100% = £80. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 7B: £80.
8A	Percentage change	A value rises from 50 to 65. What is the percentage increase?	Show answer + formula + trick 30%. Increase = 15. 15 ÷ 50 = 0.3 = 30%. Formula / rule Percentage change = change ÷ original × 100%. Applied to 8A: the worked result is 30%. Why it works here Increase = 15. 15 ÷ 50 = 0.3 = 30%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 8A: 30%.
8B	Percentage change Similar practice	A value falls from 80 to 68. What is the percentage decrease?	Show answer + formula + trick 15%. Decrease = 12. 12 ÷ 80 = 0.15 = 15%. Formula / rule Percentage change = change ÷ original × 100%. Applied to 8B: the worked result is 15%. Why it works here Decrease = 12. 12 ÷ 80 = 0.15 = 15%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 8B: 15%.
9A	Discount	A £150 item is reduced by 30%. What is the sale price?	Show answer + formula + trick £105. 30% of 150 = 45. £150 - £45 = £105. Formula / rule New amount = original × (1 − decrease rate). Applied to 9A: the worked result is £105. Why it works here 30% of 150 = 45. £150 - £45 = £105. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Discount. Checkpoint for 9A: £105.
9B	Discount Similar practice	A £240 item is reduced by 35%. What is the sale price?	Show answer + formula + trick £156. 35% of 240 = 84. £240 - £84 = £156. Formula / rule New amount = original × (1 − decrease rate). Applied to 9B: the worked result is £156. Why it works here 35% of 240 = 84. £240 - £84 = £156. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Discount. Checkpoint for 9B: £156.
10A	Find the whole	42 is 35% of a number. Find the number.	Show answer + formula + trick 120. 1% = 42 ÷ 35 = 1.2, so 100% = 120. Formula / rule If x is p% of N, then N = x ÷ (p/100). Applied to 10A: the worked result is 120. Why it works here 1% = 42 ÷ 35 = 1.2, so 100% = 120. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Find the whole. Checkpoint for 10A: 120.
10B	Find the whole Similar practice	54 is 45% of a number. Find the number.	Show answer + formula + trick 120. 1% = 54 ÷ 45 = 1.2, so 100% = 120. Formula / rule If x is p% of N, then N = x ÷ (p/100). Applied to 10B: the worked result is 120. Why it works here 1% = 54 ÷ 45 = 1.2, so 100% = 120. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Find the whole. Checkpoint for 10B: 120.

6. Ratios and Proportions

#	Question type	Example	Worked solution
1A	Simplify ratio	Simplify 42:56.	Show answer + formula + trick 3:4. Divide both parts by 14. Formula / rule Simplify a ratio by dividing every part by their highest common factor. Applied to 1A: the worked result is 3:4. Why it works here Divide both parts by 14. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 1A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 1A: 3:4.
1B	Simplify ratio Similar practice	Simplify 54:72.	Show answer + formula + trick 3:4. Divide both parts by 18. Formula / rule Simplify a ratio by dividing every part by their highest common factor. Applied to 1B: the worked result is 3:4. Why it works here Divide both parts by 18. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 1B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 1B: 3:4.
2A	Share in a ratio	Share £180 in the ratio 2:3.	Show answer + formula + trick £72 and £108. Total parts = 5. One part = £36. So 2 parts = £72 and 3 parts = £108. Formula / rule One part = total ÷ sum of ratio parts; each share = one part × its ratio number. Applied to 2A: the worked result is £72 and £108. Why it works here Total parts = 5. One part = £36. So 2 parts = £72 and 3 parts = £108. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 2A: £72 and £108.
2B	Share in a ratio Similar practice	Share £280 in the ratio 3:4.	Show answer + formula + trick £120 and £160. Total parts = 7. One part = £40. So 3 parts = £120 and 4 parts = £160. Formula / rule One part = total ÷ sum of ratio parts; each share = one part × its ratio number. Applied to 2B: the worked result is £120 and £160. Why it works here Total parts = 7. One part = £40. So 3 parts = £120 and 4 parts = £160. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 2B: £120 and £160.
3A	Find missing ratio value	3:5 = 21:x. Find x.	Show answer + formula + trick 35. 3 has been multiplied by 7, so 5 × 7 = 35. Formula / rule Equivalent ratios multiply/divide every part by the same scale factor. Applied to 3A: the worked result is 35. Why it works here 3 has been multiplied by 7, so 5 × 7 = 35. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 3A: 35.
3B	Find missing ratio value Similar practice	4:7 = 28:x. Find x.	Show answer + formula + trick 49. 4 has been multiplied by 7, so 7 × 7 = 49. Formula / rule Equivalent ratios multiply/divide every part by the same scale factor. Applied to 3B: the worked result is 49. Why it works here 4 has been multiplied by 7, so 7 × 7 = 49. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 3B: 49.
4A	Unitary method	5 notebooks cost £8.50. What do 8 cost?	Show answer + formula + trick £13.60. One notebook costs £8.50 ÷ 5 = £1.70. Then × 8 = £13.60. Formula / rule Unitary method: value of one unit = total ÷ number of units; required value = one-unit value × required units. Applied to 4A: the worked result is £13.60. Why it works here One notebook costs £8.50 ÷ 5 = £1.70. Then × 8 = £13.60. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Unitary method. Checkpoint for 4A: £13.60.
4B	Unitary method Similar practice	6 tickets cost £15.60. What do 9 cost?	Show answer + formula + trick £23.40. One ticket costs £15.60 ÷ 6 = £2.60. Then × 9 = £23.40. Formula / rule Unitary method: value of one unit = total ÷ number of units; required value = one-unit value × required units. Applied to 4B: the worked result is £23.40. Why it works here One ticket costs £15.60 ÷ 6 = £2.60. Then × 9 = £23.40. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Unitary method. Checkpoint for 4B: £23.40.
5A	Recipe proportion	A recipe for 4 uses 300 g flour. How much for 10?	Show answer + formula + trick 750 g. 300 ÷ 4 = 75 g per person. 75 × 10 = 750 g. Formula / rule Direct proportion: find the value for one unit, then multiply by the required number of units. Applied to 5A: the worked result is 750 g. Why it works here 300 ÷ 4 = 75 g per person. 75 × 10 = 750 g. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 5A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 5A: 750 g.
5B	Recipe proportion Similar practice	A recipe for 6 uses 450 g flour. How much for 14?	Show answer + formula + trick 1,050 g. 450 ÷ 6 = 75 g per person. 75 × 14 = 1,050 g. Formula / rule Direct proportion: find the value for one unit, then multiply by the required number of units. Applied to 5B: the worked result is 1,050 g. Why it works here 450 ÷ 6 = 75 g per person. 75 × 14 = 1,050 g. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 5B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 5B: 1,050 g.
6A	Scale factor	A shape side of 6 cm is enlarged by scale factor 2.5. New length?	Show answer + formula + trick 15 cm. 6 × 2.5 = 15. Formula / rule New length = original length × scale factor. Applied to 6A: the worked result is 15 cm. Why it works here 6 × 2.5 = 15. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 6A, re-check the exact numbers in this row before entering the answer. Remember Factors divide exactly. Checkpoint for 6A: 15 cm.
6B	Scale factor Similar practice	A side of 8 cm is enlarged by scale factor 1.75. New length?	Show answer + formula + trick 14 cm. 8 × 1.75 = 14. Formula / rule New length = original length × scale factor. Applied to 6B: the worked result is 14 cm. Why it works here 8 × 1.75 = 14. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 6B, re-check the exact numbers in this row before entering the answer. Remember Factors divide exactly. Checkpoint for 6B: 14 cm.
7A	Map scale	On a 1:50,000 map, two places are 6 cm apart. What is the real distance?	Show answer + formula + trick 3 km. 6 × 50,000 = 300,000 cm = 3,000 m = 3 km. Formula / rule Apply the map scale using matching units; real distance = map distance × scale factor. Applied to 7A: the worked result is 3 km. Why it works here 6 × 50,000 = 300,000 cm = 3,000 m = 3 km. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Map scale. Checkpoint for 7A: 3 km.
7B	Map scale Similar practice	On a 1:25,000 map, two places are 8 cm apart. What is the real distance?	Show answer + formula + trick 2 km. 8 × 25,000 = 200,000 cm = 2,000 m = 2 km. Formula / rule Apply the map scale using matching units; real distance = map distance × scale factor. Applied to 7B: the worked result is 2 km. Why it works here 8 × 25,000 = 200,000 cm = 2,000 m = 2 km. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Map scale. Checkpoint for 7B: 2 km.
8A	Ratio from total	Red:blue = 7:5 and there are 96 counters. How many are red?	Show answer + formula + trick 56. 12 parts in total. 96 ÷ 12 = 8 per part. Red = 7 × 8 = 56. Formula / rule Find one ratio part from the known total/part, then multiply by the required number of parts. Applied to 8A: the worked result is 56. Why it works here 12 parts in total. 96 ÷ 12 = 8 per part. Red = 7 × 8 = 56. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 8A: 56.
8B	Ratio from total Similar practice	Green:yellow = 5:3 and there are 64 counters. How many are green?	Show answer + formula + trick 40. 8 parts in total. 64 ÷ 8 = 8 per part. Green = 5 × 8 = 40. Formula / rule Find one ratio part from the known total/part, then multiply by the required number of parts. Applied to 8B: the worked result is 40. Why it works here 8 parts in total. 64 ÷ 8 = 8 per part. Green = 5 × 8 = 40. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 8B: 40.
9A	Part-to-whole	Boys:girls = 3:7. What fraction are girls?	Show answer + formula + trick 7/10. There are 10 equal parts in total; 7 are girls. Formula / rule Fraction represented by a ratio part = chosen parts ÷ total ratio parts. Applied to 9A: the worked result is 7/10. Why it works here There are 10 equal parts in total; 7 are girls. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Part-to-whole. Checkpoint for 9A: 7/10.
9B	Part-to-whole Similar practice	Cats:dogs = 2:5. What fraction are dogs?	Show answer + formula + trick 5/7. There are 7 equal parts in total; 5 are dogs. Formula / rule Fraction represented by a ratio part = chosen parts ÷ total ratio parts. Applied to 9B: the worked result is 5/7. Why it works here There are 7 equal parts in total; 5 are dogs. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Part-to-whole. Checkpoint for 9B: 5/7.
10A	Ratio after change	Red:blue = 3:2 with 18 red counters. How many blue counters?	Show answer + formula + trick 12. 3 parts = 18, so 1 part = 6. Blue = 2 parts = 12. Formula / rule Relate the numerical change to the change/difference in ratio parts to find one part. Applied to 10A: the worked result is 12. Why it works here 3 parts = 18, so 1 part = 6. Blue = 2 parts = 12. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 10A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 10A: 12.
10B	Ratio after change Similar practice	Red:blue = 4:3 with 28 red counters. How many blue counters?	Show answer + formula + trick 21. 4 parts = 28, so 1 part = 7. Blue = 3 parts = 21. Formula / rule Relate the numerical change to the change/difference in ratio parts to find one part. Applied to 10B: the worked result is 21. Why it works here 4 parts = 28, so 1 part = 7. Blue = 3 parts = 21. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 10B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 10B: 21.

7. Basic Algebra

#	Question type	Example	Worked solution
1A	Simplify like terms	Simplify 4x + 3x - 5.	Show answer + formula + trick 7x - 5. Combine only like terms. Formula / rule Combine coefficients only for like terms with the same variable and power. Applied to 1A: the worked result is 7x - 5. Why it works here Combine only like terms. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Simplify like terms. Checkpoint for 1A: 7x - 5.
1B	Simplify like terms Similar practice	Simplify 6y + 5y - 4.	Show answer + formula + trick 11y - 4. Combine only like terms. Formula / rule Combine coefficients only for like terms with the same variable and power. Applied to 1B: the worked result is 11y - 4. Why it works here Combine only like terms. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Simplify like terms. Checkpoint for 1B: 11y - 4.
2A	Substitution	Find 3a + 2 when a = 7.	Show answer + formula + trick 23. 3 × 7 + 2 = 23. Formula / rule Replace the variable with its given value, then use BODMAS. Applied to 2A: the worked result is 23. Why it works here 3 × 7 + 2 = 23. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Substitution. Checkpoint for 2A: 23.
2B	Substitution Similar practice	Find 4b - 3 when b = 6.	Show answer + formula + trick 21. 4 × 6 - 3 = 21. Formula / rule Replace the variable with its given value, then use BODMAS. Applied to 2B: the worked result is 21. Why it works here 4 × 6 - 3 = 21. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Substitution. Checkpoint for 2B: 21.
3A	One-step equation	Solve x + 17 = 43.	Show answer + formula + trick x = 26. Subtract 17 from both sides. Formula / rule Use the inverse operation on both sides to isolate the variable. Applied to 3A: the worked result is x = 26. Why it works here Subtract 17 from both sides. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 3A, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 3A: x = 26.
3B	One-step equation Similar practice	Solve x - 14 = 29.	Show answer + formula + trick x = 43. Add 14 to both sides. Formula / rule Use the inverse operation on both sides to isolate the variable. Applied to 3B: the worked result is x = 43. Why it works here Add 14 to both sides. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 3B, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 3B: x = 43.
4A	Multiplication equation	Solve 6x = 78.	Show answer + formula + trick x = 13. Divide both sides by 6. Formula / rule Use the inverse operation on both sides to isolate the variable. Applied to 4A: the worked result is x = 13. Why it works here Divide both sides by 6. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 4A, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 4A: x = 13.
4B	Multiplication equation Similar practice	Solve 9x = 117.	Show answer + formula + trick x = 13. Divide both sides by 9. Formula / rule Use the inverse operation on both sides to isolate the variable. Applied to 4B: the worked result is x = 13. Why it works here Divide both sides by 9. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 4B, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 4B: x = 13.
5A	Two-step equation	Solve 4x + 7 = 31.	Show answer + formula + trick x = 6. Subtract 7 to get 4x = 24, then divide by 4. Formula / rule Undo operations in reverse order while doing the same to both sides. Applied to 5A: the worked result is x = 6. Why it works here Subtract 7 to get 4x = 24, then divide by 4. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 5A, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 5A: x = 6.
5B	Two-step equation Similar practice	Solve 5x - 6 = 29.	Show answer + formula + trick x = 7. Add 6 to get 5x = 35, then divide by 5. Formula / rule Undo operations in reverse order while doing the same to both sides. Applied to 5B: the worked result is x = 7. Why it works here Add 6 to get 5x = 35, then divide by 5. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 5B, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 5B: x = 7.
6A	Brackets	Simplify 3(x + 4).	Show answer + formula + trick 3x + 12. Multiply every term inside the bracket by 3. Formula / rule Distributive law: a(b+c)=ab+ac and a(b−c)=ab−ac. Applied to 6A: the worked result is 3x + 12. Why it works here Multiply every term inside the bracket by 3. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Brackets. Checkpoint for 6A: 3x + 12.
6B	Brackets Similar practice	Simplify 4(x - 3).	Show answer + formula + trick 4x - 12. Multiply every term inside the bracket by 4. Formula / rule Distributive law: a(b+c)=ab+ac and a(b−c)=ab−ac. Applied to 6B: the worked result is 4x - 12. Why it works here Multiply every term inside the bracket by 4. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Brackets. Checkpoint for 6B: 4x - 12.
7A	Formula use	Use A = l × w. Find A when l = 8.5 and w = 6.	Show answer + formula + trick 51. 8.5 × 6 = 51. Formula / rule Substitute the known values into the stated formula, then calculate using BODMAS. Applied to 7A: the worked result is 51. Why it works here 8.5 × 6 = 51. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Formula use. Checkpoint for 7A: 51.
7B	Formula use Similar practice	Use P = 2(l + w). Find P when l = 9 and w = 4.5.	Show answer + formula + trick 27. P = 2(9 + 4.5) = 2 × 13.5 = 27. Formula / rule Substitute the known values into the stated formula, then calculate using BODMAS. Applied to 7B: the worked result is 27. Why it works here P = 2(9 + 4.5) = 2 × 13.5 = 27. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Formula use. Checkpoint for 7B: 27.
8A	Rearranging simple formula	If y = x + 9 and y = 25, find x.	Show answer + formula + trick 16. x = 25 - 9 = 16. Formula / rule Use inverse operations to make the required variable the subject. Applied to 8A: the worked result is 16. Why it works here x = 25 - 9 = 16. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Rearranging simple formula. Checkpoint for 8A: 16.
8B	Rearranging simple formula Similar practice	If y = x - 8 and y = 17, find x.	Show answer + formula + trick 25. x = 17 + 8 = 25. Formula / rule Use inverse operations to make the required variable the subject. Applied to 8B: the worked result is 25. Why it works here x = 17 + 8 = 25. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Rearranging simple formula. Checkpoint for 8B: 25.
9A	Word equation	A number doubled then increased by 5 equals 27. Find the number.	Show answer + formula + trick 11. 2x + 5 = 27, so 2x = 22 and x = 11. Formula / rule Undo operations in reverse order while doing the same to both sides. Applied to 9A: the worked result is 11. Why it works here 2x + 5 = 27, so 2x = 22 and x = 11. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 9A, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 9A: 11.
9B	Word equation Similar practice	A number tripled then decreased by 4 equals 29. Find the number.	Show answer + formula + trick 11. 3x - 4 = 29, so 3x = 33 and x = 11. Formula / rule Undo operations in reverse order while doing the same to both sides. Applied to 9B: the worked result is 11. Why it works here 3x - 4 = 29, so 3x = 33 and x = 11. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 9B, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 9B: 11.
10A	Function machine	Input x is multiplied by 4 then 3 is subtracted. Output is 29. Find x.	Show answer + formula + trick 8. 4x - 3 = 29, so 4x = 32 and x = 8. Formula / rule Apply function-machine operations in the stated order. Applied to 10A: the worked result is 8. Why it works here 4x - 3 = 29, so 4x = 32 and x = 8. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 10A, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 10A: 8.
10B	Function machine Similar practice	Input x is multiplied by 5 then 2 is added. Output is 42. Find x.	Show answer + formula + trick 8. 5x + 2 = 42, so 5x = 40 and x = 8. Formula / rule Apply function-machine operations in the stated order. Applied to 10B: the worked result is 8. Why it works here 5x + 2 = 42, so 5x = 40 and x = 8. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 10B, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 10B: 8.

8. Advanced Algebra

#	Question type	Example	Worked solution
1A	Word problem equation	Three times a number plus 8 is 44. Find the number.	Show answer + formula + trick 12. 3x + 8 = 44, so 3x = 36 and x = 12. Formula / rule Undo operations in reverse order while doing the same to both sides. Applied to 1A: the worked result is 12. Why it works here 3x + 8 = 44, so 3x = 36 and x = 12. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 1A, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 1A: 12.
1B	Word problem equation Similar practice	Four times a number minus 6 is 34. Find the number.	Show answer + formula + trick 10. 4x - 6 = 34, so 4x = 40 and x = 10. Formula / rule Undo operations in reverse order while doing the same to both sides. Applied to 1B: the worked result is 10. Why it works here 4x - 6 = 34, so 4x = 40 and x = 10. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 1B, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 1B: 10.
2A	Inequality	Solve x + 7 < 15.	Show answer + formula + trick x < 8. Subtract 7 from both sides. Formula / rule Solve like an equation; if multiplying/dividing by a negative number, reverse the inequality sign. Applied to 2A: the worked result is x < 8. Why it works here Subtract 7 from both sides. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Inequality. Checkpoint for 2A: x < 8.
2B	Inequality Similar practice	Solve x - 5 > 12.	Show answer + formula + trick x > 17. Add 5 to both sides. Formula / rule Solve like an equation; if multiplying/dividing by a negative number, reverse the inequality sign. Applied to 2B: the worked result is x > 17. Why it works here Add 5 to both sides. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Inequality. Checkpoint for 2B: x > 17.
3A	Arithmetic sequence	Find the next two terms: 7, 12, 17, 22, ...	Show answer + formula + trick 27, 32. The common difference is +5. Formula / rule Arithmetic sequence: add/subtract the same common difference each time. Applied to 3A: the worked result is 27, 32. Why it works here The common difference is +5. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 3A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 3A: 27, 32.
3B	Arithmetic sequence Similar practice	Find the next two terms: 11, 15, 19, 23, ...	Show answer + formula + trick 27, 31. The common difference is +4. Formula / rule Arithmetic sequence: add/subtract the same common difference each time. Applied to 3B: the worked result is 27, 31. Why it works here The common difference is +4. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 3B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 3B: 27, 31.
4A	Nth term	Find the nth term of 4, 7, 10, 13, ...	Show answer + formula + trick 3n + 1. The common difference is 3, so start with 3n. At n=1, 3n gives 3, so add 1. Formula / rule For a linear sequence, nth term = common difference × n + adjustment. Applied to 4A: the worked result is 3n + 1. Why it works here The common difference is 3, so start with 3n. At n=1, 3n gives 3, so add 1. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 4A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 4A: 3n + 1.
4B	Nth term Similar practice	Find the nth term of 6, 10, 14, 18, ...	Show answer + formula + trick 4n + 2. The common difference is 4, so start with 4n. At n=1, 4n gives 4, so add 2. Formula / rule For a linear sequence, nth term = common difference × n + adjustment. Applied to 4B: the worked result is 4n + 2. Why it works here The common difference is 4, so start with 4n. At n=1, 4n gives 4, so add 2. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 4B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 4B: 4n + 2.
5A	Use nth term	For the sequence 5n - 2, find the 12th term.	Show answer + formula + trick 58. 5 × 12 - 2 = 58. Formula / rule For a linear sequence, nth term = common difference × n + adjustment. Applied to 5A: the worked result is 58. Why it works here 5 × 12 - 2 = 58. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 5A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 5A: 58.
5B	Use nth term Similar practice	For the sequence 7n + 1, find the 9th term.	Show answer + formula + trick 64. 7 × 9 + 1 = 64. Formula / rule For a linear sequence, nth term = common difference × n + adjustment. Applied to 5B: the worked result is 64. Why it works here 7 × 9 + 1 = 64. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 5B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 5B: 64.
6A	Geometric sequence	Find the next term: 3, 6, 12, 24, ...	Show answer + formula + trick 48. Each term is multiplied by 2. Formula / rule Geometric sequence: multiply/divide by the same ratio each time. Applied to 6A: the worked result is 48. Why it works here Each term is multiplied by 2. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 6A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 6A: 48.
6B	Geometric sequence Similar practice	Find the next term: 5, 15, 45, 135, ...	Show answer + formula + trick 405. Each term is multiplied by 3. Formula / rule Geometric sequence: multiply/divide by the same ratio each time. Applied to 6B: the worked result is 405. Why it works here Each term is multiplied by 3. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 6B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 6B: 405.
7A	Missing term	Find x: 2, 6, 18, x, 162.	Show answer + formula + trick 54. The rule is ×3 each time. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 7A: the worked result is 54. Why it works here The rule is ×3 each time. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing term. Checkpoint for 7A: 54.
7B	Missing term Similar practice	Find x: 4, 12, 36, x, 324.	Show answer + formula + trick 108. The rule is ×3 each time. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 7B: the worked result is 108. Why it works here The rule is ×3 each time. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing term. Checkpoint for 7B: 108.
8A	Consecutive numbers	Three consecutive integers sum to 72. Find them.	Show answer + formula + trick 23, 24, 25. Middle number is 72 ÷ 3 = 24, so the neighbours are 23 and 25. Formula / rule Consecutive integers differ by 1; for an odd count, their average is the middle integer. Applied to 8A: the worked result is 23, 24, 25. Why it works here Middle number is 72 ÷ 3 = 24, so the neighbours are 23 and 25. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Consecutive numbers. Checkpoint for 8A: 23, 24, 25.
8B	Consecutive numbers Similar practice	Three consecutive integers sum to 96. Find them.	Show answer + formula + trick 31, 32, 33. Middle number is 96 ÷ 3 = 32, so the neighbours are 31 and 33. Formula / rule Consecutive integers differ by 1; for an odd count, their average is the middle integer. Applied to 8B: the worked result is 31, 32, 33. Why it works here Middle number is 96 ÷ 3 = 32, so the neighbours are 31 and 33. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Consecutive numbers. Checkpoint for 8B: 31, 32, 33.
9A	Age problem	Mia is 4 years older than Ava. Together they are 26. How old is Ava?	Show answer + formula + trick 11. Let Ava = x. Mia = x + 4. Then 2x + 4 = 26, so x = 11. Formula / rule For this Age problem item, use the relationship demonstrated in the worked solution and verify it with an inverse operation or estimate. Applied to 9A: the worked result is 11. Why it works here Let Ava = x. Mia = x + 4. Then 2x + 4 = 26, so x = 11. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Age problem. Checkpoint for 9A: 11.
9B	Age problem Similar practice	Noah is 6 years older than Leo. Together they are 34. How old is Leo?	Show answer + formula + trick 14. Let Leo = x. Noah = x + 6. Then 2x + 6 = 34, so x = 14. Formula / rule For this Age problem item, use the relationship demonstrated in the worked solution and verify it with an inverse operation or estimate. Applied to 9B: the worked result is 14. Why it works here Let Leo = x. Noah = x + 6. Then 2x + 6 = 34, so x = 14. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Age problem. Checkpoint for 9B: 14.
10A	Pattern formula	A matchstick pattern uses 4, 7, 10, 13 sticks. How many in figure 20?	Show answer + formula + trick 61. Nth term is 3n + 1. At n=20, 60 + 1 = 61. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 10A: the worked result is 61. Why it works here Nth term is 3n + 1. At n=20, 60 + 1 = 61. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 10A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 10A: 61.
10B	Pattern formula Similar practice	A pattern uses 6, 10, 14, 18 tiles. How many in figure 25?	Show answer + formula + trick 102. Nth term is 4n + 2. At n=25, 100 + 2 = 102. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 10B: the worked result is 102. Why it works here Nth term is 4n + 2. At n=25, 100 + 2 = 102. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 10B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 10B: 102.

9. Number Patterns and Sequences

#	Question type	Example	Worked solution
1A	Constant difference	Continue: 81, 74, 67, 60, ...	Show answer + formula + trick 53. Subtract 7 each time. Formula / rule Arithmetic sequence: add/subtract the same common difference each time. Applied to 1A: the worked result is 53. Why it works here Subtract 7 each time. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Constant difference. Checkpoint for 1A: 53.
1B	Constant difference Similar practice	Continue: 95, 87, 79, 71, ...	Show answer + formula + trick 63. Subtract 8 each time. Formula / rule Arithmetic sequence: add/subtract the same common difference each time. Applied to 1B: the worked result is 63. Why it works here Subtract 8 each time. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Constant difference. Checkpoint for 1B: 63.
2A	Increasing difference	Continue: 2, 5, 9, 14, 20, ...	Show answer + formula + trick 27. Differences are +3, +4, +5, +6, so next is +7. Formula / rule Write first differences; if they change regularly, inspect the second differences. Applied to 2A: the worked result is 27. Why it works here Differences are +3, +4, +5, +6, so next is +7. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Increasing difference. Checkpoint for 2A: 27.
2B	Increasing difference Similar practice	Continue: 3, 7, 12, 18, 25, ...	Show answer + formula + trick 33. Differences are +4, +5, +6, +7, so next is +8. Formula / rule Write first differences; if they change regularly, inspect the second differences. Applied to 2B: the worked result is 33. Why it works here Differences are +4, +5, +6, +7, so next is +8. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Increasing difference. Checkpoint for 2B: 33.
3A	Geometric	Continue: 160, 80, 40, 20, ...	Show answer + formula + trick 10. Divide by 2 each time. Formula / rule Geometric sequence: multiply/divide by the same ratio each time. Applied to 3A: the worked result is 10. Why it works here Divide by 2 each time. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Geometric. Checkpoint for 3A: 10.
3B	Geometric Similar practice	Continue: 243, 81, 27, 9, ...	Show answer + formula + trick 3. Divide by 3 each time. Formula / rule Geometric sequence: multiply/divide by the same ratio each time. Applied to 3B: the worked result is 3. Why it works here Divide by 3 each time. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Geometric. Checkpoint for 3B: 3.
4A	Fibonacci style	Continue: 4, 7, 11, 18, 29, ...	Show answer + formula + trick 47. Add the previous two terms. Formula / rule Fibonacci-style rule: each new term is the sum of the previous two. Applied to 4A: the worked result is 47. Why it works here Add the previous two terms. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Fibonacci style. Checkpoint for 4A: 47.
4B	Fibonacci style Similar practice	Continue: 5, 8, 13, 21, 34, ...	Show answer + formula + trick 55. Add the previous two terms. Formula / rule Fibonacci-style rule: each new term is the sum of the previous two. Applied to 4B: the worked result is 55. Why it works here Add the previous two terms. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Fibonacci style. Checkpoint for 4B: 55.
5A	Square numbers	What comes next: 1, 4, 9, 16, 25, ...?	Show answer + formula + trick 36. These are 1², 2², 3², 4², 5², so next is 6² = 36. Formula / rule A square number has the form n² = n×n. Applied to 5A: the worked result is 36. Why it works here These are 1², 2², 3², 4², 5², so next is 6² = 36. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Square numbers. Checkpoint for 5A: 36.
5B	Square numbers Similar practice	What comes next: 4, 9, 16, 25, 36, ...?	Show answer + formula + trick 49. These are 2², 3², 4², 5², 6², so next is 7² = 49. Formula / rule A square number has the form n² = n×n. Applied to 5B: the worked result is 49. Why it works here These are 2², 3², 4², 5², 6², so next is 7² = 49. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Square numbers. Checkpoint for 5B: 49.
6A	Triangular numbers	Continue: 1, 3, 6, 10, 15, ...	Show answer + formula + trick 21. Add 2, 3, 4, 5, then 6. Formula / rule Triangular numbers increase by consecutive integers: +2,+3,+4,+5,... Applied to 6A: the worked result is 21. Why it works here Add 2, 3, 4, 5, then 6. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Triangular numbers. Checkpoint for 6A: 21.
6B	Triangular numbers Similar practice	Continue: 3, 6, 10, 15, 21, ...	Show answer + formula + trick 28. Add 3, 4, 5, 6, then 7. Formula / rule Triangular numbers increase by consecutive integers: +2,+3,+4,+5,... Applied to 6B: the worked result is 28. Why it works here Add 3, 4, 5, 6, then 7. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Triangular numbers. Checkpoint for 6B: 28.
7A	Alternating rule	Continue: 3, 8, 6, 11, 9, 14, ...	Show answer + formula + trick 12. Alternate +5, -2. Formula / rule Look for two repeating operations or separate patterns in odd/even positions. Applied to 7A: the worked result is 12. Why it works here Alternate +5, -2. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Alternating rule. Checkpoint for 7A: 12.
7B	Alternating rule Similar practice	Continue: 10, 14, 11, 15, 12, 16, ...	Show answer + formula + trick 13. Alternate +4, -3. Formula / rule Look for two repeating operations or separate patterns in odd/even positions. Applied to 7B: the worked result is 13. Why it works here Alternate +4, -3. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Alternating rule. Checkpoint for 7B: 13.
8A	Decimal sequence	Continue: 0.4, 0.7, 1.0, 1.3, ...	Show answer + formula + trick 1.6. Add 0.3. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 8A: the worked result is 1.6. Why it works here Add 0.3. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 8A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 8A: 1.6.
8B	Decimal sequence Similar practice	Continue: 1.2, 1.6, 2.0, 2.4, ...	Show answer + formula + trick 2.8. Add 0.4. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 8B: the worked result is 2.8. Why it works here Add 0.4. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 8B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 8B: 2.8.
9A	Find missing term	5, 11, 23, ?, 95.	Show answer + formula + trick 47. Multiply by 2 and add 1 each time: 23 × 2 + 1 = 47. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 9A: the worked result is 47. Why it works here Multiply by 2 and add 1 each time: 23 × 2 + 1 = 47. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Find missing term. Checkpoint for 9A: 47.
9B	Find missing term Similar practice	7, 15, 31, ?, 127.	Show answer + formula + trick 63. Multiply by 2 and add 1 each time: 31 × 2 + 1 = 63. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 9B: the worked result is 63. Why it works here Multiply by 2 and add 1 each time: 31 × 2 + 1 = 63. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Find missing term. Checkpoint for 9B: 63.
10A	Backward sequence	100, 91, 83, 76, 70, ...	Show answer + formula + trick 65. Subtract 9, 8, 7, 6, then 5. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 10A: the worked result is 65. Why it works here Subtract 9, 8, 7, 6, then 5. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 10A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 10A: 65.
10B	Backward sequence Similar practice	120, 109, 99, 90, 82, ...	Show answer + formula + trick 75. Subtract 11, 10, 9, 8, then 7. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 10B: the worked result is 75. Why it works here Subtract 11, 10, 9, 8, then 7. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 10B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 10B: 75.

10. Basic Geometry

#	Question type	Example	Worked solution
1A	Straight line	One angle on a straight line is 128°. Find the other.	Show answer + formula + trick 52°. Angles on a straight line total 180°. Formula / rule Angles on a straight line sum to 180°. Applied to 1A: the worked result is 52°. Why it works here Angles on a straight line total 180°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Straight line. Checkpoint for 1A: 52°.
1B	Straight line Similar practice	One angle on a straight line is 143°. Find the other.	Show answer + formula + trick 37°. Angles on a straight line total 180°. Formula / rule Angles on a straight line sum to 180°. Applied to 1B: the worked result is 37°. Why it works here Angles on a straight line total 180°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Straight line. Checkpoint for 1B: 37°.
2A	Around a point	Angles around a point are 95°, 110°, 80° and x. Find x.	Show answer + formula + trick 75°. Angles around a point total 360°. 360 - 285 = 75. Formula / rule Angles around a point sum to 360°. Applied to 2A: the worked result is 75°. Why it works here Angles around a point total 360°. 360 - 285 = 75. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Around a point. Checkpoint for 2A: 75°.
2B	Around a point Similar practice	Angles around a point are 120°, 85°, 64° and x. Find x.	Show answer + formula + trick 91°. Angles around a point total 360°. 360 - 269 = 91. Formula / rule Angles around a point sum to 360°. Applied to 2B: the worked result is 91°. Why it works here Angles around a point total 360°. 360 - 269 = 91. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Around a point. Checkpoint for 2B: 91°.
3A	Triangle	A triangle has angles 48° and 67°. Find the third.	Show answer + formula + trick 65°. Angles in a triangle total 180°. 180 - 115 = 65. Formula / rule Interior angles of a triangle sum to 180°. Applied to 3A: the worked result is 65°. Why it works here Angles in a triangle total 180°. 180 - 115 = 65. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 3A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 3A: 65°.
3B	Triangle Similar practice	A triangle has angles 56° and 79°. Find the third.	Show answer + formula + trick 45°. Angles in a triangle total 180°. 180 - 135 = 45. Formula / rule Interior angles of a triangle sum to 180°. Applied to 3B: the worked result is 45°. Why it works here Angles in a triangle total 180°. 180 - 135 = 45. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 3B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 3B: 45°.
4A	Isosceles triangle	An isosceles triangle has vertex angle 38°. Find each base angle.	Show answer + formula + trick 71°. Remaining total is 142°, split equally: 71° each. Formula / rule In an isosceles triangle, the two base angles are equal and all three angles total 180°. Applied to 4A: the worked result is 71°. Why it works here Remaining total is 142°, split equally: 71° each. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 4A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 4A: 71°.
4B	Isosceles triangle Similar practice	An isosceles triangle has vertex angle 44°. Find each base angle.	Show answer + formula + trick 68°. Remaining total is 136°, split equally: 68° each. Formula / rule In an isosceles triangle, the two base angles are equal and all three angles total 180°. Applied to 4B: the worked result is 68°. Why it works here Remaining total is 136°, split equally: 68° each. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 4B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 4B: 68°.
5A	Quadrilateral	Three angles of a quadrilateral are 90°, 105° and 88°. Find the fourth.	Show answer + formula + trick 77°. Quadrilateral angles total 360°. 360 - 283 = 77. Formula / rule Interior angles of a quadrilateral sum to 360°. Applied to 5A: the worked result is 77°. Why it works here Quadrilateral angles total 360°. 360 - 283 = 77. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 5A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 5A: 77°.
5B	Quadrilateral Similar practice	Three angles of a quadrilateral are 96°, 82° and 103°. Find the fourth.	Show answer + formula + trick 79°. Quadrilateral angles total 360°. 360 - 281 = 79. Formula / rule Interior angles of a quadrilateral sum to 360°. Applied to 5B: the worked result is 79°. Why it works here Quadrilateral angles total 360°. 360 - 281 = 79. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 5B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 5B: 79°.
6A	Vertically opposite	Two straight lines cross. One angle is 63°. Find the vertically opposite angle.	Show answer + formula + trick 63°. Vertically opposite angles are equal. Formula / rule Vertically opposite angles are equal. Applied to 6A: the worked result is 63°. Why it works here Vertically opposite angles are equal. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Vertically opposite. Checkpoint for 6A: 63°.
6B	Vertically opposite Similar practice	Two straight lines cross. One angle is 117°. Find the vertically opposite angle.	Show answer + formula + trick 117°. Vertically opposite angles are equal. Formula / rule Vertically opposite angles are equal. Applied to 6B: the worked result is 117°. Why it works here Vertically opposite angles are equal. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Vertically opposite. Checkpoint for 6B: 117°.
7A	Corresponding angles	Two parallel lines are crossed by a transversal. One corresponding angle is 112°. Find the matching corresponding angle.	Show answer + formula + trick 112°. Corresponding angles on parallel lines are equal. Formula / rule Corresponding angles on parallel lines are equal. Applied to 7A: the worked result is 112°. Why it works here Corresponding angles on parallel lines are equal. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 7A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 7A: 112°.
7B	Corresponding angles Similar practice	Two parallel lines are crossed by a transversal. One corresponding angle is 74°. Find the matching corresponding angle.	Show answer + formula + trick 74°. Corresponding angles on parallel lines are equal. Formula / rule Corresponding angles on parallel lines are equal. Applied to 7B: the worked result is 74°. Why it works here Corresponding angles on parallel lines are equal. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 7B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 7B: 74°.
8A	Alternate angles	A pair of alternate interior angles lie on parallel lines. One is 68°. Find the other.	Show answer + formula + trick 68°. Alternate interior angles are equal. Formula / rule Alternate interior angles on parallel lines are equal. Applied to 8A: the worked result is 68°. Why it works here Alternate interior angles are equal. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 8A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 8A: 68°.
8B	Alternate angles Similar practice	A pair of alternate interior angles lie on parallel lines. One is 121°. Find the other.	Show answer + formula + trick 121°. Alternate interior angles are equal. Formula / rule Alternate interior angles on parallel lines are equal. Applied to 8B: the worked result is 121°. Why it works here Alternate interior angles are equal. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 8B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 8B: 121°.
9A	Exterior triangle angle	The two opposite interior angles of a triangle are 42° and 73°. Find the exterior angle.	Show answer + formula + trick 115°. An exterior angle equals the sum of the two opposite interior angles: 42 + 73 = 115. Formula / rule A triangle exterior angle equals the sum of the two opposite interior angles. Applied to 9A: the worked result is 115°. Why it works here An exterior angle equals the sum of the two opposite interior angles: 42 + 73 = 115. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 9A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 9A: 115°.
9B	Exterior triangle angle Similar practice	The two opposite interior angles of a triangle are 58° and 47°. Find the exterior angle.	Show answer + formula + trick 105°. An exterior angle equals the sum of the two opposite interior angles: 58 + 47 = 105. Formula / rule A triangle exterior angle equals the sum of the two opposite interior angles. Applied to 9B: the worked result is 105°. Why it works here An exterior angle equals the sum of the two opposite interior angles: 58 + 47 = 105. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 9B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 9B: 105°.
10A	Angle type	Classify 235°.	Show answer + formula + trick Reflex angle. A reflex angle is greater than 180° and less than 360°. Formula / rule Acute <90°, right=90°, obtuse 90°–180°, straight=180°, reflex 180°–360°. Applied to 10A: the worked result is Reflex angle. Why it works here A reflex angle is greater than 180° and less than 360°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 10A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 10A: Reflex angle.
10B	Angle type Similar practice	Classify 152°.	Show answer + formula + trick Obtuse angle. An obtuse angle is greater than 90° and less than 180°. Formula / rule Acute <90°, right=90°, obtuse 90°–180°, straight=180°, reflex 180°–360°. Applied to 10B: the worked result is Obtuse angle. Why it works here An obtuse angle is greater than 90° and less than 180°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 10B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 10B: Obtuse angle.

11. Shapes and Symmetry

#	Question type	Example	Worked solution
1A	Rectangle perimeter	A rectangle is 12 cm by 7 cm. Find its perimeter.	Show answer + formula + trick 38 cm. P = 2(l + w) = 2(12 + 7) = 38. Formula / rule Rectangle perimeter P=2(L+W). Applied to 1A: the worked result is 38 cm. Why it works here P = 2(l + w) = 2(12 + 7) = 38. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 1A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 1A: 38 cm.
1B	Rectangle perimeter Similar practice	A rectangle is 15 cm by 9 cm. Find its perimeter.	Show answer + formula + trick 48 cm. P = 2(l + w) = 2(15 + 9) = 48. Formula / rule Rectangle perimeter P=2(L+W). Applied to 1B: the worked result is 48 cm. Why it works here P = 2(l + w) = 2(15 + 9) = 48. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 1B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 1B: 48 cm.
2A	Rectangle area	A rectangle is 13 cm by 8 cm. Find its area.	Show answer + formula + trick 104 cm². A = l × w = 13 × 8. Formula / rule Rectangle area A=L×W. Applied to 2A: the worked result is 104 cm². Why it works here A = l × w = 13 × 8. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 2A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 2A: 104 cm².
2B	Rectangle area Similar practice	A rectangle is 17 cm by 6 cm. Find its area.	Show answer + formula + trick 102 cm². A = l × w = 17 × 6. Formula / rule Rectangle area A=L×W. Applied to 2B: the worked result is 102 cm². Why it works here A = l × w = 17 × 6. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 2B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 2B: 102 cm².
3A	Triangle area	A triangle has base 14 cm and perpendicular height 9 cm. Find its area.	Show answer + formula + trick 63 cm². A = 1/2 × base × height = 1/2 × 14 × 9. Formula / rule Triangle area A=½×base×perpendicular height. Applied to 3A: the worked result is 63 cm². Why it works here A = 1/2 × base × height = 1/2 × 14 × 9. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 3A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 3A: 63 cm².
3B	Triangle area Similar practice	A triangle has base 18 cm and perpendicular height 7 cm. Find its area.	Show answer + formula + trick 63 cm². A = 1/2 × 18 × 7 = 63. Formula / rule Triangle area A=½×base×perpendicular height. Applied to 3B: the worked result is 63 cm². Why it works here A = 1/2 × 18 × 7 = 63. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 3B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 3B: 63 cm².
4A	Parallelogram area	Base 11 cm, perpendicular height 6 cm. Area?	Show answer + formula + trick 66 cm². A = base × perpendicular height. Formula / rule Parallelogram area A=base×perpendicular height. Applied to 4A: the worked result is 66 cm². Why it works here A = base × perpendicular height. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 4A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 4A: 66 cm².
4B	Parallelogram area Similar practice	Base 13 cm, perpendicular height 8 cm. Area?	Show answer + formula + trick 104 cm². A = base × perpendicular height. Formula / rule Parallelogram area A=base×perpendicular height. Applied to 4B: the worked result is 104 cm². Why it works here A = base × perpendicular height. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 4B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 4B: 104 cm².
5A	Trapezium area	Parallel sides are 8 cm and 14 cm, height 5 cm. Area?	Show answer + formula + trick 55 cm². A = 1/2 × (8 + 14) × 5 = 55. Formula / rule Trapezium area A=½×(sum of parallel sides)×perpendicular height. Applied to 5A: the worked result is 55 cm². Why it works here A = 1/2 × (8 + 14) × 5 = 55. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 5A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 5A: 55 cm².
5B	Trapezium area Similar practice	Parallel sides are 10 cm and 16 cm, height 6 cm. Area?	Show answer + formula + trick 78 cm². A = 1/2 × (10 + 16) × 6 = 78. Formula / rule Trapezium area A=½×(sum of parallel sides)×perpendicular height. Applied to 5B: the worked result is 78 cm². Why it works here A = 1/2 × (10 + 16) × 6 = 78. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 5B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 5B: 78 cm².
6A	Regular polygon perimeter	A regular octagon has side 6.5 cm. Perimeter?	Show answer + formula + trick 52 cm. 8 × 6.5 = 52. Formula / rule Perimeter of a regular polygon = number of sides × side length. Applied to 6A: the worked result is 52 cm. Why it works here 8 × 6.5 = 52. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 6A, re-check the exact numbers in this row before entering the answer. Remember Perimeter traces the outside boundary. Checkpoint for 6A: 52 cm.
6B	Regular polygon perimeter Similar practice	A regular decagon has side 4.2 cm. Perimeter?	Show answer + formula + trick 42 cm. 10 × 4.2 = 42. Formula / rule Perimeter of a regular polygon = number of sides × side length. Applied to 6B: the worked result is 42 cm. Why it works here 10 × 4.2 = 42. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 6B, re-check the exact numbers in this row before entering the answer. Remember Perimeter traces the outside boundary. Checkpoint for 6B: 42 cm.
7A	Line symmetry	How many lines of symmetry does a square have?	Show answer + formula + trick 4. Two diagonals and two lines through opposite side midpoints. Formula / rule A line of symmetry divides a shape into mirror-image halves. Applied to 7A: the worked result is 4. Why it works here Two diagonals and two lines through opposite side midpoints. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Line symmetry. Checkpoint for 7A: 4.
7B	Line symmetry Similar practice	How many lines of symmetry does an equilateral triangle have?	Show answer + formula + trick 3. Each line goes from a vertex to the midpoint of the opposite side. Formula / rule A line of symmetry divides a shape into mirror-image halves. Applied to 7B: the worked result is 3. Why it works here Each line goes from a vertex to the midpoint of the opposite side. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Line symmetry. Checkpoint for 7B: 3.
8A	Rotational symmetry	What is the order of rotational symmetry of a regular hexagon?	Show answer + formula + trick 6. It matches itself 6 times in a full turn. Formula / rule Rotational symmetry order = number of matches in one full 360° turn. Applied to 8A: the worked result is 6. Why it works here It matches itself 6 times in a full turn. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 8A, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 8A: 6.
8B	Rotational symmetry Similar practice	What is the order of rotational symmetry of a rectangle that is not a square?	Show answer + formula + trick 2. It matches itself after 180° and 360°. Formula / rule Rotational symmetry order = number of matches in one full 360° turn. Applied to 8B: the worked result is 2. Why it works here It matches itself after 180° and 360°. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 8B, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 8B: 2.
9A	Reflection coordinates	Reflect (3, -2) in the y-axis.	Show answer + formula + trick (-3, -2). Reflection in the y-axis changes the sign of x only. Formula / rule Reflection in y-axis: (x,y)→(−x,y). Applied to 9A: the worked result is (-3, -2). Why it works here Reflection in the y-axis changes the sign of x only. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 9A, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 9A: (-3, -2).
9B	Reflection coordinates Similar practice	Reflect (-5, 4) in the y-axis.	Show answer + formula + trick (5, 4). Reflection in the y-axis changes the sign of x only. Formula / rule Reflection in y-axis: (x,y)→(−x,y). Applied to 9B: the worked result is (5, 4). Why it works here Reflection in the y-axis changes the sign of x only. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 9B, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 9B: (5, 4).
10A	Composite area	A 10 cm by 8 cm rectangle has a 4 cm by 3 cm corner removed. Find remaining area.	Show answer + formula + trick 68 cm². 80 - 12 = 68. Formula / rule Composite area = add non-overlapping parts or subtract cut-outs from a larger simple shape. Applied to 10A: the worked result is 68 cm². Why it works here 80 - 12 = 68. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 10A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 10A: 68 cm².
10B	Composite area Similar practice	A 12 cm by 9 cm rectangle has a 5 cm by 2 cm corner removed. Find remaining area.	Show answer + formula + trick 98 cm². 108 - 10 = 98. Formula / rule Composite area = add non-overlapping parts or subtract cut-outs from a larger simple shape. Applied to 10B: the worked result is 98 cm². Why it works here 108 - 10 = 98. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 10B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 10B: 98 cm².

12. 3D Shapes and Volume

#	Question type	Example	Worked solution
1A	Cuboid volume	Find the volume of a cuboid 8 cm × 5 cm × 3 cm.	Show answer + formula + trick 120 cm³. V = l × w × h. Formula / rule Cuboid volume V=L×W×H. Applied to 1A: the worked result is 120 cm³. Why it works here V = l × w × h. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 1A, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 1A: 120 cm³.
1B	Cuboid volume Similar practice	Find the volume of a cuboid 9 cm × 4 cm × 5 cm.	Show answer + formula + trick 180 cm³. V = l × w × h. Formula / rule Cuboid volume V=L×W×H. Applied to 1B: the worked result is 180 cm³. Why it works here V = l × w × h. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 1B, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 1B: 180 cm³.
2A	Cube volume	A cube has side 6 cm. Find its volume.	Show answer + formula + trick 216 cm³. V = 6³ = 216. Formula / rule Cube volume V=s³. Applied to 2A: the worked result is 216 cm³. Why it works here V = 6³ = 216. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 2A, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 2A: 216 cm³.
2B	Cube volume Similar practice	A cube has side 7 cm. Find its volume.	Show answer + formula + trick 343 cm³. V = 7³ = 343. Formula / rule Cube volume V=s³. Applied to 2B: the worked result is 343 cm³. Why it works here V = 7³ = 343. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 2B, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 2B: 343 cm³.
3A	Cube surface area	A cube has side 4 cm. Find its surface area.	Show answer + formula + trick 96 cm². Six faces, each 4 × 4 = 16. Total 6 × 16 = 96. Formula / rule Cube surface area =6s². Applied to 3A: the worked result is 96 cm². Why it works here Six faces, each 4 × 4 = 16. Total 6 × 16 = 96. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 3A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 3A: 96 cm².
3B	Cube surface area Similar practice	A cube has side 5 cm. Find its surface area.	Show answer + formula + trick 150 cm². Six faces, each 5 × 5 = 25. Total 6 × 25 = 150. Formula / rule Cube surface area =6s². Applied to 3B: the worked result is 150 cm². Why it works here Six faces, each 5 × 5 = 25. Total 6 × 25 = 150. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 3B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 3B: 150 cm².
4A	Cuboid surface area	Find the surface area of a 3 cm × 4 cm × 5 cm cuboid.	Show answer + formula + trick 94 cm². 2(3×4 + 3×5 + 4×5) = 2(12+15+20) = 94. Formula / rule Cuboid surface area =2(LW+LH+WH). Applied to 4A: the worked result is 94 cm². Why it works here 2(3×4 + 3×5 + 4×5) = 2(12+15+20) = 94. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 4A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 4A: 94 cm².
4B	Cuboid surface area Similar practice	Find the surface area of a 2 cm × 6 cm × 7 cm cuboid.	Show answer + formula + trick 136 cm². 2(2×6 + 2×7 + 6×7) = 2(12+14+42) = 136. Formula / rule Cuboid surface area =2(LW+LH+WH). Applied to 4B: the worked result is 136 cm². Why it works here 2(2×6 + 2×7 + 6×7) = 2(12+14+42) = 136. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 4B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 4B: 136 cm².
5A	Faces edges vertices	How many faces, edges and vertices does a cube have?	Show answer + formula + trick 6 faces, 12 edges, 8 vertices. Memorise this core fact. Formula / rule Cube/cuboid: 6 faces, 12 edges, 8 vertices. Applied to 5A: the worked result is 6 faces, 12 edges, 8 vertices. Why it works here Memorise this core fact. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Faces edges vertices. Checkpoint for 5A: 6 faces, 12 edges, 8 vertices.
5B	Faces edges vertices Similar practice	How many faces, edges and vertices does a cuboid have?	Show answer + formula + trick 6 faces, 12 edges, 8 vertices. A cuboid has the same counts as a cube. Formula / rule Cube/cuboid: 6 faces, 12 edges, 8 vertices. Applied to 5B: the worked result is 6 faces, 12 edges, 8 vertices. Why it works here A cuboid has the same counts as a cube. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Faces edges vertices. Checkpoint for 5B: 6 faces, 12 edges, 8 vertices.
6A	Triangular prism	How many faces does a triangular prism have?	Show answer + formula + trick 5. Two triangular ends and three rectangular faces. Formula / rule Triangular prism: 2 triangular ends +3 rectangular faces; 6 vertices, 9 edges. Applied to 6A: the worked result is 5. Why it works here Two triangular ends and three rectangular faces. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Triangular prism. Checkpoint for 6A: 5.
6B	Triangular prism Similar practice	How many vertices does a triangular prism have?	Show answer + formula + trick 6. Three vertices on each triangular end. Formula / rule Triangular prism: 2 triangular ends +3 rectangular faces; 6 vertices, 9 edges. Applied to 6B: the worked result is 6. Why it works here Three vertices on each triangular end. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Triangular prism. Checkpoint for 6B: 6.
7A	Pyramid recognition	A square-based pyramid has how many vertices?	Show answer + formula + trick 5. Four on the base plus the apex. Formula / rule Use the defining faces/edges/vertices and imagine folding the net into the solid. Applied to 7A: the worked result is 5. Why it works here Four on the base plus the apex. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Pyramid recognition. Checkpoint for 7A: 5.
7B	Pyramid recognition Similar practice	A triangular-based pyramid has how many faces?	Show answer + formula + trick 4. One triangular base plus three triangular side faces. Formula / rule Use the defining faces/edges/vertices and imagine folding the net into the solid. Applied to 7B: the worked result is 4. Why it works here One triangular base plus three triangular side faces. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Pyramid recognition. Checkpoint for 7B: 4.
8A	Cylinder properties	How many flat circular faces does a cylinder have?	Show answer + formula + trick 2. Top and bottom are flat circles; the side is curved. Formula / rule Use the defining faces/edges/vertices and imagine folding the net into the solid. Applied to 8A: the worked result is 2. Why it works here Top and bottom are flat circles; the side is curved. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Cylinder properties. Checkpoint for 8A: 2.
8B	Cylinder properties Similar practice	How many curved surfaces does a cylinder have?	Show answer + formula + trick 1. The side wraps around as one curved surface. Formula / rule Use the defining faces/edges/vertices and imagine folding the net into the solid. Applied to 8B: the worked result is 1. Why it works here The side wraps around as one curved surface. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Cylinder properties. Checkpoint for 8B: 1.
9A	Net reasoning	A valid cube net must contain how many equal squares?	Show answer + formula + trick 6. A cube has six square faces. Formula / rule Use the defining faces/edges/vertices and imagine folding the net into the solid. Applied to 9A: the worked result is 6. Why it works here A cube has six square faces. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Net reasoning. Checkpoint for 9A: 6.
9B	Net reasoning Similar practice	A valid triangular prism net must contain how many faces altogether?	Show answer + formula + trick 5. Two triangles and three rectangles. Formula / rule Use the defining faces/edges/vertices and imagine folding the net into the solid. Applied to 9B: the worked result is 5. Why it works here Two triangles and three rectangles. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Net reasoning. Checkpoint for 9B: 5.
10A	Volume reverse problem	A cuboid has volume 240 cm³, length 10 cm and width 6 cm. Find height.	Show answer + formula + trick 4 cm. h = 240 ÷ (10 × 6) = 4. Formula / rule From V=LWH: missing dimension = V ÷ product of the other two dimensions. Applied to 10A: the worked result is 4 cm. Why it works here h = 240 ÷ (10 × 6) = 4. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 10A, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 10A: 4 cm.
10B	Volume reverse problem Similar practice	A cuboid has volume 360 cm³, length 12 cm and width 5 cm. Find height.	Show answer + formula + trick 6 cm. h = 360 ÷ (12 × 5) = 6. Formula / rule From V=LWH: missing dimension = V ÷ product of the other two dimensions. Applied to 10B: the worked result is 6 cm. Why it works here h = 360 ÷ (12 × 5) = 6. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 10B, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 10B: 6 cm.

13. Measurement and Units

#	Question type	Example	Worked solution
1A	Length conversion	Convert 3.45 km to metres.	Show answer + formula + trick 3,450 m. 1 km = 1,000 m. Formula / rule 1 km=1000 m; 1 m=100 cm=1000 mm. Applied to 1A: the worked result is 3,450 m. Why it works here 1 km = 1,000 m. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 1A, re-check the exact numbers in this row before entering the answer. Remember Convert before calculating. Checkpoint for 1A: 3,450 m.
1B	Length conversion Similar practice	Convert 5.08 km to metres.	Show answer + formula + trick 5,080 m. 1 km = 1,000 m. Formula / rule 1 km=1000 m; 1 m=100 cm=1000 mm. Applied to 1B: the worked result is 5,080 m. Why it works here 1 km = 1,000 m. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 1B, re-check the exact numbers in this row before entering the answer. Remember Convert before calculating. Checkpoint for 1B: 5,080 m.
2A	Mass conversion	Convert 2.75 kg to grams.	Show answer + formula + trick 2,750 g. 1 kg = 1,000 g. Formula / rule 1 kg=1000 g. Applied to 2A: the worked result is 2,750 g. Why it works here 1 kg = 1,000 g. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 2A, re-check the exact numbers in this row before entering the answer. Remember Convert before calculating. Checkpoint for 2A: 2,750 g.
2B	Mass conversion Similar practice	Convert 3.6 kg to grams.	Show answer + formula + trick 3,600 g. 1 kg = 1,000 g. Formula / rule 1 kg=1000 g. Applied to 2B: the worked result is 3,600 g. Why it works here 1 kg = 1,000 g. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 2B, re-check the exact numbers in this row before entering the answer. Remember Convert before calculating. Checkpoint for 2B: 3,600 g.
3A	Capacity conversion	Convert 4.2 litres to millilitres.	Show answer + formula + trick 4,200 ml. 1 litre = 1,000 ml. Formula / rule 1 litre=1000 ml. Applied to 3A: the worked result is 4,200 ml. Why it works here 1 litre = 1,000 ml. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 3A, re-check the exact numbers in this row before entering the answer. Remember Convert before calculating. Checkpoint for 3A: 4,200 ml.
3B	Capacity conversion Similar practice	Convert 2.35 litres to millilitres.	Show answer + formula + trick 2,350 ml. 1 litre = 1,000 ml. Formula / rule 1 litre=1000 ml. Applied to 3B: the worked result is 2,350 ml. Why it works here 1 litre = 1,000 ml. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 3B, re-check the exact numbers in this row before entering the answer. Remember Convert before calculating. Checkpoint for 3B: 2,350 ml.
4A	Time conversion	Convert 2 hours 35 minutes to minutes.	Show answer + formula + trick 155 minutes. 2 hours = 120 minutes; 120 + 35 = 155. Formula / rule 1 hour=60 minutes; 1 minute=60 seconds. Applied to 4A: the worked result is 155 minutes. Why it works here 2 hours = 120 minutes; 120 + 35 = 155. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 4A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 4A: 155 minutes.
4B	Time conversion Similar practice	Convert 3 hours 42 minutes to minutes.	Show answer + formula + trick 222 minutes. 3 hours = 180 minutes; 180 + 42 = 222. Formula / rule 1 hour=60 minutes; 1 minute=60 seconds. Applied to 4B: the worked result is 222 minutes. Why it works here 3 hours = 180 minutes; 180 + 42 = 222. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 4B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 4B: 222 minutes.
5A	24-hour time	Write 7:45 pm in 24-hour time.	Show answer + formula + trick 19:45. For pm times after noon, add 12 to the hour. Formula / rule For pm times after 12 noon add 12 to the hour; 12 am becomes 00 and 12 pm remains 12. Applied to 5A: the worked result is 19:45. Why it works here For pm times after noon, add 12 to the hour. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 5A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 5A: 19:45.
5B	24-hour time Similar practice	Write 6:20 am in 24-hour time.	Show answer + formula + trick 06:20. For am times before noon, keep the hour and use two digits. Formula / rule For pm times after 12 noon add 12 to the hour; 12 am becomes 00 and 12 pm remains 12. Applied to 5B: the worked result is 06:20. Why it works here For am times before noon, keep the hour and use two digits. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 5B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 5B: 06:20.
6A	Duration	A train leaves at 13:48 and arrives at 16:17. Journey time?	Show answer + formula + trick 2 h 29 min. 13:48 to 14:00 = 12 min, to 16:00 = 2 h, then 17 min. Total 2 h 29 min. Formula / rule Elapsed time = finish time − start time, often easiest by bridging through whole hours. Applied to 6A: the worked result is 2 h 29 min. Why it works here 13:48 to 14:00 = 12 min, to 16:00 = 2 h, then 17 min. Total 2 h 29 min. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 6A: 2 h 29 min.
6B	Duration Similar practice	A bus leaves at 09:37 and arrives at 12:05. Journey time?	Show answer + formula + trick 2 h 28 min. 09:37 to 10:00 = 23 min, to 12:00 = 2 h, then 5 min. Total 2 h 28 min. Formula / rule Elapsed time = finish time − start time, often easiest by bridging through whole hours. Applied to 6B: the worked result is 2 h 28 min. Why it works here 09:37 to 10:00 = 23 min, to 12:00 = 2 h, then 5 min. Total 2 h 28 min. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 6B: 2 h 28 min.
7A	Money	Three books cost £7.85 each. You pay £25. Change?	Show answer + formula + trick £1.45. 3 × £7.85 = £23.55; £25 - £23.55 = £1.45. Formula / rule Total cost = quantity × unit price; change = amount paid − total cost. Applied to 7A: the worked result is £1.45. Why it works here 3 × £7.85 = £23.55; £25 - £23.55 = £1.45. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Money. Checkpoint for 7A: £1.45.
7B	Money Similar practice	Four tickets cost £6.75 each. You pay £30. Change?	Show answer + formula + trick £3.00. 4 × £6.75 = £27.00; £30 - £27 = £3. Formula / rule Total cost = quantity × unit price; change = amount paid − total cost. Applied to 7B: the worked result is £3.00. Why it works here 4 × £6.75 = £27.00; £30 - £27 = £3. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Money. Checkpoint for 7B: £3.00.
8A	Reading scales	A scale has 5 equal intervals between 2 kg and 3 kg. What is each interval?	Show answer + formula + trick 0.2 kg. Difference is 1 kg. 1 ÷ 5 = 0.2 kg. Formula / rule Value per interval = labelled range ÷ number of equal intervals. Applied to 8A: the worked result is 0.2 kg. Why it works here Difference is 1 kg. 1 ÷ 5 = 0.2 kg. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Reading scales. Checkpoint for 8A: 0.2 kg.
8B	Reading scales Similar practice	A scale has 8 equal intervals between 4 kg and 6 kg. What is each interval?	Show answer + formula + trick 0.25 kg. Difference is 2 kg. 2 ÷ 8 = 0.25 kg. Formula / rule Value per interval = labelled range ÷ number of equal intervals. Applied to 8B: the worked result is 0.25 kg. Why it works here Difference is 2 kg. 2 ÷ 8 = 0.25 kg. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Reading scales. Checkpoint for 8B: 0.25 kg.
9A	Area units	Convert 3.2 m² to cm².	Show answer + formula + trick 32,000 cm². 1 m² = 10,000 cm², so 3.2 × 10,000 = 32,000. Formula / rule 1 m²=10,000 cm² because 1 m=100 cm and area conversion is squared. Applied to 9A: the worked result is 32,000 cm². Why it works here 1 m² = 10,000 cm², so 3.2 × 10,000 = 32,000. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 9A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 9A: 32,000 cm².
9B	Area units Similar practice	Convert 1.7 m² to cm².	Show answer + formula + trick 17,000 cm². 1 m² = 10,000 cm², so 1.7 × 10,000 = 17,000. Formula / rule 1 m²=10,000 cm² because 1 m=100 cm and area conversion is squared. Applied to 9B: the worked result is 17,000 cm². Why it works here 1 m² = 10,000 cm², so 1.7 × 10,000 = 17,000. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 9B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 9B: 17,000 cm².
10A	Speed	A car travels 150 km in 2.5 hours. Average speed?	Show answer + formula + trick 60 km/h. Speed = distance ÷ time = 150 ÷ 2.5 = 60. Formula / rule Speed = distance ÷ time. Applied to 10A: the worked result is 60 km/h. Why it works here Speed = distance ÷ time = 150 ÷ 2.5 = 60. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 10A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 10A: 60 km/h.
10B	Speed Similar practice	A cyclist travels 84 km in 3.5 hours. Average speed?	Show answer + formula + trick 24 km/h. Speed = distance ÷ time = 84 ÷ 3.5 = 24. Formula / rule Speed = distance ÷ time. Applied to 10B: the worked result is 24 km/h. Why it works here Speed = distance ÷ time = 84 ÷ 3.5 = 24. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 10B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 10B: 24 km/h.

14. Data Handling

#	Question type	Example	Worked solution
1A	Read a table	A table shows 18, 24, 17 and 31 visitors over four days. Total?	Show answer + formula + trick 90. 18 + 24 + 17 + 31 = 90. Formula / rule Add the relevant table entries once each. Applied to 1A: the worked result is 90. Why it works here 18 + 24 + 17 + 31 = 90. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Read a table. Checkpoint for 1A: 90.
1B	Read a table Similar practice	A table shows 26, 19, 34 and 21 visitors over four days. Total?	Show answer + formula + trick 100. 26 + 19 + 34 + 21 = 100. Formula / rule Add the relevant table entries once each. Applied to 1B: the worked result is 100. Why it works here 26 + 19 + 34 + 21 = 100. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Read a table. Checkpoint for 1B: 100.
2A	Difference from bar chart	One bar shows 48 and another 35. What is the difference?	Show answer + formula + trick 13. 48 - 35 = 13. Formula / rule Change/difference = larger value − smaller value. Applied to 2A: the worked result is 13. Why it works here 48 - 35 = 13. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Difference from bar chart. Checkpoint for 2A: 13.
2B	Difference from bar chart Similar practice	One bar shows 63 and another 47. What is the difference?	Show answer + formula + trick 16. 63 - 47 = 16. Formula / rule Change/difference = larger value − smaller value. Applied to 2B: the worked result is 16. Why it works here 63 - 47 = 16. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Difference from bar chart. Checkpoint for 2B: 16.
3A	Line graph change	A value rises from 22 to 37. By how much?	Show answer + formula + trick 15. 37 - 22 = 15. Formula / rule Change/difference = larger value − smaller value. Applied to 3A: the worked result is 15. Why it works here 37 - 22 = 15. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Line graph change. Checkpoint for 3A: 15.
3B	Line graph change Similar practice	A value falls from 54 to 38. By how much?	Show answer + formula + trick 16. 54 - 38 = 16. Formula / rule Change/difference = larger value − smaller value. Applied to 3B: the worked result is 16. Why it works here 54 - 38 = 16. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Line graph change. Checkpoint for 3B: 16.
4A	Pie chart angle	A category is 25% of a pie chart. What angle represents it?	Show answer + formula + trick 90°. 25% of 360° = 90°. Formula / rule Sector angle = part/whole ×360°. Applied to 4A: the worked result is 90°. Why it works here 25% of 360° = 90°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 4A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 4A: 90°.
4B	Pie chart angle Similar practice	A category is 35% of a pie chart. What angle represents it?	Show answer + formula + trick 126°. 35% of 360° = 0.35 × 360 = 126°. Formula / rule Sector angle = part/whole ×360°. Applied to 4B: the worked result is 126°. Why it works here 35% of 360° = 0.35 × 360 = 126°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 4B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 4B: 126°.
5A	Pie chart from angle	A sector is 72°. What fraction of the whole is this?	Show answer + formula + trick 1/5. 72/360 = 1/5. Formula / rule Fraction of whole = sector angle ÷360°, then simplify. Applied to 5A: the worked result is 1/5. Why it works here 72/360 = 1/5. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 5A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 5A: 1/5.
5B	Pie chart from angle Similar practice	A sector is 45°. What fraction of the whole is this?	Show answer + formula + trick 1/8. 45/360 = 1/8. Formula / rule Fraction of whole = sector angle ÷360°, then simplify. Applied to 5B: the worked result is 1/8. Why it works here 45/360 = 1/8. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 5B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 5B: 1/8.
6A	Pictogram	One symbol represents 8 pupils. What do 3.5 symbols represent?	Show answer + formula + trick 28 pupils. 3.5 × 8 = 28. Formula / rule Quantity = number of symbols × value represented by one symbol. Applied to 6A: the worked result is 28 pupils. Why it works here 3.5 × 8 = 28. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Pictogram. Checkpoint for 6A: 28 pupils.
6B	Pictogram Similar practice	One symbol represents 6 pupils. What do 4.5 symbols represent?	Show answer + formula + trick 27 pupils. 4.5 × 6 = 27. Formula / rule Quantity = number of symbols × value represented by one symbol. Applied to 6B: the worked result is 27 pupils. Why it works here 4.5 × 6 = 27. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Pictogram. Checkpoint for 6B: 27 pupils.
7A	Missing table value	A total is 120. Three categories are 35, 28 and 41. Find the fourth.	Show answer + formula + trick 16. 120 - (35 + 28 + 41) = 16. Formula / rule Missing amount = required total − sum of known amounts. Applied to 7A: the worked result is 16. Why it works here 120 - (35 + 28 + 41) = 16. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing table value. Checkpoint for 7A: 16.
7B	Missing table value Similar practice	A total is 150. Three categories are 42, 37 and 51. Find the fourth.	Show answer + formula + trick 20. 150 - (42 + 37 + 51) = 20. Formula / rule Missing amount = required total − sum of known amounts. Applied to 7B: the worked result is 20. Why it works here 150 - (42 + 37 + 51) = 20. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing table value. Checkpoint for 7B: 20.
8A	Two-way total	There are 18 girls and 14 boys in Club A, and 11 girls and 17 boys in Club B. Total pupils?	Show answer + formula + trick 60. 18 + 14 + 11 + 17 = 60. Formula / rule Add the relevant table entries once each. Applied to 8A: the worked result is 60. Why it works here 18 + 14 + 11 + 17 = 60. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Two-way total. Checkpoint for 8A: 60.
8B	Two-way total Similar practice	There are 22 girls and 16 boys in Club A, and 13 girls and 19 boys in Club B. Total pupils?	Show answer + formula + trick 70. 22 + 16 + 13 + 19 = 70. Formula / rule Add the relevant table entries once each. Applied to 8B: the worked result is 70. Why it works here 22 + 16 + 13 + 19 = 70. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Two-way total. Checkpoint for 8B: 70.
9A	Compare proportions	Class A: 12 of 30 choose chess. Class B: 15 of 50. Which has the higher proportion?	Show answer + formula + trick Class A. A = 40%; B = 30%. Formula / rule Convert each part/whole comparison to the same fraction, decimal or percentage. Applied to 9A: the worked result is Class A. Why it works here A = 40%; B = 30%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 9A: Class A.
9B	Compare proportions Similar practice	Class A: 18 of 45 choose chess. Class B: 14 of 35. Which has the higher proportion?	Show answer + formula + trick They are equal. 18/45 = 40%; 14/35 = 40%. Formula / rule Convert each part/whole comparison to the same fraction, decimal or percentage. Applied to 9B: the worked result is They are equal. Why it works here 18/45 = 40%; 14/35 = 40%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 9B: They are equal.
10A	Scale on graph	A graph axis marks 0, 20, 40 with 4 equal small intervals between labelled marks. Each small interval?	Show answer + formula + trick 5. 20 ÷ 4 = 5. Formula / rule Minor interval value = difference between labelled values ÷ number of equal intervals. Applied to 10A: the worked result is 5. Why it works here 20 ÷ 4 = 5. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Scale on graph. Checkpoint for 10A: 5.
10B	Scale on graph Similar practice	A graph axis marks 0, 30, 60 with 6 equal small intervals between labelled marks. Each small interval?	Show answer + formula + trick 5. 30 ÷ 6 = 5. Formula / rule Minor interval value = difference between labelled values ÷ number of equal intervals. Applied to 10B: the worked result is 5. Why it works here 30 ÷ 6 = 5. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Scale on graph. Checkpoint for 10B: 5.

15. Advanced Data Handling

#	Question type	Example	Worked solution
1A	Mean	Find the mean of 7, 9, 12, 16, 21.	Show answer + formula + trick 13. Total = 65; 65 ÷ 5 = 13. Formula / rule Mean = sum of values ÷ number of values. Applied to 1A: the worked result is 13. Why it works here Total = 65; 65 ÷ 5 = 13. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 1A, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 1A: 13.
1B	Mean Similar practice	Find the mean of 8, 10, 15, 17, 20.	Show answer + formula + trick 14. Total = 70; 70 ÷ 5 = 14. Formula / rule Mean = sum of values ÷ number of values. Applied to 1B: the worked result is 14. Why it works here Total = 70; 70 ÷ 5 = 14. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 1B, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 1B: 14.
2A	Median odd set	Find the median of 3, 11, 7, 8, 20.	Show answer + formula + trick 8. Order: 3, 7, 8, 11, 20. Middle is 8. Formula / rule Order the data; median is the middle value, or the mean of the two middle values for an even count. Applied to 2A: the worked result is 8. Why it works here Order: 3, 7, 8, 11, 20. Middle is 8. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 2A, re-check the exact numbers in this row before entering the answer. Remember Order the data first. Checkpoint for 2A: 8.
2B	Median odd set Similar practice	Find the median of 14, 5, 9, 21, 11.	Show answer + formula + trick 11. Order: 5, 9, 11, 14, 21. Middle is 11. Formula / rule Order the data; median is the middle value, or the mean of the two middle values for an even count. Applied to 2B: the worked result is 11. Why it works here Order: 5, 9, 11, 14, 21. Middle is 11. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 2B, re-check the exact numbers in this row before entering the answer. Remember Order the data first. Checkpoint for 2B: 11.
3A	Median even set	Find the median of 4, 7, 10, 15, 21, 30.	Show answer + formula + trick 12.5. Middle pair is 10 and 15. Mean = 12.5. Formula / rule Order the data; median is the middle value, or the mean of the two middle values for an even count. Applied to 3A: the worked result is 12.5. Why it works here Middle pair is 10 and 15. Mean = 12.5. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 3A, re-check the exact numbers in this row before entering the answer. Remember Order the data first. Checkpoint for 3A: 12.5.
3B	Median even set Similar practice	Find the median of 6, 8, 13, 17, 22, 28.	Show answer + formula + trick 15. Middle pair is 13 and 17. Mean = 15. Formula / rule Order the data; median is the middle value, or the mean of the two middle values for an even count. Applied to 3B: the worked result is 15. Why it works here Middle pair is 13 and 17. Mean = 15. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 3B, re-check the exact numbers in this row before entering the answer. Remember Order the data first. Checkpoint for 3B: 15.
4A	Mode	Find the mode: 4, 6, 6, 7, 8, 6, 9.	Show answer + formula + trick 6. 6 occurs most often. Formula / rule Mode/modal class = value or class with the greatest frequency. Applied to 4A: the worked result is 6. Why it works here 6 occurs most often. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 4A, re-check the exact numbers in this row before entering the answer. Remember Mode means most frequent. Checkpoint for 4A: 6.
4B	Mode Similar practice	Find the mode: 3, 5, 7, 5, 8, 5, 9.	Show answer + formula + trick 5. 5 occurs most often. Formula / rule Mode/modal class = value or class with the greatest frequency. Applied to 4B: the worked result is 5. Why it works here 5 occurs most often. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 4B, re-check the exact numbers in this row before entering the answer. Remember Mode means most frequent. Checkpoint for 4B: 5.
5A	Range	Find the range of 18, 4, 27, 11, 9.	Show answer + formula + trick 23. 27 - 4 = 23. Formula / rule Range = maximum − minimum. Applied to 5A: the worked result is 23. Why it works here 27 - 4 = 23. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 5A, re-check the exact numbers in this row before entering the answer. Remember Range = max − min. Checkpoint for 5A: 23.
5B	Range Similar practice	Find the range of 31, 12, 45, 28, 19.	Show answer + formula + trick 33. 45 - 12 = 33. Formula / rule Range = maximum − minimum. Applied to 5B: the worked result is 33. Why it works here 45 - 12 = 33. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 5B, re-check the exact numbers in this row before entering the answer. Remember Range = max − min. Checkpoint for 5B: 33.
6A	Missing value from mean	The mean of 5 numbers is 12. Four are 9, 11, 13, 15. Find the fifth.	Show answer + formula + trick 12. Required total = 5 × 12 = 60. Known total = 48. Missing = 12. Formula / rule Required total = mean×count; missing value = required total − known total. Applied to 6A: the worked result is 12. Why it works here Required total = 5 × 12 = 60. Known total = 48. Missing = 12. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 6A, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 6A: 12.
6B	Missing value from mean Similar practice	The mean of 6 numbers is 15. Five are 10, 12, 14, 18, 20. Find the sixth.	Show answer + formula + trick 16. Required total = 6 × 15 = 90. Known total = 74. Missing = 16. Formula / rule Required total = mean×count; missing value = required total − known total. Applied to 6B: the worked result is 16. Why it works here Required total = 6 × 15 = 90. Known total = 74. Missing = 16. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 6B, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 6B: 16.
7A	Frequency mean	Scores 1, 2, 3 occur with frequencies 2, 3, 5. Find the mean.	Show answer + formula + trick 2.3. Weighted total = 1×2 + 2×3 + 3×5 = 23; total frequency = 10; mean = 2.3. Formula / rule Mean from frequency table = Σ(value×frequency) ÷ Σfrequency. Applied to 7A: the worked result is 2.3. Why it works here Weighted total = 1×2 + 2×3 + 3×5 = 23; total frequency = 10; mean = 2.3. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 7A, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 7A: 2.3.
7B	Frequency mean Similar practice	Scores 2, 3, 4 occur with frequencies 3, 2, 5. Find the mean.	Show answer + formula + trick 3.2. Weighted total = 2×3 + 3×2 + 4×5 = 32; total frequency = 10; mean = 3.2. Formula / rule Mean from frequency table = Σ(value×frequency) ÷ Σfrequency. Applied to 7B: the worked result is 3.2. Why it works here Weighted total = 2×3 + 3×2 + 4×5 = 32; total frequency = 10; mean = 3.2. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 7B, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 7B: 3.2.
8A	Simple probability	A bag has 3 red, 5 blue and 2 green counters. Probability of blue?	Show answer + formula + trick 1/2. 5 blue out of 10 total = 5/10 = 1/2. Formula / rule P(event)=favourable equally likely outcomes ÷ total equally likely outcomes. Applied to 8A: the worked result is 1/2. Why it works here 5 blue out of 10 total = 5/10 = 1/2. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 8A, re-check the exact numbers in this row before entering the answer. Remember Probability stays between 0 and 1. Checkpoint for 8A: 1/2.
8B	Simple probability Similar practice	A bag has 4 red, 3 blue and 5 green counters. Probability of red?	Show answer + formula + trick 1/3. 4 red out of 12 total = 4/12 = 1/3. Formula / rule P(event)=favourable equally likely outcomes ÷ total equally likely outcomes. Applied to 8B: the worked result is 1/3. Why it works here 4 red out of 12 total = 4/12 = 1/3. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 8B, re-check the exact numbers in this row before entering the answer. Remember Probability stays between 0 and 1. Checkpoint for 8B: 1/3.
9A	Scatter graph interpretation	As x increases, y generally decreases. What type of correlation?	Show answer + formula + trick Negative correlation. The variables move in opposite directions. Formula / rule Positive correlation rises together; negative correlation moves in opposite directions; no correlation has no clear trend. Applied to 9A: the worked result is Negative correlation. Why it works here The variables move in opposite directions. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Scatter graph interpretation. Checkpoint for 9A: Negative correlation.
9B	Scatter graph interpretation Similar practice	As x increases, y generally increases. What type of correlation?	Show answer + formula + trick Positive correlation. The variables tend to increase together. Formula / rule Positive correlation rises together; negative correlation moves in opposite directions; no correlation has no clear trend. Applied to 9B: the worked result is Positive correlation. Why it works here The variables tend to increase together. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Scatter graph interpretation. Checkpoint for 9B: Positive correlation.
10A	Frequency reasoning	The modal class in a grouped frequency table is the class with what property?	Show answer + formula + trick The greatest frequency. Mode means most frequent. Formula / rule Mode/modal class = value or class with the greatest frequency. Applied to 10A: the worked result is The greatest frequency. Why it works here Mode means most frequent. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Frequency reasoning. Checkpoint for 10A: The greatest frequency.
10B	Frequency reasoning Similar practice	Which class is the modal class: 0-10 frequency 4, 10-20 frequency 9, 20-30 frequency 6?	Show answer + formula + trick 10-20. It has the greatest frequency, 9. Formula / rule Mode/modal class = value or class with the greatest frequency. Applied to 10B: the worked result is 10-20. Why it works here It has the greatest frequency, 9. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Frequency reasoning. Checkpoint for 10B: 10-20.

16. Venn Diagrams and Probability

#	Question type	Example	Worked solution
1A	Intersection	18 pupils like Maths, 12 like Science and 5 like both. How many like Maths only?	Show answer + formula + trick 13. 18 - 5 = 13. Formula / rule Set only = set total − overlap; overlap belongs to both sets. Applied to 1A: the worked result is 13. Why it works here 18 - 5 = 13. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Intersection. Checkpoint for 1A: 13.
1B	Intersection Similar practice	24 pupils like Art, 15 like Music and 7 like both. How many like Art only?	Show answer + formula + trick 17. 24 - 7 = 17. Formula / rule Set only = set total − overlap; overlap belongs to both sets. Applied to 1B: the worked result is 17. Why it works here 24 - 7 = 17. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Intersection. Checkpoint for 1B: 17.
2A	Union	Using the same data, how many like Maths or Science?	Show answer + formula + trick 25. 18 + 12 - 5 = 25. Formula / rule n(A∪B)=n(A)+n(B)−n(A∩B). Applied to 2A: the worked result is 25. Why it works here 18 + 12 - 5 = 25. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Union. Checkpoint for 2A: 25.
2B	Union Similar practice	Using the same data, how many like Art or Music?	Show answer + formula + trick 32. 24 + 15 - 7 = 32. Formula / rule n(A∪B)=n(A)+n(B)−n(A∩B). Applied to 2B: the worked result is 32. Why it works here 24 + 15 - 7 = 32. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Union. Checkpoint for 2B: 32.
3A	Neither	In a class of 30, 25 like Maths or Science. How many like neither?	Show answer + formula + trick 5. 30 - 25 = 5. Formula / rule Neither = universal total − number in the union. Applied to 3A: the worked result is 5. Why it works here 30 - 25 = 5. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Neither. Checkpoint for 3A: 5.
3B	Neither Similar practice	In a class of 36, 32 like Art or Music. How many like neither?	Show answer + formula + trick 4. 36 - 32 = 4. Formula / rule Neither = universal total − number in the union. Applied to 3B: the worked result is 4. Why it works here 36 - 32 = 4. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Neither. Checkpoint for 3B: 4.
4A	Set total	Maths only 9, Science only 7, both 6, neither 3. Class size?	Show answer + formula + trick 25. 9 + 7 + 6 + 3 = 25. Formula / rule Universal total = A-only + B-only + overlap + neither. Applied to 4A: the worked result is 25. Why it works here 9 + 7 + 6 + 3 = 25. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Set total. Checkpoint for 4A: 25.
4B	Set total Similar practice	Art only 11, Music only 8, both 5, neither 4. Class size?	Show answer + formula + trick 28. 11 + 8 + 5 + 4 = 28. Formula / rule Universal total = A-only + B-only + overlap + neither. Applied to 4B: the worked result is 28. Why it works here 11 + 8 + 5 + 4 = 28. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Set total. Checkpoint for 4B: 28.
5A	Probability complement	Probability of rain is 0.35. Probability of no rain?	Show answer + formula + trick 0.65. Probabilities of complementary events sum to 1. Formula / rule P(not A)=1−P(A). Applied to 5A: the worked result is 0.65. Why it works here Probabilities of complementary events sum to 1. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 5A, re-check the exact numbers in this row before entering the answer. Remember Probability stays between 0 and 1. Checkpoint for 5A: 0.65.
5B	Probability complement Similar practice	Probability of winning is 0.28. Probability of not winning?	Show answer + formula + trick 0.72. Complementary probabilities sum to 1. Formula / rule P(not A)=1−P(A). Applied to 5B: the worked result is 0.72. Why it works here Complementary probabilities sum to 1. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 5B, re-check the exact numbers in this row before entering the answer. Remember Probability stays between 0 and 1. Checkpoint for 5B: 0.72.
6A	Probability scale	What does probability 0 mean?	Show answer + formula + trick Impossible. 0 = impossible, 1 = certain. Formula / rule Probability ranges from 0 (impossible) to 1 (certain). Applied to 6A: the worked result is Impossible. Why it works here 0 = impossible, 1 = certain. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 6A, re-check the exact numbers in this row before entering the answer. Remember Probability stays between 0 and 1. Checkpoint for 6A: Impossible.
6B	Probability scale Similar practice	What does probability 1 mean?	Show answer + formula + trick Certain. 1 = certain, 0 = impossible. Formula / rule Probability ranges from 0 (impossible) to 1 (certain). Applied to 6B: the worked result is Certain. Why it works here 1 = certain, 0 = impossible. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 6B, re-check the exact numbers in this row before entering the answer. Remember Probability stays between 0 and 1. Checkpoint for 6B: Certain.
7A	Two independent events	A fair coin is tossed twice. Probability of two heads?	Show answer + formula + trick 1/4. 1/2 × 1/2 = 1/4. Formula / rule For independent successive events, multiply the probabilities. Applied to 7A: the worked result is 1/4. Why it works here 1/2 × 1/2 = 1/4. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Two independent events. Checkpoint for 7A: 1/4.
7B	Two independent events Similar practice	A fair coin is tossed three times. Probability of three heads?	Show answer + formula + trick 1/8. 1/2 × 1/2 × 1/2 = 1/8. Formula / rule For independent successive events, multiply the probabilities. Applied to 7B: the worked result is 1/8. Why it works here 1/2 × 1/2 × 1/2 = 1/8. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Two independent events. Checkpoint for 7B: 1/8.
8A	Dice compound event	A fair die is rolled. Probability of an even number greater than 3?	Show answer + formula + trick 1/3. Favourable outcomes are 4 and 6: 2/6 = 1/3. Formula / rule P(event)=favourable equally likely outcomes ÷ total equally likely outcomes. Applied to 8A: the worked result is 1/3. Why it works here Favourable outcomes are 4 and 6: 2/6 = 1/3. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Dice compound event. Checkpoint for 8A: 1/3.
8B	Dice compound event Similar practice	A fair die is rolled. Probability of a prime number greater than 2?	Show answer + formula + trick 1/3. Favourable outcomes are 3 and 5: 2/6 = 1/3. Formula / rule P(event)=favourable equally likely outcomes ÷ total equally likely outcomes. Applied to 8B: the worked result is 1/3. Why it works here Favourable outcomes are 3 and 5: 2/6 = 1/3. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Dice compound event. Checkpoint for 8B: 1/3.
9A	Without replacement	A bag has 3 red and 2 blue counters. Two are drawn without replacement. Probability both are red?	Show answer + formula + trick 3/10. 3/5 × 2/4 = 6/20 = 3/10. Formula / rule Multiply successive probabilities, reducing the total after the first draw because there is no replacement. Applied to 9A: the worked result is 3/10. Why it works here 3/5 × 2/4 = 6/20 = 3/10. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Without replacement. Checkpoint for 9A: 3/10.
9B	Without replacement Similar practice	A bag has 4 red and 3 blue counters. Two are drawn without replacement. Probability both are red?	Show answer + formula + trick 2/7. 4/7 × 3/6 = 12/42 = 2/7. Formula / rule Multiply successive probabilities, reducing the total after the first draw because there is no replacement. Applied to 9B: the worked result is 2/7. Why it works here 4/7 × 3/6 = 12/42 = 2/7. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Without replacement. Checkpoint for 9B: 2/7.
10A	At least one	Two fair coins are tossed. Probability of at least one head?	Show answer + formula + trick 3/4. Use complement: only no head is TT = 1/4. So 1 - 1/4 = 3/4. Formula / rule P(at least one)=1−P(none), when the complement is easier. Applied to 10A: the worked result is 3/4. Why it works here Use complement: only no head is TT = 1/4. So 1 - 1/4 = 3/4. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: At least one. Checkpoint for 10A: 3/4.
10B	At least one Similar practice	Two fair coins are tossed. Probability of at least one tail?	Show answer + formula + trick 3/4. Use complement: only no tail is HH = 1/4. So 1 - 1/4 = 3/4. Formula / rule P(at least one)=1−P(none), when the complement is easier. Applied to 10B: the worked result is 3/4. Why it works here Use complement: only no tail is HH = 1/4. So 1 - 1/4 = 3/4. Tip / trick Count the sample space/set regions carefully before calculating. Use the complement when 'at least one' is awkward. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: At least one. Checkpoint for 10B: 3/4.

17. Word Problems and Applied Maths

#	Question type	Example	Worked solution
1A	Shopping	4 pens at £1.35 and 3 notebooks at £2.40. Total cost?	Show answer + formula + trick £12.60. Pens £5.40, notebooks £7.20, total £12.60. Formula / rule Total cost = sum of item costs; change = amount paid − total cost. Applied to 1A: the worked result is £12.60. Why it works here Pens £5.40, notebooks £7.20, total £12.60. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Shopping. Checkpoint for 1A: £12.60.
1B	Shopping Similar practice	5 pens at £1.25 and 2 notebooks at £3.40. Total cost?	Show answer + formula + trick £13.05. Pens £6.25, notebooks £6.80, total £13.05. Formula / rule Total cost = sum of item costs; change = amount paid − total cost. Applied to 1B: the worked result is £13.05. Why it works here Pens £6.25, notebooks £6.80, total £13.05. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Shopping. Checkpoint for 1B: £13.05.
2A	Travel	A train covers 168 miles in 3 hours. Average speed?	Show answer + formula + trick 56 mph. 168 ÷ 3 = 56. Formula / rule Speed = distance ÷ time. Applied to 2A: the worked result is 56 mph. Why it works here 168 ÷ 3 = 56. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Travel. Checkpoint for 2A: 56 mph.
2B	Travel Similar practice	A car covers 210 miles in 3.5 hours. Average speed?	Show answer + formula + trick 60 mph. 210 ÷ 3.5 = 60. Formula / rule Speed = distance ÷ time. Applied to 2B: the worked result is 60 mph. Why it works here 210 ÷ 3.5 = 60. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Travel. Checkpoint for 2B: 60 mph.
3A	Consecutive numbers	Two consecutive numbers sum to 71. Find them.	Show answer + formula + trick 35 and 36. Half of 71 is 35.5, so the consecutive integers are 35 and 36. Formula / rule Consecutive integers differ by 1; for an odd count, their average is the middle integer. Applied to 3A: the worked result is 35 and 36. Why it works here Half of 71 is 35.5, so the consecutive integers are 35 and 36. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Consecutive numbers. Checkpoint for 3A: 35 and 36.
3B	Consecutive numbers Similar practice	Two consecutive numbers sum to 93. Find them.	Show answer + formula + trick 46 and 47. Half of 93 is 46.5, so the consecutive integers are 46 and 47. Formula / rule Consecutive integers differ by 1; for an odd count, their average is the middle integer. Applied to 3B: the worked result is 46 and 47. Why it works here Half of 93 is 46.5, so the consecutive integers are 46 and 47. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Consecutive numbers. Checkpoint for 3B: 46 and 47.
4A	Ages	A mother is 28 years older than her daughter. Together they are 54. Ages?	Show answer + formula + trick Daughter 13, mother 41. Let daughter = x. Then x + x + 28 = 54, so 2x = 26 and x = 13. Formula / rule For this Ages item, use the relationship demonstrated in the worked solution and verify it with an inverse operation or estimate. Applied to 4A: the worked result is Daughter 13, mother 41. Why it works here Let daughter = x. Then x + x + 28 = 54, so 2x = 26 and x = 13. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Ages. Checkpoint for 4A: Daughter 13, mother 41.
4B	Ages Similar practice	A father is 30 years older than his son. Together they are 58. Ages?	Show answer + formula + trick Son 14, father 44. Let son = x. Then x + x + 30 = 58, so 2x = 28 and x = 14. Formula / rule For this Ages item, use the relationship demonstrated in the worked solution and verify it with an inverse operation or estimate. Applied to 4B: the worked result is Son 14, father 44. Why it works here Let son = x. Then x + x + 30 = 58, so 2x = 28 and x = 14. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Ages. Checkpoint for 4B: Son 14, father 44.
5A	Area in context	A garden is 12 m by 9 m. A 3 m by 2 m shed covers part of it. Uncovered area?	Show answer + formula + trick 102 m². Garden = 108 m², shed = 6 m², difference = 102 m². Formula / rule Composite area = add non-overlapping parts or subtract cut-outs from a larger simple shape. Applied to 5A: the worked result is 102 m². Why it works here Garden = 108 m², shed = 6 m², difference = 102 m². Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 5A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 5A: 102 m².
5B	Area in context Similar practice	A patio is 15 m by 10 m. A 4 m by 3 m pond covers part of it. Uncovered area?	Show answer + formula + trick 138 m². Patio = 150 m², pond = 12 m², difference = 138 m². Formula / rule Composite area = add non-overlapping parts or subtract cut-outs from a larger simple shape. Applied to 5B: the worked result is 138 m². Why it works here Patio = 150 m², pond = 12 m², difference = 138 m². Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 5B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 5B: 138 m².
6A	Time	A film starts at 18:35 and lasts 1 h 48 min. Finish time?	Show answer + formula + trick 20:23. 18:35 + 1 h = 19:35, +48 min = 20:23. Formula / rule Time = total distance/amount ÷ speed/rate. Applied to 6A: the worked result is 20:23. Why it works here 18:35 + 1 h = 19:35, +48 min = 20:23. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 6A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 6A: 20:23.
6B	Time Similar practice	A concert starts at 19:20 and lasts 2 h 15 min. Finish time?	Show answer + formula + trick 21:35. 19:20 + 2 h = 21:20, +15 min = 21:35. Formula / rule Time = total distance/amount ÷ speed/rate. Applied to 6B: the worked result is 21:35. Why it works here 19:20 + 2 h = 21:20, +15 min = 21:35. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 6B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 6B: 21:35.
7A	Packing	487 sweets are packed equally into bags of 24. How many full bags and how many left?	Show answer + formula + trick 20 full bags, 7 left. 24 × 20 = 480, remainder 7. Formula / rule Whole groups = integer quotient; leftover = remainder. Applied to 7A: the worked result is 20 full bags, 7 left. Why it works here 24 × 20 = 480, remainder 7. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Packing. Checkpoint for 7A: 20 full bags, 7 left.
7B	Packing Similar practice	562 biscuits are packed equally into boxes of 25. How many full boxes and how many left?	Show answer + formula + trick 22 full boxes, 12 left. 25 × 22 = 550, remainder 12. Formula / rule Whole groups = integer quotient; leftover = remainder. Applied to 7B: the worked result is 22 full boxes, 12 left. Why it works here 25 × 22 = 550, remainder 12. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Packing. Checkpoint for 7B: 22 full boxes, 12 left.
8A	Mixture ratio	Orange:apple juice = 3:2. A jug contains 1.5 litres. How much orange juice?	Show answer + formula + trick 0.9 litres. 5 parts total. 1.5 ÷ 5 = 0.3 L per part. Orange = 3 × 0.3 = 0.9 L. Formula / rule One part = total mixture ÷ total ratio parts; required amount = one part × required parts. Applied to 8A: the worked result is 0.9 litres. Why it works here 5 parts total. 1.5 ÷ 5 = 0.3 L per part. Orange = 3 × 0.3 = 0.9 L. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 8A: 0.9 litres.
8B	Mixture ratio Similar practice	Red:yellow paint = 2:3. A tin contains 2 litres. How much yellow paint?	Show answer + formula + trick 1.2 litres. 5 parts total. 2 ÷ 5 = 0.4 L per part. Yellow = 3 × 0.4 = 1.2 L. Formula / rule One part = total mixture ÷ total ratio parts; required amount = one part × required parts. Applied to 8B: the worked result is 1.2 litres. Why it works here 5 parts total. 2 ÷ 5 = 0.4 L per part. Yellow = 3 × 0.4 = 1.2 L. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 8B: 1.2 litres.
9A	Logic	Three boxes contain 12, 18 and 30 balls. Balls are redistributed equally among the three boxes. How many per box?	Show answer + formula + trick 20. Total 60; 60 ÷ 3 = 20. Formula / rule Combine the given totals first, then apply the stated equal-sharing or logical condition. Applied to 9A: the worked result is 20. Why it works here Total 60; 60 ÷ 3 = 20. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Logic. Checkpoint for 9A: 20.
9B	Logic Similar practice	Four boxes contain 14, 18, 22 and 26 balls. Balls are redistributed equally among the four boxes. How many per box?	Show answer + formula + trick 20. Total 80; 80 ÷ 4 = 20. Formula / rule Combine the given totals first, then apply the stated equal-sharing or logical condition. Applied to 9B: the worked result is 20. Why it works here Total 80; 80 ÷ 4 = 20. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Logic. Checkpoint for 9B: 20.
10A	Multi-step percentage	A £240 bicycle is reduced by 15%, then £12 delivery is added. Final cost?	Show answer + formula + trick £216. 15% of £240 = £36. Sale price £204. Add £12 = £216. Formula / rule Apply the percentage change first, then perform the next stated charge/addition. Applied to 10A: the worked result is £216. Why it works here 15% of £240 = £36. Sale price £204. Add £12 = £216. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 10A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 10A: £216.
10B	Multi-step percentage Similar practice	A £360 tablet is reduced by 20%, then £15 delivery is added. Final cost?	Show answer + formula + trick £303. 20% of £360 = £72. Sale price £288. Add £15 = £303. Formula / rule Apply the percentage change first, then perform the next stated charge/addition. Applied to 10B: the worked result is £303. Why it works here 20% of £360 = £72. Sale price £288. Add £15 = £303. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 10B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 10B: £303.

18. Spatial Reasoning

#	Question type	Example	Worked solution
1A	Translation	Translate point (2, 3) by vector (4, -1).	Show answer + formula + trick (6, 2). Add 4 to x and subtract 1 from y. Formula / rule Translation by vector (a,b): (x,y)→(x+a,y+b). Applied to 1A: the worked result is (6, 2). Why it works here Add 4 to x and subtract 1 from y. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 1A, re-check the exact numbers in this row before entering the answer. Remember A translation slides without turning. Checkpoint for 1A: (6, 2).
1B	Translation Similar practice	Translate point (-3, 4) by vector (5, -2).	Show answer + formula + trick (2, 2). Add 5 to x and subtract 2 from y. Formula / rule Translation by vector (a,b): (x,y)→(x+a,y+b). Applied to 1B: the worked result is (2, 2). Why it works here Add 5 to x and subtract 2 from y. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 1B, re-check the exact numbers in this row before entering the answer. Remember A translation slides without turning. Checkpoint for 1B: (2, 2).
2A	Reflection in x-axis	Reflect (-5, 3) in the x-axis.	Show answer + formula + trick (-5, -3). Keep x, change the sign of y. Formula / rule Reflection in x-axis: (x,y)→(x,−y). Applied to 2A: the worked result is (-5, -3). Why it works here Keep x, change the sign of y. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 2A, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 2A: (-5, -3).
2B	Reflection in x-axis Similar practice	Reflect (6, -4) in the x-axis.	Show answer + formula + trick (6, 4). Keep x, change the sign of y. Formula / rule Reflection in x-axis: (x,y)→(x,−y). Applied to 2B: the worked result is (6, 4). Why it works here Keep x, change the sign of y. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 2B, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 2B: (6, 4).
3A	Reflection in y-axis	Reflect (4, -2) in the y-axis.	Show answer + formula + trick (-4, -2). Change the sign of x, keep y. Formula / rule Reflection in y-axis: (x,y)→(−x,y). Applied to 3A: the worked result is (-4, -2). Why it works here Change the sign of x, keep y. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 3A, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 3A: (-4, -2).
3B	Reflection in y-axis Similar practice	Reflect (-7, 3) in the y-axis.	Show answer + formula + trick (7, 3). Change the sign of x, keep y. Formula / rule Reflection in y-axis: (x,y)→(−x,y). Applied to 3B: the worked result is (7, 3). Why it works here Change the sign of x, keep y. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 3B, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 3B: (7, 3).
4A	Quarter-turn rotation	A point lies 3 units to the right of the centre. After a 90° anticlockwise turn, where is it relative to the centre?	Show answer + formula + trick 3 units above. A quarter-turn anticlockwise sends right to up. Formula / rule A quarter-turn is 90°; track clockwise/anticlockwise direction carefully. Applied to 4A: the worked result is 3 units above. Why it works here A quarter-turn anticlockwise sends right to up. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 4A, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 4A: 3 units above.
4B	Quarter-turn rotation Similar practice	A point lies 4 units above the centre. After a 90° clockwise turn, where is it relative to the centre?	Show answer + formula + trick 4 units to the right. A quarter-turn clockwise sends up to right. Formula / rule A quarter-turn is 90°; track clockwise/anticlockwise direction carefully. Applied to 4B: the worked result is 4 units to the right. Why it works here A quarter-turn clockwise sends up to right. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 4B, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 4B: 4 units to the right.
5A	Half-turn rotation	What happens to a point 2 right and 5 up after a 180° rotation about the centre?	Show answer + formula + trick 2 left and 5 down. A half-turn reverses both directions. Formula / rule A half-turn is 180° and reverses both directions relative to the centre. Applied to 5A: the worked result is 2 left and 5 down. Why it works here A half-turn reverses both directions. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 5A, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 5A: 2 left and 5 down.
5B	Half-turn rotation Similar practice	What happens to a point 3 left and 2 down after a 180° rotation about the centre?	Show answer + formula + trick 3 right and 2 up. A half-turn reverses both directions. Formula / rule A half-turn is 180° and reverses both directions relative to the centre. Applied to 5B: the worked result is 3 right and 2 up. Why it works here A half-turn reverses both directions. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 5B, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 5B: 3 right and 2 up.
6A	Tessellation	Can regular hexagons tessellate on their own?	Show answer + formula + trick Yes. Their 120° interior angles fit three around a point: 3 × 120° = 360°. Formula / rule Shapes tessellate around a point when their meeting angles can total exactly 360° without gaps/overlaps. Applied to 6A: the worked result is Yes. Why it works here Their 120° interior angles fit three around a point: 3 × 120° = 360°. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Tessellation. Checkpoint for 6A: Yes.
6B	Tessellation Similar practice	Can regular pentagons tessellate on their own?	Show answer + formula + trick No. Their interior angle is 108°, and 360 ÷ 108 is not a whole number. Formula / rule Shapes tessellate around a point when their meeting angles can total exactly 360° without gaps/overlaps. Applied to 6B: the worked result is No. Why it works here Their interior angle is 108°, and 360 ÷ 108 is not a whole number. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Tessellation. Checkpoint for 6B: No.
7A	Cube net	How many square faces must a cube net contain?	Show answer + formula + trick 6. One for each face of the cube. Formula / rule Use the defining faces/edges/vertices and imagine folding the net into the solid. Applied to 7A: the worked result is 6. Why it works here One for each face of the cube. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Cube net. Checkpoint for 7A: 6.
7B	Cube net Similar practice	How many edges does a cube have?	Show answer + formula + trick 12. A cube has 12 edges. Formula / rule Use the defining faces/edges/vertices and imagine folding the net into the solid. Applied to 7B: the worked result is 12. Why it works here A cube has 12 edges. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Cube net. Checkpoint for 7B: 12.
8A	Viewpoint	A stack has two cubes one behind the other at the same height. From the front, how many cubes may be visible?	Show answer + formula + trick 1. The rear cube can be hidden directly behind the front cube. Formula / rule In 3D views, cubes directly behind others can be hidden; count only what is visible from the stated direction. Applied to 8A: the worked result is 1. Why it works here The rear cube can be hidden directly behind the front cube. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Viewpoint. Checkpoint for 8A: 1.
8B	Viewpoint Similar practice	A stack has three cubes directly one behind another at the same height. From the front, how many cubes may be visible?	Show answer + formula + trick 1. The two rear cubes can be completely hidden behind the front cube. Formula / rule In 3D views, cubes directly behind others can be hidden; count only what is visible from the stated direction. Applied to 8B: the worked result is 1. Why it works here The two rear cubes can be completely hidden behind the front cube. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Viewpoint. Checkpoint for 8B: 1.
9A	Symmetry after reflection	A shape is reflected twice in the same mirror line. What is the final position?	Show answer + formula + trick The original position. Two identical reflections cancel each other. Formula / rule Two reflections in the same mirror line return a shape to its original position. Applied to 9A: the worked result is The original position. Why it works here Two identical reflections cancel each other. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 9A, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 9A: The original position.
9B	Symmetry after reflection Similar practice	A shape is reflected in the x-axis and then reflected in the x-axis again. What is the final position?	Show answer + formula + trick The original position. Two identical reflections cancel each other. Formula / rule Two reflections in the same mirror line return a shape to its original position. Applied to 9B: the worked result is The original position. Why it works here Two identical reflections cancel each other. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 9B, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 9B: The original position.
10A	Coordinate movement	Move (-2, 5) 7 units right and 3 units down.	Show answer + formula + trick (5, 2). x: -2 + 7 = 5; y: 5 - 3 = 2. Formula / rule Combine horizontal movement in x and vertical movement in y; opposite directions cancel. Applied to 10A: the worked result is (5, 2). Why it works here x: -2 + 7 = 5; y: 5 - 3 = 2. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Coordinate movement. Checkpoint for 10A: (5, 2).
10B	Coordinate movement Similar practice	Move (4, -1) 6 units left and 5 units up.	Show answer + formula + trick (-2, 4). x: 4 - 6 = -2; y: -1 + 5 = 4. Formula / rule Combine horizontal movement in x and vertical movement in y; opposite directions cancel. Applied to 10B: the worked result is (-2, 4). Why it works here x: 4 - 6 = -2; y: -1 + 5 = 4. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Coordinate movement. Checkpoint for 10B: (-2, 4).

19. Financial Mathematics

#	Question type	Example	Worked solution
1A	Profit	An item costs £36 and sells for £49. Profit?	Show answer + formula + trick £13. Selling price - cost price = £49 - £36. Formula / rule Profit = selling price − cost price. Applied to 1A: the worked result is £13. Why it works here Selling price - cost price = £49 - £36. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Profit. Checkpoint for 1A: £13.
1B	Profit Similar practice	An item costs £52 and sells for £68. Profit?	Show answer + formula + trick £16. Selling price - cost price = £68 - £52. Formula / rule Profit = selling price − cost price. Applied to 1B: the worked result is £16. Why it works here Selling price - cost price = £68 - £52. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Profit. Checkpoint for 1B: £16.
2A	Loss	An item costs £85 and sells for £72. Loss?	Show answer + formula + trick £13. Cost price - selling price = £13. Formula / rule Loss = cost price − selling price. Applied to 2A: the worked result is £13. Why it works here Cost price - selling price = £13. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Loss. Checkpoint for 2A: £13.
2B	Loss Similar practice	An item costs £96 and sells for £81. Loss?	Show answer + formula + trick £15. Cost price - selling price = £15. Formula / rule Loss = cost price − selling price. Applied to 2B: the worked result is £15. Why it works here Cost price - selling price = £15. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Loss. Checkpoint for 2B: £15.
3A	Percentage profit	A toy costs £40 and sells for £50. Profit percentage?	Show answer + formula + trick 25%. Profit = £10. £10 ÷ £40 = 25%. Formula / rule Profit % = profit ÷ cost price ×100%. Applied to 3A: the worked result is 25%. Why it works here Profit = £10. £10 ÷ £40 = 25%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 3A: 25%.
3B	Percentage profit Similar practice	A book costs £24 and sells for £30. Profit percentage?	Show answer + formula + trick 25%. Profit = £6. £6 ÷ £24 = 25%. Formula / rule Profit % = profit ÷ cost price ×100%. Applied to 3B: the worked result is 25%. Why it works here Profit = £6. £6 ÷ £24 = 25%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 3B: 25%.
4A	Simple interest	£500 earns 4% simple interest per year for 3 years. Interest earned?	Show answer + formula + trick £60. 4% of £500 = £20 per year. Over 3 years = £60. Formula / rule Simple interest = principal × annual rate × time. Applied to 4A: the worked result is £60. Why it works here 4% of £500 = £20 per year. Over 3 years = £60. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Simple interest. Checkpoint for 4A: £60.
4B	Simple interest Similar practice	£800 earns 3% simple interest per year for 4 years. Interest earned?	Show answer + formula + trick £96. 3% of £800 = £24 per year. Over 4 years = £96. Formula / rule Simple interest = principal × annual rate × time. Applied to 4B: the worked result is £96. Why it works here 3% of £800 = £24 per year. Over 4 years = £96. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Simple interest. Checkpoint for 4B: £96.
5A	Savings balance	You save £35 per month for 8 months, then spend £90. Balance?	Show answer + formula + trick £190. 35 × 8 = 280; 280 - 90 = 190. Formula / rule Balance = total saved/income − spending. Applied to 5A: the worked result is £190. Why it works here 35 × 8 = 280; 280 - 90 = 190. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Savings balance. Checkpoint for 5A: £190.
5B	Savings balance Similar practice	You save £42 per month for 6 months, then spend £75. Balance?	Show answer + formula + trick £177. 42 × 6 = 252; 252 - 75 = 177. Formula / rule Balance = total saved/income − spending. Applied to 5B: the worked result is £177. Why it works here 42 × 6 = 252; 252 - 75 = 177. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Savings balance. Checkpoint for 5B: £177.
6A	Budget	Weekly budget £60. Spending is £18.50, £12.75 and £21.40. Money left?	Show answer + formula + trick £7.35. Total spent £52.65; £60 - £52.65 = £7.35. Formula / rule Money left = budget − total spending. Applied to 6A: the worked result is £7.35. Why it works here Total spent £52.65; £60 - £52.65 = £7.35. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Budget. Checkpoint for 6A: £7.35.
6B	Budget Similar practice	Weekly budget £85. Spending is £24.60, £18.35 and £31.20. Money left?	Show answer + formula + trick £10.85. Total spent £74.15; £85 - £74.15 = £10.85. Formula / rule Money left = budget − total spending. Applied to 6B: the worked result is £10.85. Why it works here Total spent £74.15; £85 - £74.15 = £10.85. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Budget. Checkpoint for 6B: £10.85.
7A	Best buy	750 g cereal costs £3.00; 1 kg costs £3.80. Which is cheaper per kg?	Show answer + formula + trick 1 kg pack. 750 g pack costs £4.00 per kg. The 1 kg pack costs £3.80 per kg. Formula / rule Unit price = pack price ÷ pack quantity; compare using the same unit. Applied to 7A: the worked result is 1 kg pack. Why it works here 750 g pack costs £4.00 per kg. The 1 kg pack costs £3.80 per kg. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Best buy. Checkpoint for 7A: 1 kg pack.
7B	Best buy Similar practice	600 g cereal costs £2.70; 1 kg costs £4.20. Which is cheaper per kg?	Show answer + formula + trick 1 kg pack. 600 g pack costs £4.50 per kg. The 1 kg pack costs £4.20 per kg. Formula / rule Unit price = pack price ÷ pack quantity; compare using the same unit. Applied to 7B: the worked result is 1 kg pack. Why it works here 600 g pack costs £4.50 per kg. The 1 kg pack costs £4.20 per kg. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Best buy. Checkpoint for 7B: 1 kg pack.
8A	Discount then price	A £90 coat has 20% off. Sale price?	Show answer + formula + trick £72. 20% of £90 = £18; £90 - £18 = £72. Formula / rule Sale price = original×(1−discount rate); then apply any payment/change step. Applied to 8A: the worked result is £72. Why it works here 20% of £90 = £18; £90 - £18 = £72. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Discount then price. Checkpoint for 8A: £72.
8B	Discount then price Similar practice	A £125 jacket has 16% off. Sale price?	Show answer + formula + trick £105. 16% of £125 = £20; £125 - £20 = £105. Formula / rule Sale price = original×(1−discount rate); then apply any payment/change step. Applied to 8B: the worked result is £105. Why it works here 16% of £125 = £20; £125 - £20 = £105. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Discount then price. Checkpoint for 8B: £105.
9A	Bill total	Electricity £42.60, water £18.75, internet £27.99. Total?	Show answer + formula + trick £89.34. 42.60 + 18.75 + 27.99 = 89.34. Formula / rule Total bill = sum of all listed charges. Applied to 9A: the worked result is £89.34. Why it works here 42.60 + 18.75 + 27.99 = 89.34. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Bill total. Checkpoint for 9A: £89.34.
9B	Bill total Similar practice	Gas £35.80, water £22.45, internet £29.95. Total?	Show answer + formula + trick £88.20. 35.80 + 22.45 + 29.95 = 88.20. Formula / rule Total bill = sum of all listed charges. Applied to 9B: the worked result is £88.20. Why it works here 35.80 + 22.45 + 29.95 = 88.20. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Bill total. Checkpoint for 9B: £88.20.
10A	Change and check	You buy items costing £6.85, £4.70 and £3.95 and pay £20. Change?	Show answer + formula + trick £4.50. Total = £15.50. Change = £20 - £15.50 = £4.50. Formula / rule Total cost = sum of item costs; change = amount paid − total cost. Applied to 10A: the worked result is £4.50. Why it works here Total = £15.50. Change = £20 - £15.50 = £4.50. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Change and check. Checkpoint for 10A: £4.50.
10B	Change and check Similar practice	You buy items costing £8.65, £5.75 and £4.10 and pay £25. Change?	Show answer + formula + trick £6.50. Total = £18.50. Change = £25 - £18.50 = £6.50. Formula / rule Total cost = sum of item costs; change = amount paid − total cost. Applied to 10B: the worked result is £6.50. Why it works here Total = £18.50. Change = £25 - £18.50 = £6.50. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Change and check. Checkpoint for 10B: £6.50.

20. Tiffin Focus: Special Numbers & Divisibility

Tiffin-focused extension: these are additional high-value styles aligned with the school's published Stage One English/Maths format and with question categories reported by independent 11+ preparation sources. They are practice, not official past-paper questions.

#	Question type	Example	Worked solution
1A	Divisibility by 3 and 9	Which of 4,572 and 4,573 is divisible by 9?	Show answer + formula + trick 4,572. Digit sum 4+5+7+2=18, which is divisible by 9. The other digit sum is 19. Formula / rule Divisible by 3 or 9 when the digit sum is divisible by 3 or 9 respectively. Applied to 1A: the worked result is 4,572. Why it works here Digit sum 4+5+7+2=18, which is divisible by 9. The other digit sum is 19. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 1A, re-check the exact numbers in this row before entering the answer. Remember Use the divisibility test, not long division. Checkpoint for 1A: 4,572.
1B	Divisibility by 3 and 9 Similar practice	Which of 6,381 and 6,382 is divisible by 3?	Show answer + formula + trick 6,381. Digit sum 6+3+8+1=18, which is divisible by 3. Formula / rule Divisible by 3 or 9 when the digit sum is divisible by 3 or 9 respectively. Applied to 1B: the worked result is 6,381. Why it works here Digit sum 6+3+8+1=18, which is divisible by 3. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 1B, re-check the exact numbers in this row before entering the answer. Remember Use the divisibility test, not long division. Checkpoint for 1B: 6,381.
2A	Divisibility by 4	Is 7,316 divisible by 4?	Show answer + formula + trick Yes. Only the last two digits matter. 16 is divisible by 4. Formula / rule A number is divisible by 4 when its last two digits form a number divisible by 4. Applied to 2A: the worked result is Yes. Why it works here Only the last two digits matter. 16 is divisible by 4. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 2A, re-check the exact numbers in this row before entering the answer. Remember Use the divisibility test, not long division. Checkpoint for 2A: Yes.
2B	Divisibility by 4 Similar practice	Is 9,542 divisible by 4?	Show answer + formula + trick No. Only the last two digits matter. 42 is not divisible by 4. Formula / rule A number is divisible by 4 when its last two digits form a number divisible by 4. Applied to 2B: the worked result is No. Why it works here Only the last two digits matter. 42 is not divisible by 4. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 2B, re-check the exact numbers in this row before entering the answer. Remember Use the divisibility test, not long division. Checkpoint for 2B: No.
3A	Divisibility by 6	Is 2,154 divisible by 6?	Show answer + formula + trick Yes. It is even, so divisible by 2, and its digit sum is 12, so divisible by 3. Formula / rule A number is divisible by 6 only when it is divisible by both 2 and 3. Applied to 3A: the worked result is Yes. Why it works here It is even, so divisible by 2, and its digit sum is 12, so divisible by 3. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 3A, re-check the exact numbers in this row before entering the answer. Remember Use the divisibility test, not long division. Checkpoint for 3A: Yes.
3B	Divisibility by 6 Similar practice	Is 3,428 divisible by 6?	Show answer + formula + trick No. It is even, but its digit sum is 17, not divisible by 3. Formula / rule A number is divisible by 6 only when it is divisible by both 2 and 3. Applied to 3B: the worked result is No. Why it works here It is even, but its digit sum is 17, not divisible by 3. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 3B, re-check the exact numbers in this row before entering the answer. Remember Use the divisibility test, not long division. Checkpoint for 3B: No.
4A	Prime factor reasoning	Which is prime: 51, 53, 55, 57?	Show answer + formula + trick 53. 51 and 57 are divisible by 3, and 55 is divisible by 5. Formula / rule A prime number has exactly two positive factors: 1 and itself. Applied to 4A: the worked result is 53. Why it works here 51 and 57 are divisible by 3, and 55 is divisible by 5. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 4A, re-check the exact numbers in this row before entering the answer. Remember Factors divide exactly. Checkpoint for 4A: 53.
4B	Prime factor reasoning Similar practice	Which is prime: 61, 63, 65, 69?	Show answer + formula + trick 61. 63 and 69 are divisible by 3; 65 is divisible by 5. Formula / rule A prime number has exactly two positive factors: 1 and itself. Applied to 4B: the worked result is 61. Why it works here 63 and 69 are divisible by 3; 65 is divisible by 5. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 4B, re-check the exact numbers in this row before entering the answer. Remember Factors divide exactly. Checkpoint for 4B: 61.
5A	Common factor puzzle	A number is a factor of both 48 and 72 and is greater than 10. What is the greatest possible number?	Show answer + formula + trick 24. The HCF of 48 and 72 is 24. Formula / rule HCF = the greatest factor that divides every given number exactly. Applied to 5A: the worked result is 24. Why it works here The HCF of 48 and 72 is 24. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 5A, re-check the exact numbers in this row before entering the answer. Remember Factors divide exactly. Checkpoint for 5A: 24.
5B	Common factor puzzle Similar practice	A number is a factor of both 60 and 84 and is greater than 10. What is the greatest possible number?	Show answer + formula + trick 12. The HCF of 60 and 84 is 12. Formula / rule HCF = the greatest factor that divides every given number exactly. Applied to 5B: the worked result is 12. Why it works here The HCF of 60 and 84 is 12. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 5B, re-check the exact numbers in this row before entering the answer. Remember Factors divide exactly. Checkpoint for 5B: 12.
6A	Common multiple puzzle	What is the smallest number greater than 20 that is a multiple of both 6 and 8?	Show answer + formula + trick 24. The LCM of 6 and 8 is 24. Formula / rule LCM = the smallest positive multiple shared by all given numbers. Applied to 6A: the worked result is 24. Why it works here The LCM of 6 and 8 is 24. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 6A, re-check the exact numbers in this row before entering the answer. Remember Multiples come from times tables. Checkpoint for 6A: 24.
6B	Common multiple puzzle Similar practice	What is the smallest number greater than 30 that is a multiple of both 9 and 12?	Show answer + formula + trick 36. The LCM of 9 and 12 is 36. Formula / rule LCM = the smallest positive multiple shared by all given numbers. Applied to 6B: the worked result is 36. Why it works here The LCM of 9 and 12 is 36. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 6B, re-check the exact numbers in this row before entering the answer. Remember Multiples come from times tables. Checkpoint for 6B: 36.
7A	Square and cube recognition	Which number is both a square number and less than 50: 18, 25, 32, 45?	Show answer + formula + trick 25. 25=5². Formula / rule Square numbers are n²; cube numbers are n³. Applied to 7A: the worked result is 25. Why it works here 25=5². Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Square and cube recognition. Checkpoint for 7A: 25.
7B	Square and cube recognition Similar practice	Which number is a cube number: 24, 27, 30, 36?	Show answer + formula + trick 27. 27=3³. Formula / rule Square numbers are n²; cube numbers are n³. Applied to 7B: the worked result is 27. Why it works here 27=3³. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Square and cube recognition. Checkpoint for 7B: 27.
8A	Remainder pattern	What remainder is left when 1,003 is divided by 10?	Show answer + formula + trick 3. Division by 10 leaves the units digit as the remainder. Formula / rule Dividend = divisor × quotient + remainder, with 0 ≤ remainder < divisor. Applied to 8A: the worked result is 3. Why it works here Division by 10 leaves the units digit as the remainder. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 8A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 8A: 3.
8B	Remainder pattern Similar practice	What remainder is left when 2,047 is divided by 100?	Show answer + formula + trick 47. Division by 100 leaves the final two digits as the remainder. Formula / rule Dividend = divisor × quotient + remainder, with 0 ≤ remainder < divisor. Applied to 8B: the worked result is 47. Why it works here Division by 100 leaves the final two digits as the remainder. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 8B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 8B: 47.
9A	Consecutive integer properties	The sum of three consecutive integers is 75. What is the largest?	Show answer + formula + trick 26. The middle is 75÷3=25, so the integers are 24,25,26. Formula / rule Consecutive integers differ by 1; for an odd count, their average is the middle integer. Applied to 9A: the worked result is 26. Why it works here The middle is 75÷3=25, so the integers are 24,25,26. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Consecutive integer properties. Checkpoint for 9A: 26.
9B	Consecutive integer properties Similar practice	The sum of five consecutive integers is 110. What is the middle integer?	Show answer + formula + trick 22. The middle is the average: 110÷5=22. Formula / rule Consecutive integers differ by 1; for an odd count, their average is the middle integer. Applied to 9B: the worked result is 22. Why it works here The middle is the average: 110÷5=22. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Consecutive integer properties. Checkpoint for 9B: 22.
10A	Digit-sum puzzle	A two-digit number has digits that total 11. Its tens digit is 3 more than its units digit. What is the number?	Show answer + formula + trick 74. Let units=u, tens=u+3. Then 2u+3=11, so u=4, tens=7. Formula / rule A two-digit number with tens digit a and units digit b has value 10a+b. Applied to 10A: the worked result is 74. Why it works here Let units=u, tens=u+3. Then 2u+3=11, so u=4, tens=7. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Digit-sum puzzle. Checkpoint for 10A: 74.
10B	Digit-sum puzzle Similar practice	A two-digit number has digits that total 13. Its tens digit is 5 more than its units digit. What is the number?	Show answer + formula + trick 94. Let units=u, tens=u+5. Then 2u+5=13, so u=4, tens=9. Formula / rule A two-digit number with tens digit a and units digit b has value 10a+b. Applied to 10B: the worked result is 94. Why it works here Let units=u, tens=u+5. Then 2u+5=13, so u=4, tens=9. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Digit-sum puzzle. Checkpoint for 10B: 94.

21. Tiffin Focus: Missing Numbers, Digits & Arithmetic Puzzles

#	Question type	Example	Worked solution
1A	Missing digit in addition	Fill the digit: 4□7 + 268 = 735.	Show answer + formula + trick 6. 735-268=467, so the missing digit is 6. Formula / rule Missing addend = total − known addend. Applied to 1A: the worked result is 6. Why it works here 735-268=467, so the missing digit is 6. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing digit in addition. Checkpoint for 1A: 6.
1B	Missing digit in addition Similar practice	Fill the digit: 5□4 + 379 = 913.	Show answer + formula + trick 3. 913-379=534, so the missing digit is 3. Formula / rule Missing addend = total − known addend. Applied to 1B: the worked result is 3. Why it works here 913-379=534, so the missing digit is 3. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing digit in addition. Checkpoint for 1B: 3.
2A	Missing digit in subtraction	Fill the digit: 8□2 - 347 = 475.	Show answer + formula + trick 2. 475+347=822, so the missing digit is 2. Formula / rule For A−B=C, recover the missing starting number with A=B+C. Applied to 2A: the worked result is 2. Why it works here 475+347=822, so the missing digit is 2. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing digit in subtraction. Checkpoint for 2A: 2.
2B	Missing digit in subtraction Similar practice	Fill the digit: 9□5 - 468 = 457.	Show answer + formula + trick 2. 457+468=925, so the missing digit is 2. Formula / rule For A−B=C, recover the missing starting number with A=B+C. Applied to 2B: the worked result is 2. Why it works here 457+468=925, so the missing digit is 2. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing digit in subtraction. Checkpoint for 2B: 2.
3A	Missing digit in multiplication	Fill the digit: 3□ × 7 = 238.	Show answer + formula + trick 4. 238÷7=34. Formula / rule Missing factor = product ÷ known factor. Applied to 3A: the worked result is 4. Why it works here 238÷7=34. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing digit in multiplication. Checkpoint for 3A: 4.
3B	Missing digit in multiplication Similar practice	Fill the digit: 4□ × 6 = 288.	Show answer + formula + trick 8. 288÷6=48. Formula / rule Missing factor = product ÷ known factor. Applied to 3B: the worked result is 8. Why it works here 288÷6=48. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing digit in multiplication. Checkpoint for 3B: 8.
4A	Missing divisor	432 ÷ □ = 18. Find □.	Show answer + formula + trick 24. 432÷18=24. Formula / rule For A÷B=C, the missing divisor is B=A÷C. Applied to 4A: the worked result is 24. Why it works here 432÷18=24. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing divisor. Checkpoint for 4A: 24.
4B	Missing divisor Similar practice	588 ÷ □ = 21. Find □.	Show answer + formula + trick 28. 588÷21=28. Formula / rule For A÷B=C, the missing divisor is B=A÷C. Applied to 4B: the worked result is 28. Why it works here 588÷21=28. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing divisor. Checkpoint for 4B: 28.
5A	Digit rearrangement	Using digits 2, 5 and 8 once each, make the largest even number.	Show answer + formula + trick 852. An even number must end in 2 or 8. To maximise it, put 8 in hundreds and 2 in units: 852. Formula / rule Meet the digit-condition first (such as an even units digit), then order remaining digits to maximise or minimise. Applied to 5A: the worked result is 852. Why it works here An even number must end in 2 or 8. To maximise it, put 8 in hundreds and 2 in units: 852. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 5A, re-check the exact numbers in this row before entering the answer. Remember Range = max − min. Checkpoint for 5A: 852.
5B	Digit rearrangement Similar practice	Using digits 3, 6 and 9 once each, make the smallest even number.	Show answer + formula + trick 396. The units digit must be 6, then use the smallest remaining digit first. Formula / rule Meet the digit-condition first (such as an even units digit), then order remaining digits to maximise or minimise. Applied to 5B: the worked result is 396. Why it works here The units digit must be 6, then use the smallest remaining digit first. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 5B, re-check the exact numbers in this row before entering the answer. Remember Range = max − min. Checkpoint for 5B: 396.
6A	Reverse operation chain	I think of a number, multiply by 5, then add 7 to get 52. What was the number?	Show answer + formula + trick 9. Work backwards: 52-7=45, then 45÷5=9. Formula / rule Inverse pairs: + ↔ − and × ↔ ÷. Undo operations in reverse order. Applied to 6A: the worked result is 9. Why it works here Work backwards: 52-7=45, then 45÷5=9. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 6A: 9.
6B	Reverse operation chain Similar practice	I think of a number, divide by 4, then subtract 3 to get 8. What was the number?	Show answer + formula + trick 44. Work backwards: 8+3=11, then 11×4=44. Formula / rule Inverse pairs: + ↔ − and × ↔ ÷. Undo operations in reverse order. Applied to 6B: the worked result is 44. Why it works here Work backwards: 8+3=11, then 11×4=44. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 6B: 44.
7A	Balance puzzle	3 identical boxes weigh 27 kg altogether. What do 5 boxes weigh?	Show answer + formula + trick 45 kg. One box=27÷3=9 kg. Five=45 kg. Formula / rule Unitary method: value of one unit = total ÷ number of units; required value = one-unit value × required units. Applied to 7A: the worked result is 45 kg. Why it works here One box=27÷3=9 kg. Five=45 kg. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Balance puzzle. Checkpoint for 7A: 45 kg.
7B	Balance puzzle Similar practice	4 identical parcels weigh 30 kg altogether. What do 10 parcels weigh?	Show answer + formula + trick 75 kg. One parcel=30÷4=7.5 kg. Ten=75 kg. Formula / rule Unitary method: value of one unit = total ÷ number of units; required value = one-unit value × required units. Applied to 7B: the worked result is 75 kg. Why it works here One parcel=30÷4=7.5 kg. Ten=75 kg. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Balance puzzle. Checkpoint for 7B: 75 kg.
8A	Operation symbol puzzle	Which operation makes 48 □ 6 = 8 true?	Show answer + formula + trick ÷. 48÷6=8. Formula / rule Test +, −, ×, ÷ using BODMAS and the required target. Applied to 8A: the worked result is ÷. Why it works here 48÷6=8. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 8A: ÷.
8B	Operation symbol puzzle Similar practice	Which operation makes 15 □ 4 = 60 true?	Show answer + formula + trick ×. 15×4=60. Formula / rule Test +, −, ×, ÷ using BODMAS and the required target. Applied to 8B: the worked result is ×. Why it works here 15×4=60. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 8B: ×.
9A	Target-number puzzle	Use 8, 4 and 2 once each with +, −, × or ÷ to make 20.	Show answer + formula + trick 8×2+4=20. 8×2=16, then +4=20. Formula / rule Use each stated number exactly once, respecting BODMAS; compare candidate expressions with the target. Applied to 9A: the worked result is 8×2+4=20. Why it works here 8×2=16, then +4=20. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Target-number puzzle. Checkpoint for 9A: 8×2+4=20.
9B	Target-number puzzle Similar practice	Use 9, 3 and 2 once each to make 21.	Show answer + formula + trick 9×2+3=21. 9×2=18, then +3=21. Formula / rule Use each stated number exactly once, respecting BODMAS; compare candidate expressions with the target. Applied to 9B: the worked result is 9×2+3=21. Why it works here 9×2=18, then +3=21. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Target-number puzzle. Checkpoint for 9B: 9×2+3=21.
10A	Largest possible result	Choose + or × in 7 □ 6 □ 2 to make the largest result, using normal order of operations.	Show answer + formula + trick 84. 7×6×2=84. Formula / rule Use each stated number exactly once, respecting BODMAS; compare candidate expressions with the target. Applied to 10A: the worked result is 84. Why it works here 7×6×2=84. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Largest possible result. Checkpoint for 10A: 84.
10B	Largest possible result Similar practice	Choose + or × in 9 □ 4 □ 3 to make the largest result.	Show answer + formula + trick 108. 9×4×3=108. Formula / rule Use each stated number exactly once, respecting BODMAS; compare candidate expressions with the target. Applied to 10B: the worked result is 108. Why it works here 9×4×3=108. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Largest possible result. Checkpoint for 10B: 108.

22. Tiffin Focus: Fractions, Decimals & Percentages Challenge

#	Question type	Example	Worked solution
1A	Ordering mixed forms	Order from smallest to largest: 0.6, 5/8, 62%.	Show answer + formula + trick 0.6, 62%, 5/8. Convert to percentages: 60%, 62%, 62.5%. Formula / rule Convert fractions, decimals and percentages to one common form before ordering. Applied to 1A: the worked result is 0.6, 62%, 5/8. Why it works here Convert to percentages: 60%, 62%, 62.5%. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Ordering mixed forms. Checkpoint for 1A: 0.6, 62%, 5/8.
1B	Ordering mixed forms Similar practice	Order from smallest to largest: 3/4, 0.72, 73%.	Show answer + formula + trick 0.72, 73%, 3/4. Convert to percentages: 75%,72%,73%. Formula / rule Convert fractions, decimals and percentages to one common form before ordering. Applied to 1B: the worked result is 0.72, 73%, 3/4. Why it works here Convert to percentages: 75%,72%,73%. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Ordering mixed forms. Checkpoint for 1B: 0.72, 73%, 3/4.
2A	Fraction remaining	A tank is 3/5 full. One quarter of the water is used. What fraction of the tank remains full?	Show answer + formula + trick 9/20. Used=1/4 of 3/5=3/20. Remaining=12/20-3/20=9/20. Formula / rule When taking a fraction of an existing fraction, multiply; subtract the used part from the starting fraction. Applied to 2A: the worked result is 9/20. Why it works here Used=1/4 of 3/5=3/20. Remaining=12/20-3/20=9/20. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 2A: 9/20.
2B	Fraction remaining Similar practice	A box is 7/8 full. Two sevenths of its contents are removed. What fraction of the box remains full?	Show answer + formula + trick 5/8. Removed=2/7×7/8=1/4. Remaining=7/8-2/8=5/8. Formula / rule When taking a fraction of an existing fraction, multiply; subtract the used part from the starting fraction. Applied to 2B: the worked result is 5/8. Why it works here Removed=2/7×7/8=1/4. Remaining=7/8-2/8=5/8. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 2B: 5/8.
3A	Successive percentage changes	A price rises by 20% then falls by 20%. Starting at £100, what is the final price?	Show answer + formula + trick £96. £100×1.2=£120; £120×0.8=£96. Formula / rule Percentage change = change ÷ original × 100%. Applied to 3A: the worked result is £96. Why it works here £100×1.2=£120; £120×0.8=£96. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 3A: £96.
3B	Successive percentage changes Similar practice	A value falls by 10% then rises by 10%. Starting at 200, what is the final value?	Show answer + formula + trick 198. 200×0.9=180; 180×1.1=198. Formula / rule Percentage change = change ÷ original × 100%. Applied to 3B: the worked result is 198. Why it works here 200×0.9=180; 180×1.1=198. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 3B: 198.
4A	Reverse fraction	3/7 of a number is 24. Find the number.	Show answer + formula + trick 56. One seventh=24÷3=8; seven sevenths=56. Formula / rule If a/b of N = x, then N = x÷a×b. Applied to 4A: the worked result is 56. Why it works here One seventh=24÷3=8; seven sevenths=56. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 4A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 4A: 56.
4B	Reverse fraction Similar practice	5/8 of a number is 45. Find the number.	Show answer + formula + trick 72. One eighth=45÷5=9; eight eighths=72. Formula / rule If a/b of N = x, then N = x÷a×b. Applied to 4B: the worked result is 72. Why it works here One eighth=45÷5=9; eight eighths=72. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 4B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 4B: 72.
5A	Recurring relationship	Which is greater: 2/3 or 0.66?	Show answer + formula + trick 2/3. 2/3≈0.666..., which is greater than 0.66. Formula / rule Convert the fraction to a decimal (or compare by cross-multiplication) before deciding which is greater. Applied to 5A: the worked result is 2/3. Why it works here 2/3≈0.666..., which is greater than 0.66. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Recurring relationship. Checkpoint for 5A: 2/3.
5B	Recurring relationship Similar practice	Which is greater: 5/6 or 0.83?	Show answer + formula + trick 5/6. 5/6≈0.833..., which is greater than 0.83. Formula / rule Convert the fraction to a decimal (or compare by cross-multiplication) before deciding which is greater. Applied to 5B: the worked result is 5/6. Why it works here 5/6≈0.833..., which is greater than 0.83. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Recurring relationship. Checkpoint for 5B: 5/6.
6A	Percentage points vs percentage	A score rises from 60% to 75%. By how many percentage points?	Show answer + formula + trick 15 percentage points. 75-60=15. Formula / rule Percentage-point change = new percentage − old percentage. Applied to 6A: the worked result is 15 percentage points. Why it works here 75-60=15. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 6A: 15 percentage points.
6B	Percentage points vs percentage Similar practice	A score falls from 82% to 70%. By how many percentage points?	Show answer + formula + trick 12 percentage points. 82-70=12. Formula / rule Percentage-point change = new percentage − old percentage. Applied to 6B: the worked result is 12 percentage points. Why it works here 82-70=12. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 6B: 12 percentage points.
7A	Mixed-number proportion	Calculate 1 1/2 × 2 2/3.	Show answer + formula + trick 4. 3/2×8/3=4. Formula / rule Multiply fractions straight across after cancelling common factors; convert mixed numbers to improper fractions first. Applied to 7A: the worked result is 4. Why it works here 3/2×8/3=4. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 7A: 4.
7B	Mixed-number proportion Similar practice	Calculate 2 1/4 × 1 1/3.	Show answer + formula + trick 3. 9/4×4/3=3. Formula / rule Multiply fractions straight across after cancelling common factors; convert mixed numbers to improper fractions first. Applied to 7B: the worked result is 3. Why it works here 9/4×4/3=3. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 7B: 3.
8A	Decimal scaling	A 0.75 kg pack costs £4.80. What is the cost per kg?	Show answer + formula + trick £6.40. £4.80÷0.75=£6.40. Formula / rule Unit value = total value ÷ decimal quantity; scale numerator and denominator equally if that makes division easier. Applied to 8A: the worked result is £6.40. Why it works here £4.80÷0.75=£6.40. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Decimal scaling. Checkpoint for 8A: £6.40.
8B	Decimal scaling Similar practice	A 1.25 L bottle costs £2.50. What is the cost per litre?	Show answer + formula + trick £2.00. £2.50÷1.25=£2.00. Formula / rule Unit value = total value ÷ decimal quantity; scale numerator and denominator equally if that makes division easier. Applied to 8B: the worked result is £2.00. Why it works here £2.50÷1.25=£2.00. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Decimal scaling. Checkpoint for 8B: £2.00.
9A	Fraction difference	How much greater is 7/8 than 2/3?	Show answer + formula + trick 5/24. 7/8=21/24 and 2/3=16/24; difference=5/24. Formula / rule Compare/subtract fractions using a common denominator or cross-multiplication. Applied to 9A: the worked result is 5/24. Why it works here 7/8=21/24 and 2/3=16/24; difference=5/24. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 9A: 5/24.
9B	Fraction difference Similar practice	How much greater is 5/6 than 3/5?	Show answer + formula + trick 7/30. 25/30-18/30=7/30. Formula / rule Compare/subtract fractions using a common denominator or cross-multiplication. Applied to 9B: the worked result is 7/30. Why it works here 25/30-18/30=7/30. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 9B: 7/30.
10A	Percentage of percentage	40% of a class are boys. 25% of the boys wear glasses. What percentage of the whole class are boys who wear glasses?	Show answer + formula + trick 10%. 40%×25%=0.4×0.25=0.10=10%. Formula / rule A percentage of a percentage: convert both to decimals and multiply. Applied to 10A: the worked result is 10%. Why it works here 40%×25%=0.4×0.25=0.10=10%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 10A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 10A: 10%.
10B	Percentage of percentage Similar practice	60% of a group are adults. 30% of the adults are teachers. What percentage of the whole group are adult teachers?	Show answer + formula + trick 18%. 0.6×0.3=0.18=18%. Formula / rule A percentage of a percentage: convert both to decimals and multiply. Applied to 10B: the worked result is 18%. Why it works here 0.6×0.3=0.18=18%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 10B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 10B: 18%.

23. Tiffin Focus: Ratio, Proportion & Scale Problems

#	Question type	Example	Worked solution
1A	Ratio chain	A:B=2:3 and B:C=6:5. Find A:B:C.	Show answer + formula + trick 4:6:5. Scale 2:3 by 2 to make B=6. Formula / rule For linked ratios A:B and B:C, scale them so the shared term B is equal, then combine. Applied to 1A: the worked result is 4:6:5. Why it works here Scale 2:3 by 2 to make B=6. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 1A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 1A: 4:6:5.
1B	Ratio chain Similar practice	P:Q=3:4 and Q:R=8:7. Find P:Q:R.	Show answer + formula + trick 6:8:7. Scale 3:4 by 2 to make Q=8. Formula / rule For linked ratios A:B and B:C, scale them so the shared term B is equal, then combine. Applied to 1B: the worked result is 6:8:7. Why it works here Scale 3:4 by 2 to make Q=8. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 1B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 1B: 6:8:7.
2A	Changing a ratio	Red:blue=3:2. After adding 4 blue counters, the numbers are equal. How many red counters were there originally?	Show answer + formula + trick 12. Let one part=4 because blue needs one extra part to equal red. Red=3×4=12. Formula / rule Relate the numerical change to the change/difference in ratio parts to find one part. Applied to 2A: the worked result is 12. Why it works here Let one part=4 because blue needs one extra part to equal red. Red=3×4=12. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 2A: 12.
2B	Changing a ratio Similar practice	Cats:dogs=5:3. After adding 8 dogs, numbers are equal. How many cats originally?	Show answer + formula + trick 20. The difference is 2 parts=8, so 1 part=4; cats=20. Formula / rule Relate the numerical change to the change/difference in ratio parts to find one part. Applied to 2B: the worked result is 20. Why it works here The difference is 2 parts=8, so 1 part=4; cats=20. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 2B: 20.
3A	Proportion with inverse thinking	8 workers complete a task in 6 days at the same rate. How many worker-days is that?	Show answer + formula + trick 48 worker-days. 8×6=48. Formula / rule For a fixed task at constant rate, work = workers × time; more workers means less time. Applied to 3A: the worked result is 48 worker-days. Why it works here 8×6=48. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 3A: 48 worker-days.
3B	Proportion with inverse thinking Similar practice	5 machines run for 12 hours. How many machine-hours?	Show answer + formula + trick 60 machine-hours. 5×12=60. Formula / rule For a fixed task at constant rate, work = workers × time; more workers means less time. Applied to 3B: the worked result is 60 machine-hours. Why it works here 5×12=60. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 3B: 60 machine-hours.
4A	Recipe scaling non-integer	A recipe for 8 uses 600 g flour. How much for 14?	Show answer + formula + trick 1,050 g. 600÷8=75 g each; 75×14=1,050. Formula / rule Direct proportion: find the value for one unit, then multiply by the required number of units. Applied to 4A: the worked result is 1,050 g. Why it works here 600÷8=75 g each; 75×14=1,050. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Recipe scaling non-integer. Checkpoint for 4A: 1,050 g.
4B	Recipe scaling non-integer Similar practice	A recipe for 5 uses 350 ml milk. How much for 12?	Show answer + formula + trick 840 ml. 350÷5=70 ml each; 70×12=840. Formula / rule Direct proportion: find the value for one unit, then multiply by the required number of units. Applied to 4B: the worked result is 840 ml. Why it works here 350÷5=70 ml each; 70×12=840. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Recipe scaling non-integer. Checkpoint for 4B: 840 ml.
5A	Scale drawing reverse	A model is made at scale 1:50. A real length is 4 m. How long is it on the model?	Show answer + formula + trick 8 cm. 4 m=400 cm; 400÷50=8 cm. Formula / rule Drawing length = real length ÷ scale factor, after converting to the same units. Applied to 5A: the worked result is 8 cm. Why it works here 4 m=400 cm; 400÷50=8 cm. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Scale drawing reverse. Checkpoint for 5A: 8 cm.
5B	Scale drawing reverse Similar practice	A plan uses scale 1:200. A real wall is 12 m. Length on plan?	Show answer + formula + trick 6 cm. 12 m=1,200 cm; 1,200÷200=6 cm. Formula / rule Drawing length = real length ÷ scale factor, after converting to the same units. Applied to 5B: the worked result is 6 cm. Why it works here 12 m=1,200 cm; 1,200÷200=6 cm. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Scale drawing reverse. Checkpoint for 5B: 6 cm.
6A	Ratio and fraction conversion	In a bag red:green=4:7. What fraction are red?	Show answer + formula + trick 4/11. Total parts=11; red=4 parts. Formula / rule Fraction represented by a ratio part = chosen parts ÷ total ratio parts. Applied to 6A: the worked result is 4/11. Why it works here Total parts=11; red=4 parts. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 6A: 4/11.
6B	Ratio and fraction conversion Similar practice	Boys:girls=5:8. What fraction are boys?	Show answer + formula + trick 5/13. Total parts=13; boys=5 parts. Formula / rule Fraction represented by a ratio part = chosen parts ÷ total ratio parts. Applied to 6B: the worked result is 5/13. Why it works here Total parts=13; boys=5 parts. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 6B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 6B: 5/13.
7A	Ratio from difference	A:B=7:4 and A is 18 more than B. Find A.	Show answer + formula + trick 42. Difference=3 parts=18, so 1 part=6; A=7×6=42. Formula / rule One ratio part = known numerical difference ÷ difference in ratio parts. Applied to 7A: the worked result is 42. Why it works here Difference=3 parts=18, so 1 part=6; A=7×6=42. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 7A: 42.
7B	Ratio from difference Similar practice	P:Q=9:5 and P is 28 more than Q. Find Q.	Show answer + formula + trick 35. Difference=4 parts=28, so 1 part=7; Q=5×7=35. Formula / rule One ratio part = known numerical difference ÷ difference in ratio parts. Applied to 7B: the worked result is 35. Why it works here Difference=4 parts=28, so 1 part=7; Q=5×7=35. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 7B: 35.
8A	Ratio from one amount	Apples:pears=5:3. There are 45 apples. How many fruit altogether?	Show answer + formula + trick 72. 5 parts=45 so 1 part=9; total 8 parts=72. Formula / rule Find one ratio part from the known total/part, then multiply by the required number of parts. Applied to 8A: the worked result is 72. Why it works here 5 parts=45 so 1 part=9; total 8 parts=72. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 8A: 72.
8B	Ratio from one amount Similar practice	Red:blue=4:7. There are 28 red. How many counters altogether?	Show answer + formula + trick 77. 4 parts=28 so 1 part=7; total 11 parts=77. Formula / rule Find one ratio part from the known total/part, then multiply by the required number of parts. Applied to 8B: the worked result is 77. Why it works here 4 parts=28 so 1 part=7; total 11 parts=77. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 8B: 77.
9A	Proportional cost	7 metres of fabric cost £31.50. What do 12 metres cost?	Show answer + formula + trick £54. £31.50÷7=£4.50 per m; ×12=£54. Formula / rule Direct proportion: find the value for one unit, then multiply by the required number of units. Applied to 9A: the worked result is £54. Why it works here £31.50÷7=£4.50 per m; ×12=£54. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 9A: £54.
9B	Proportional cost Similar practice	9 kg of rice costs £24.30. What do 5 kg cost?	Show answer + formula + trick £13.50. £24.30÷9=£2.70 per kg; ×5=£13.50. Formula / rule Direct proportion: find the value for one unit, then multiply by the required number of units. Applied to 9B: the worked result is £13.50. Why it works here £24.30÷9=£2.70 per kg; ×5=£13.50. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 9B: £13.50.
10A	Map journey	A map scale is 1 cm:4 km. A route is 7.5 cm. What is the real distance?	Show answer + formula + trick 30 km. 7.5×4=30. Formula / rule Apply the map scale using matching units; real distance = map distance × scale factor. Applied to 10A: the worked result is 30 km. Why it works here 7.5×4=30. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Map journey. Checkpoint for 10A: 30 km.
10B	Map journey Similar practice	A map scale is 2 cm:5 km. A route is 8 cm. What is the real distance?	Show answer + formula + trick 20 km. 8÷2=4 scale units; 4×5=20 km. Formula / rule Apply the map scale using matching units; real distance = map distance × scale factor. Applied to 10B: the worked result is 20 km. Why it works here 8÷2=4 scale units; 4×5=20 km. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Map journey. Checkpoint for 10B: 20 km.

24. Tiffin Focus: Time, Timetables & Time Zones

#	Question type	Example	Worked solution
1A	Elapsed time across hour	A lesson starts at 09:47 and ends at 11:13. How long is it?	Show answer + formula + trick 1 h 26 min. 09:47→10:00=13 min, →11:00=1 h, →11:13=13 min. Formula / rule Elapsed time = finish time − start time, often easiest by bridging through whole hours. Applied to 1A: the worked result is 1 h 26 min. Why it works here 09:47→10:00=13 min, →11:00=1 h, →11:13=13 min. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 1A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 1A: 1 h 26 min.
1B	Elapsed time across hour Similar practice	A journey starts at 14:38 and ends at 16:05. How long is it?	Show answer + formula + trick 1 h 27 min. 14:38→15:00=22 min, →16:00=1 h, →16:05=5 min. Formula / rule Elapsed time = finish time − start time, often easiest by bridging through whole hours. Applied to 1B: the worked result is 1 h 27 min. Why it works here 14:38→15:00=22 min, →16:00=1 h, →16:05=5 min. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 1B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 1B: 1 h 27 min.
2A	Find start time	A film ends at 21:35 and lasts 1 h 48 min. When did it start?	Show answer + formula + trick 19:47. 21:35-1 h=20:35; subtract 48 min=19:47. Formula / rule Start time = finish time − duration. Applied to 2A: the worked result is 19:47. Why it works here 21:35-1 h=20:35; subtract 48 min=19:47. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 2A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 2A: 19:47.
2B	Find start time Similar practice	A train arrives at 17:12 after 2 h 37 min. When did it leave?	Show answer + formula + trick 14:35. 17:12-2 h=15:12; subtract 37 min=14:35. Formula / rule Start time = finish time − duration. Applied to 2B: the worked result is 14:35. Why it works here 17:12-2 h=15:12; subtract 37 min=14:35. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 2B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 2B: 14:35.
3A	Find finish time	A test starts at 10:25 and lasts 1 h 55 min. When does it finish?	Show answer + formula + trick 12:20. 10:25+1 h=11:25; +55 min=12:20. Formula / rule Finish time = start time + duration. Applied to 3A: the worked result is 12:20. Why it works here 10:25+1 h=11:25; +55 min=12:20. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 3A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 3A: 12:20.
3B	Find finish time Similar practice	A match starts at 13:48 and lasts 2 h 36 min. Finish?	Show answer + formula + trick 16:24. 13:48+2 h=15:48; +36 min=16:24. Formula / rule Finish time = start time + duration. Applied to 3B: the worked result is 16:24. Why it works here 13:48+2 h=15:48; +36 min=16:24. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 3B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 3B: 16:24.
4A	Timetable wait	A bus arrives at 08:42. The next train leaves at 09:07. Waiting time?	Show answer + formula + trick 25 min. 08:42→09:00=18 min, plus 7=25 min. Formula / rule Elapsed time = finish time − start time, often easiest by bridging through whole hours. Applied to 4A: the worked result is 25 min. Why it works here 08:42→09:00=18 min, plus 7=25 min. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 4A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 4A: 25 min.
4B	Timetable wait Similar practice	A train arrives at 15:53. A connection leaves at 16:21. Waiting time?	Show answer + formula + trick 28 min. 15:53→16:00=7 min, plus 21=28 min. Formula / rule Elapsed time = finish time − start time, often easiest by bridging through whole hours. Applied to 4B: the worked result is 28 min. Why it works here 15:53→16:00=7 min, plus 21=28 min. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 4B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 4B: 28 min.
5A	Time interval frequency	A bus leaves every 18 minutes. One leaves at 09:12. What time does the fourth bus leave, counting 09:12 as the first?	Show answer + formula + trick 10:06. The fourth is 3 intervals later: 3×18=54 min; 09:12+54=10:06. Formula / rule The nth event occurs (n−1) intervals after the first event. Applied to 5A: the worked result is 10:06. Why it works here The fourth is 3 intervals later: 3×18=54 min; 09:12+54=10:06. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 5A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 5A: 10:06.
5B	Time interval frequency Similar practice	A train leaves every 25 minutes. One leaves at 14:10. When is the fifth, counting 14:10 as first?	Show answer + formula + trick 15:50. Four intervals later: 4×25=100 min; 14:10+1 h 40=15:50. Formula / rule The nth event occurs (n−1) intervals after the first event. Applied to 5B: the worked result is 15:50. Why it works here Four intervals later: 4×25=100 min; 14:10+1 h 40=15:50. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 5B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 5B: 15:50.
6A	12-hour to 24-hour	Write 12:35 am in 24-hour time.	Show answer + formula + trick 00:35. After midnight, 12:xx am becomes 00:xx. Formula / rule For pm times after 12 noon add 12 to the hour; 12 am becomes 00 and 12 pm remains 12. Applied to 6A: the worked result is 00:35. Why it works here After midnight, 12:xx am becomes 00:xx. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: 12-hour to 24-hour. Checkpoint for 6A: 00:35.
6B	12-hour to 24-hour Similar practice	Write 12:20 pm in 24-hour time.	Show answer + formula + trick 12:20. Noon is 12:00, so 12:20 pm stays 12:20. Formula / rule For pm times after 12 noon add 12 to the hour; 12 am becomes 00 and 12 pm remains 12. Applied to 6B: the worked result is 12:20. Why it works here Noon is 12:00, so 12:20 pm stays 12:20. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: 12-hour to 24-hour. Checkpoint for 6B: 12:20.
7A	Time zones east	London is 2 hours behind Athens. If it is 14:35 in London, what time is it in Athens?	Show answer + formula + trick 16:35. Athens is ahead, so add 2 hours. Formula / rule If the destination is stated to be ahead, add the time difference; if behind, subtract. Applied to 7A: the worked result is 16:35. Why it works here Athens is ahead, so add 2 hours. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 7A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 7A: 16:35.
7B	Time zones east Similar practice	London is 5 hours behind Dubai. If it is 08:20 in London, what time is it in Dubai?	Show answer + formula + trick 13:20. Dubai is ahead, so add 5 hours. Formula / rule If the destination is stated to be ahead, add the time difference; if behind, subtract. Applied to 7B: the worked result is 13:20. Why it works here Dubai is ahead, so add 5 hours. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 7B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 7B: 13:20.
8A	Time zones crossing midnight	City B is 7 hours ahead of City A. It is 20:30 in A. What time is it in B?	Show answer + formula + trick 03:30 next day. 20:30+7 h=27:30, which is 03:30 next day. Formula / rule Apply the time difference; if hours reach 24, subtract 24 and move to the next day. Applied to 8A: the worked result is 03:30 next day. Why it works here 20:30+7 h=27:30, which is 03:30 next day. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 8A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 8A: 03:30 next day.
8B	Time zones crossing midnight Similar practice	City Y is 9 hours ahead of City X. It is 18:15 in X. Time in Y?	Show answer + formula + trick 03:15 next day. 18:15+9 h=27:15=03:15 next day. Formula / rule Apply the time difference; if hours reach 24, subtract 24 and move to the next day. Applied to 8B: the worked result is 03:15 next day. Why it works here 18:15+9 h=27:15=03:15 next day. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 8B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 8B: 03:15 next day.
9A	Minutes as fraction of hour	What fraction of an hour is 45 minutes?	Show answer + formula + trick 3/4. 45/60=3/4. Formula / rule Fraction of an hour = minutes/60, simplified. Applied to 9A: the worked result is 3/4. Why it works here 45/60=3/4. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 9A: 3/4.
9B	Minutes as fraction of hour Similar practice	What fraction of an hour is 20 minutes?	Show answer + formula + trick 1/3. 20/60=1/3. Formula / rule Fraction of an hour = minutes/60, simplified. Applied to 9B: the worked result is 1/3. Why it works here 20/60=1/3. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 9B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 9B: 1/3.
10A	Decimal hours	Convert 2.5 hours to hours and minutes.	Show answer + formula + trick 2 h 30 min. 0.5 hour=30 min. Formula / rule Minutes = decimal part of an hour ×60. Applied to 10A: the worked result is 2 h 30 min. Why it works here 0.5 hour=30 min. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Decimal hours. Checkpoint for 10A: 2 h 30 min.
10B	Decimal hours Similar practice	Convert 1.75 hours to hours and minutes.	Show answer + formula + trick 1 h 45 min. 0.75 hour=45 min. Formula / rule Minutes = decimal part of an hour ×60. Applied to 10B: the worked result is 1 h 45 min. Why it works here 0.75 hour=45 min. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Decimal hours. Checkpoint for 10B: 1 h 45 min.

25. Tiffin Focus: Speed, Distance, Rates & Unitary Method

#	Question type	Example	Worked solution
1A	Speed	A car travels 216 km in 3 hours. What is its average speed?	Show answer + formula + trick 72 km/h. 216÷3=72. Formula / rule Speed = distance ÷ time. Applied to 1A: the worked result is 72 km/h. Why it works here 216÷3=72. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 1A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 1A: 72 km/h.
1B	Speed Similar practice	A train travels 315 km in 4.5 hours. Average speed?	Show answer + formula + trick 70 km/h. 315÷4.5=70. Formula / rule Speed = distance ÷ time. Applied to 1B: the worked result is 70 km/h. Why it works here 315÷4.5=70. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 1B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 1B: 70 km/h.
2A	Distance	A cyclist travels at 18 km/h for 2.5 hours. Distance?	Show answer + formula + trick 45 km. 18×2.5=45. Formula / rule Distance = speed × time. Applied to 2A: the worked result is 45 km. Why it works here 18×2.5=45. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 2A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 2A: 45 km.
2B	Distance Similar practice	A boat travels at 24 km/h for 3.25 hours. Distance?	Show answer + formula + trick 78 km. 24×3.25=78. Formula / rule Distance = speed × time. Applied to 2B: the worked result is 78 km. Why it works here 24×3.25=78. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 2B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 2B: 78 km.
3A	Time	How long does it take to travel 150 km at 60 km/h?	Show answer + formula + trick 2.5 h. 150÷60=2.5. Formula / rule Time = total distance/amount ÷ speed/rate. Applied to 3A: the worked result is 2.5 h. Why it works here 150÷60=2.5. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 3A, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 3A: 2.5 h.
3B	Time Similar practice	How long to travel 96 km at 32 km/h?	Show answer + formula + trick 3 h. 96÷32=3. Formula / rule Time = total distance/amount ÷ speed/rate. Applied to 3B: the worked result is 3 h. Why it works here 96÷32=3. Tip / trick Bridge through the next whole hour and mark whether you cross noon or midnight; this is safer than mental borrowing. For 3B, re-check the exact numbers in this row before entering the answer. Remember 60 minutes = 1 hour. Checkpoint for 3B: 3 h.
4A	Minutes in speed problem	A car travels 50 km/h for 36 minutes. How far?	Show answer + formula + trick 30 km. 36 min=0.6 h; 50×0.6=30. Formula / rule Convert minutes to hours first: hours = minutes÷60; then distance=speed×time. Applied to 4A: the worked result is 30 km. Why it works here 36 min=0.6 h; 50×0.6=30. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 4A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 4A: 30 km.
4B	Minutes in speed problem Similar practice	A cyclist travels 24 km/h for 45 minutes. How far?	Show answer + formula + trick 18 km. 45 min=0.75 h; 24×0.75=18. Formula / rule Convert minutes to hours first: hours = minutes÷60; then distance=speed×time. Applied to 4B: the worked result is 18 km. Why it works here 45 min=0.75 h; 24×0.75=18. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 4B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 4B: 18 km.
5A	Unit rate	12 oranges cost £4.80. Cost per orange?	Show answer + formula + trick £0.40. £4.80÷12=£0.40. Formula / rule Unit rate = total amount ÷ number of units. Applied to 5A: the worked result is £0.40. Why it works here £4.80÷12=£0.40. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 5A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 5A: £0.40.
5B	Unit rate Similar practice	8 pencils cost £3.60. Cost per pencil?	Show answer + formula + trick £0.45. £3.60÷8=£0.45. Formula / rule Unit rate = total amount ÷ number of units. Applied to 5B: the worked result is £0.45. Why it works here £3.60÷8=£0.45. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 5B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 5B: £0.45.
6A	Rate comparison	Machine A makes 120 items in 4 h. Machine B makes 140 in 5 h. Which is faster?	Show answer + formula + trick Machine A. A=30/h, B=28/h. Formula / rule Convert each option to the same unit rate before comparing. Applied to 6A: the worked result is Machine A. Why it works here A=30/h, B=28/h. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 6A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 6A: Machine A.
6B	Rate comparison Similar practice	Runner A covers 18 km in 2 h. Runner B covers 25 km in 3 h. Who has greater average speed?	Show answer + formula + trick Runner A. A=9 km/h; B≈8.33 km/h. Formula / rule Convert each option to the same unit rate before comparing. Applied to 6B: the worked result is Runner A. Why it works here A=9 km/h; B≈8.33 km/h. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 6B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 6B: Runner A.
7A	Filling rate	A tap fills 15 litres in 3 minutes. How much in 8 minutes at the same rate?	Show answer + formula + trick 40 L. 5 L/min ×8=40. Formula / rule Amount = rate per unit time × time. Applied to 7A: the worked result is 40 L. Why it works here 5 L/min ×8=40. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 7A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 7A: 40 L.
7B	Filling rate Similar practice	A machine packs 42 boxes in 6 minutes. How many in 15 minutes?	Show answer + formula + trick 105. 7 boxes/min ×15=105. Formula / rule Amount = rate per unit time × time. Applied to 7B: the worked result is 105. Why it works here 7 boxes/min ×15=105. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 7B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 7B: 105.
8A	Combined rate interpretation	Two taps each fill 6 L/min. Together, how much in 7 min?	Show answer + formula + trick 84 L. Combined rate=12 L/min; 12×7=84. Formula / rule For simultaneous independent sources, combined rate = sum of individual rates. Applied to 8A: the worked result is 84 L. Why it works here Combined rate=12 L/min; 12×7=84. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 8A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 8A: 84 L.
8B	Combined rate interpretation Similar practice	Three identical machines each make 8 parts/min. How many parts in 5 min?	Show answer + formula + trick 120. Combined rate=24/min; ×5=120. Formula / rule For simultaneous independent sources, combined rate = sum of individual rates. Applied to 8B: the worked result is 120. Why it works here Combined rate=24/min; ×5=120. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 8B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 8B: 120.
9A	Average speed two equal distances	A car travels 60 km at 30 km/h then 60 km at 60 km/h. Total time?	Show answer + formula + trick 3 h. First leg 2 h, second 1 h; total 3 h. Formula / rule For a multi-leg trip, total time = sum of leg times; calculate each leg with time=distance÷speed. Applied to 9A: the worked result is 3 h. Why it works here First leg 2 h, second 1 h; total 3 h. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 9A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 9A: 3 h.
9B	Average speed two equal distances Similar practice	A cyclist travels 20 km at 10 km/h then 20 km at 20 km/h. Total time?	Show answer + formula + trick 3 h. 2 h +1 h =3 h. Formula / rule For a multi-leg trip, total time = sum of leg times; calculate each leg with time=distance÷speed. Applied to 9B: the worked result is 3 h. Why it works here 2 h +1 h =3 h. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 9B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 9B: 3 h.
10A	Rate word problem	A printer produces 18 pages per minute. How long for 450 pages?	Show answer + formula + trick 25 min. 450÷18=25. Formula / rule Time = total distance/amount ÷ speed/rate. Applied to 10A: the worked result is 25 min. Why it works here 450÷18=25. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 10A, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 10A: 25 min.
10B	Rate word problem Similar practice	A pump moves 24 L/min. How long for 600 L?	Show answer + formula + trick 25 min. 600÷24=25. Formula / rule Time = total distance/amount ÷ speed/rate. Applied to 10B: the worked result is 25 min. Why it works here 600÷24=25. Tip / trick Make the units compatible first. If time is in minutes but speed is per hour, convert before using the formula. For 10B, re-check the exact numbers in this row before entering the answer. Remember Keep distance, time and rate units compatible. Checkpoint for 10B: 25 min.

26. Tiffin Focus: Geometry, Angles & Shape Properties

#	Question type	Example	Worked solution
1A	Angles in parallel lines	Two parallel lines are crossed by a transversal. An acute angle is 64°. What is the adjacent obtuse angle?	Show answer + formula + trick 116°. Adjacent angles on a straight line total 180°: 180-64=116. Formula / rule Use equal corresponding/alternate angles and supplementary adjacent angles (sum 180°). Applied to 1A: the worked result is 116°. Why it works here Adjacent angles on a straight line total 180°: 180-64=116. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 1A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 1A: 116°.
1B	Angles in parallel lines Similar practice	An acute angle formed by parallel lines is 73°. Adjacent obtuse angle?	Show answer + formula + trick 107°. 180-73=107. Formula / rule Use equal corresponding/alternate angles and supplementary adjacent angles (sum 180°). Applied to 1B: the worked result is 107°. Why it works here 180-73=107. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 1B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 1B: 107°.
2A	Triangle angle ratio	Angles in a triangle are in ratio 2:3:4. Find the largest angle.	Show answer + formula + trick 80°. Total parts=9; one part=180÷9=20; largest=4×20=80. Formula / rule Share the triangle total 180° in the stated ratio. Applied to 2A: the worked result is 80°. Why it works here Total parts=9; one part=180÷9=20; largest=4×20=80. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 2A: 80°.
2B	Triangle angle ratio Similar practice	Triangle angles are in ratio 3:4:5. Find the smallest.	Show answer + formula + trick 45°. 12 parts; one=15°; smallest=45°. Formula / rule Share the triangle total 180° in the stated ratio. Applied to 2B: the worked result is 45°. Why it works here 12 parts; one=15°; smallest=45°. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 2B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 2B: 45°.
3A	Quadrilateral angle ratio	Angles of a quadrilateral are in ratio 2:3:4:3. Find the largest.	Show answer + formula + trick 120°. 12 parts; one=30°; largest=4×30=120°. Formula / rule Share the quadrilateral total 360° in the stated ratio. Applied to 3A: the worked result is 120°. Why it works here 12 parts; one=30°; largest=4×30=120°. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3A, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 3A: 120°.
3B	Quadrilateral angle ratio Similar practice	Angles are in ratio 1:2:3:4. Find the second largest.	Show answer + formula + trick 108°. 10 parts; one=36°; second largest=3×36=108°. Formula / rule Share the quadrilateral total 360° in the stated ratio. Applied to 3B: the worked result is 108°. Why it works here 10 parts; one=36°; second largest=3×36=108°. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 3B, re-check the exact numbers in this row before entering the answer. Remember Find one part or one unit before scaling. Checkpoint for 3B: 108°.
4A	Regular polygon exterior angle	What is each exterior angle of a regular hexagon?	Show answer + formula + trick 60°. 360÷6=60°. Formula / rule Each exterior angle of a regular n-gon = 360°÷n. Applied to 4A: the worked result is 60°. Why it works here 360÷6=60°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 4A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 4A: 60°.
4B	Regular polygon exterior angle Similar practice	What is each exterior angle of a regular decagon?	Show answer + formula + trick 36°. 360÷10=36°. Formula / rule Each exterior angle of a regular n-gon = 360°÷n. Applied to 4B: the worked result is 36°. Why it works here 360÷10=36°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 4B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 4B: 36°.
5A	Regular polygon interior angle	Find each interior angle of a regular pentagon.	Show answer + formula + trick 108°. Exterior=360÷5=72; interior=180-72=108. Formula / rule Each interior angle of a regular n-gon = 180°−360°/n. Applied to 5A: the worked result is 108°. Why it works here Exterior=360÷5=72; interior=180-72=108. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 5A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 5A: 108°.
5B	Regular polygon interior angle Similar practice	Find each interior angle of a regular octagon.	Show answer + formula + trick 135°. Exterior=360÷8=45; interior=180-45=135. Formula / rule Each interior angle of a regular n-gon = 180°−360°/n. Applied to 5B: the worked result is 135°. Why it works here Exterior=360÷8=45; interior=180-45=135. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 5B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 5B: 135°.
6A	Isosceles exterior	An isosceles triangle has equal base angles. One exterior angle at the vertex is 130°. Find each base angle.	Show answer + formula + trick 65°. Interior vertex=50°. Remaining 130° shared equally=65°. Formula / rule Exterior + adjacent interior =180°; then use triangle total 180° and equal base angles. Applied to 6A: the worked result is 65°. Why it works here Interior vertex=50°. Remaining 130° shared equally=65°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Isosceles exterior. Checkpoint for 6A: 65°.
6B	Isosceles exterior Similar practice	Vertex exterior angle is 100°. Find each base angle.	Show answer + formula + trick 50°. Interior vertex=80°, remaining 100°, each base=50°. Formula / rule Exterior + adjacent interior =180°; then use triangle total 180° and equal base angles. Applied to 6B: the worked result is 50°. Why it works here Interior vertex=80°, remaining 100°, each base=50°. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Isosceles exterior. Checkpoint for 6B: 50°.
7A	Shape property identification	Which quadrilateral always has exactly one pair of parallel sides in the usual UK definition?	Show answer + formula + trick Trapezium. A trapezium has one pair of parallel sides. Formula / rule Identify the quadrilateral from properties that must always hold, not from appearance. Applied to 7A: the worked result is Trapezium. Why it works here A trapezium has one pair of parallel sides. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Shape property identification. Checkpoint for 7A: Trapezium.
7B	Shape property identification Similar practice	Which quadrilateral has four equal sides but need not have four right angles?	Show answer + formula + trick Rhombus. A rhombus has four equal sides. Formula / rule Identify the quadrilateral from properties that must always hold, not from appearance. Applied to 7B: the worked result is Rhombus. Why it works here A rhombus has four equal sides. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Shape property identification. Checkpoint for 7B: Rhombus.
8A	Diagonal properties	Which quadrilateral has diagonals that are equal and bisect each other but need not meet at right angles?	Show answer + formula + trick Rectangle. Rectangle diagonals are equal and bisect each other. Formula / rule Identify the quadrilateral from properties that must always hold, not from appearance. Applied to 8A: the worked result is Rectangle. Why it works here Rectangle diagonals are equal and bisect each other. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Diagonal properties. Checkpoint for 8A: Rectangle.
8B	Diagonal properties Similar practice	Which quadrilateral has diagonals that cross at right angles and bisect each other but need not be equal?	Show answer + formula + trick Rhombus. Rhombus diagonals are perpendicular bisectors. Formula / rule Identify the quadrilateral from properties that must always hold, not from appearance. Applied to 8B: the worked result is Rhombus. Why it works here Rhombus diagonals are perpendicular bisectors. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Diagonal properties. Checkpoint for 8B: Rhombus.
9A	Symmetry properties	A regular pentagon has how many lines of symmetry?	Show answer + formula + trick 5. A regular n-gon has n lines of symmetry. Formula / rule For this Symmetry properties item, use the relationship demonstrated in the worked solution and verify it with an inverse operation or estimate. Applied to 9A: the worked result is 5. Why it works here A regular n-gon has n lines of symmetry. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Symmetry properties. Checkpoint for 9A: 5.
9B	Symmetry properties Similar practice	A regular octagon has rotational symmetry of what order?	Show answer + formula + trick 8. A regular n-gon has rotational symmetry order n. Formula / rule For this Symmetry properties item, use the relationship demonstrated in the worked solution and verify it with an inverse operation or estimate. Applied to 9B: the worked result is 8. Why it works here A regular n-gon has rotational symmetry order n. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Symmetry properties. Checkpoint for 9B: 8.
10A	Angle chase multi-step	A triangle has one angle 52°. The other two are equal. Find each equal angle.	Show answer + formula + trick 64°. 180-52=128; 128÷2=64. Formula / rule Use the relevant total (180° or 360°), subtract known angles, then apply any equality condition. Applied to 10A: the worked result is 64°. Why it works here 180-52=128; 128÷2=64. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 10A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 10A: 64°.
10B	Angle chase multi-step Similar practice	A quadrilateral has two right angles and the other two angles are equal. Find each equal angle.	Show answer + formula + trick 90°. 360-180=180; 180÷2=90. Formula / rule Use the relevant total (180° or 360°), subtract known angles, then apply any equality condition. Applied to 10B: the worked result is 90°. Why it works here 360-180=180; 180÷2=90. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 10B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 10B: 90°.

27. Tiffin Focus: Perimeter, Area & Volume Reverse Problems

#	Question type	Example	Worked solution
1A	Rectangle missing side from perimeter	A rectangle has perimeter 46 cm and length 15 cm. Find width.	Show answer + formula + trick 8 cm. 2(L+W)=46, so L+W=23; W=8. Formula / rule From P=2(L+W): missing side = P/2 − known side. Applied to 1A: the worked result is 8 cm. Why it works here 2(L+W)=46, so L+W=23; W=8. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 1A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 1A: 8 cm.
1B	Rectangle missing side from perimeter Similar practice	Perimeter 62 cm, length 19 cm. Width?	Show answer + formula + trick 12 cm. L+W=31, so W=12. Formula / rule From P=2(L+W): missing side = P/2 − known side. Applied to 1B: the worked result is 12 cm. Why it works here L+W=31, so W=12. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 1B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 1B: 12 cm.
2A	Rectangle missing side from area	A rectangle has area 96 cm² and length 12 cm. Find width.	Show answer + formula + trick 8 cm. 96÷12=8. Formula / rule From A=L×W: missing side = area ÷ known side. Applied to 2A: the worked result is 8 cm. Why it works here 96÷12=8. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 2A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 2A: 8 cm.
2B	Rectangle missing side from area Similar practice	Area 135 cm², length 15 cm. Width?	Show answer + formula + trick 9 cm. 135÷15=9. Formula / rule From A=L×W: missing side = area ÷ known side. Applied to 2B: the worked result is 9 cm. Why it works here 135÷15=9. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 2B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 2B: 9 cm.
3A	Triangle missing height	A triangle has area 54 cm² and base 12 cm. Find perpendicular height.	Show answer + formula + trick 9 cm. 54=1/2×12×h=6h, so h=9. Formula / rule From A=½bh: height h=2A÷b. Applied to 3A: the worked result is 9 cm. Why it works here 54=1/2×12×h=6h, so h=9. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 3A, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 3A: 9 cm.
3B	Triangle missing height Similar practice	Area 70 cm², base 14 cm. Height?	Show answer + formula + trick 10 cm. 70=7h, so h=10. Formula / rule From A=½bh: height h=2A÷b. Applied to 3B: the worked result is 10 cm. Why it works here 70=7h, so h=10. Tip / trick Write the governing angle total/equality next to the diagram first, then subtract or share. Never trust the drawing's appearance. For 3B, re-check the exact numbers in this row before entering the answer. Remember State the angle fact first. Checkpoint for 3B: 10 cm.
4A	Composite L-shape area	A 12×10 rectangle has a 4×3 rectangle removed. Remaining area?	Show answer + formula + trick 108 cm². 120-12=108. Formula / rule Composite area = add non-overlapping parts or subtract cut-outs from a larger simple shape. Applied to 4A: the worked result is 108 cm². Why it works here 120-12=108. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 4A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 4A: 108 cm².
4B	Composite L-shape area Similar practice	A 15×8 rectangle has a 5×2 rectangle removed. Remaining area?	Show answer + formula + trick 110 cm². 120-10=110. Formula / rule Composite area = add non-overlapping parts or subtract cut-outs from a larger simple shape. Applied to 4B: the worked result is 110 cm². Why it works here 120-10=110. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 4B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 4B: 110 cm².
5A	Composite perimeter	A 10 cm by 6 cm rectangle has a 3 cm by 2 cm corner cut from one corner. What is the perimeter of the resulting L-shape?	Show answer + formula + trick 32 cm. The removed corner replaces two outer lengths with equal inner lengths, so perimeter remains 2(10+6)=32. Formula / rule Perimeter = sum of every outer boundary segment exactly once. Applied to 5A: the worked result is 32 cm. Why it works here The removed corner replaces two outer lengths with equal inner lengths, so perimeter remains 2(10+6)=32. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 5A, re-check the exact numbers in this row before entering the answer. Remember Perimeter traces the outside boundary. Checkpoint for 5A: 32 cm.
5B	Composite perimeter Similar practice	A 14 cm by 9 cm rectangle has a 4 cm by 3 cm corner cut out. Perimeter?	Show answer + formula + trick 46 cm. Perimeter remains 2(14+9)=46. Formula / rule Perimeter = sum of every outer boundary segment exactly once. Applied to 5B: the worked result is 46 cm. Why it works here Perimeter remains 2(14+9)=46. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 5B, re-check the exact numbers in this row before entering the answer. Remember Perimeter traces the outside boundary. Checkpoint for 5B: 46 cm.
6A	Cuboid missing dimension	Volume=360 cm³, length=12 cm, width=5 cm. Height?	Show answer + formula + trick 6 cm. h=360÷(12×5)=6. Formula / rule From V=LWH: missing dimension = V ÷ product of the other two dimensions. Applied to 6A: the worked result is 6 cm. Why it works here h=360÷(12×5)=6. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Cuboid missing dimension. Checkpoint for 6A: 6 cm.
6B	Cuboid missing dimension Similar practice	Volume=504 cm³, length=9 cm, width=7 cm. Height?	Show answer + formula + trick 8 cm. 504÷63=8. Formula / rule From V=LWH: missing dimension = V ÷ product of the other two dimensions. Applied to 6B: the worked result is 8 cm. Why it works here 504÷63=8. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Cuboid missing dimension. Checkpoint for 6B: 8 cm.
7A	Cube side from volume	A cube has volume 512 cm³. Side length?	Show answer + formula + trick 8 cm. 8³=512. Formula / rule Cube side s = cube root of volume. Applied to 7A: the worked result is 8 cm. Why it works here 8³=512. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 7A, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 7A: 8 cm.
7B	Cube side from volume Similar practice	A cube has volume 729 cm³. Side length?	Show answer + formula + trick 9 cm. 9³=729. Formula / rule Cube side s = cube root of volume. Applied to 7B: the worked result is 9 cm. Why it works here 9³=729. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 7B, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 7B: 9 cm.
8A	Area scaling	A rectangle's length and width are both doubled. By what factor does its area change?	Show answer + formula + trick 4. Area scale factor=2×2=4. Formula / rule If lengths scale by k, areas scale by k². Applied to 8A: the worked result is 4. Why it works here Area scale factor=2×2=4. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 8A, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 8A: 4.
8B	Area scaling Similar practice	A square's side is tripled. By what factor does area change?	Show answer + formula + trick 9. 3²=9. Formula / rule If lengths scale by k, areas scale by k². Applied to 8B: the worked result is 9. Why it works here 3²=9. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 8B, re-check the exact numbers in this row before entering the answer. Remember Area uses square units. Checkpoint for 8B: 9.
9A	Volume scaling	A cube's side length doubles. By what factor does volume change?	Show answer + formula + trick 8. 2³=8. Formula / rule If all lengths scale by k, volume scales by k³. Applied to 9A: the worked result is 8. Why it works here 2³=8. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 9A, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 9A: 8.
9B	Volume scaling Similar practice	All dimensions of a cuboid are tripled. Volume factor?	Show answer + formula + trick 27. 3³=27. Formula / rule If all lengths scale by k, volume scales by k³. Applied to 9B: the worked result is 27. Why it works here 3³=27. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 9B, re-check the exact numbers in this row before entering the answer. Remember Volume uses cubic units. Checkpoint for 9B: 27.
10A	Perimeter comparison	A square and a rectangle both have perimeter 40 cm. The square side is 10 cm. The rectangle length is 14 cm. Find its width.	Show answer + formula + trick 6 cm. L+W=20, so W=6. Formula / rule Find the common perimeter, then use the relevant perimeter formula for the unknown shape. Applied to 10A: the worked result is 6 cm. Why it works here L+W=20, so W=6. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 10A, re-check the exact numbers in this row before entering the answer. Remember Perimeter traces the outside boundary. Checkpoint for 10A: 6 cm.
10B	Perimeter comparison Similar practice	A rectangle has the same perimeter as a square of side 12 cm. If rectangle length=16 cm, width?	Show answer + formula + trick 8 cm. Square perimeter=48; rectangle L+W=24; W=8. Formula / rule Find the common perimeter, then use the relevant perimeter formula for the unknown shape. Applied to 10B: the worked result is 8 cm. Why it works here Square perimeter=48; rectangle L+W=24; W=8. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 10B, re-check the exact numbers in this row before entering the answer. Remember Perimeter traces the outside boundary. Checkpoint for 10B: 8 cm.

28. Tiffin Focus: Position, Direction, Reflection & Rotation

#	Question type	Example	Worked solution
1A	Coordinate translation	Translate (−2,5) by vector (6,−3).	Show answer + formula + trick (4,2). Add 6 to x and subtract 3 from y. Formula / rule Translation by vector (a,b): (x,y)→(x+a,y+b). Applied to 1A: the worked result is (4,2). Why it works here Add 6 to x and subtract 3 from y. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 1A, re-check the exact numbers in this row before entering the answer. Remember A translation slides without turning. Checkpoint for 1A: (4,2).
1B	Coordinate translation Similar practice	Translate (4,−1) by vector (−7,5).	Show answer + formula + trick (−3,4). Subtract 7 from x and add 5 to y. Formula / rule Translation by vector (a,b): (x,y)→(x+a,y+b). Applied to 1B: the worked result is (−3,4). Why it works here Subtract 7 from x and add 5 to y. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 1B, re-check the exact numbers in this row before entering the answer. Remember A translation slides without turning. Checkpoint for 1B: (−3,4).
2A	Reflection in x-axis	Reflect (−6,4) in the x-axis.	Show answer + formula + trick (−6,−4). x stays, y changes sign. Formula / rule Reflection in x-axis: (x,y)→(x,−y). Applied to 2A: the worked result is (−6,−4). Why it works here x stays, y changes sign. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 2A, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 2A: (−6,−4).
2B	Reflection in x-axis Similar practice	Reflect (3,−7) in the x-axis.	Show answer + formula + trick (3,7). x stays, y changes sign. Formula / rule Reflection in x-axis: (x,y)→(x,−y). Applied to 2B: the worked result is (3,7). Why it works here x stays, y changes sign. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 2B, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 2B: (3,7).
3A	Reflection in y-axis	Reflect (−5,8) in the y-axis.	Show answer + formula + trick (5,8). x changes sign, y stays. Formula / rule Reflection in y-axis: (x,y)→(−x,y). Applied to 3A: the worked result is (5,8). Why it works here x changes sign, y stays. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 3A, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 3A: (5,8).
3B	Reflection in y-axis Similar practice	Reflect (7,−2) in the y-axis.	Show answer + formula + trick (−7,−2). x changes sign, y stays. Formula / rule Reflection in y-axis: (x,y)→(−x,y). Applied to 3B: the worked result is (−7,−2). Why it works here x changes sign, y stays. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 3B, re-check the exact numbers in this row before entering the answer. Remember A reflection keeps equal distance from the mirror line. Checkpoint for 3B: (−7,−2).
4A	180° rotation about origin	Rotate (3,−4) by 180° about the origin.	Show answer + formula + trick (−3,4). Both coordinate signs change. Formula / rule 180° rotation about origin: (x,y)→(−x,−y). Applied to 4A: the worked result is (−3,4). Why it works here Both coordinate signs change. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 4A, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 4A: (−3,4).
4B	180° rotation about origin Similar practice	Rotate (−6,2) by 180° about the origin.	Show answer + formula + trick (6,−2). Both signs change. Formula / rule 180° rotation about origin: (x,y)→(−x,−y). Applied to 4B: the worked result is (6,−2). Why it works here Both signs change. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 4B, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 4B: (6,−2).
5A	90° direction reasoning	You face north and turn 90° clockwise. Which direction now?	Show answer + formula + trick East. Clockwise quarter-turn from north points east. Formula / rule A quarter-turn is 90°; track clockwise/anticlockwise direction carefully. Applied to 5A: the worked result is East. Why it works here Clockwise quarter-turn from north points east. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: 90° direction reasoning. Checkpoint for 5A: East.
5B	90° direction reasoning Similar practice	You face west and turn 90° anticlockwise. Which direction now?	Show answer + formula + trick South. Anticlockwise from west points south. Formula / rule A quarter-turn is 90°; track clockwise/anticlockwise direction carefully. Applied to 5B: the worked result is South. Why it works here Anticlockwise from west points south. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: 90° direction reasoning. Checkpoint for 5B: South.
6A	Three-quarter turn	You face south and turn 270° clockwise. Which direction now?	Show answer + formula + trick East. 270° clockwise equals 90° anticlockwise; south→east. Formula / rule 270° one way is equivalent to 90° the opposite way. Applied to 6A: the worked result is East. Why it works here 270° clockwise equals 90° anticlockwise; south→east. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Three-quarter turn. Checkpoint for 6A: East.
6B	Three-quarter turn Similar practice	You face east and turn 270° anticlockwise. Direction?	Show answer + formula + trick South. 270° anticlockwise equals 90° clockwise; east→south. Formula / rule 270° one way is equivalent to 90° the opposite way. Applied to 6B: the worked result is South. Why it works here 270° anticlockwise equals 90° clockwise; east→south. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Three-quarter turn. Checkpoint for 6B: South.
7A	Grid displacement	Move 5 right, 3 up, then 2 left from (1,1). Final coordinate?	Show answer + formula + trick (4,4). x:1+5−2=4; y:1+3=4. Formula / rule Combine horizontal movement in x and vertical movement in y; opposite directions cancel. Applied to 7A: the worked result is (4,4). Why it works here x:1+5−2=4; y:1+3=4. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Grid displacement. Checkpoint for 7A: (4,4).
7B	Grid displacement Similar practice	Move 4 left, 6 down, then 7 right from (2,5). Final coordinate?	Show answer + formula + trick (5,−1). x:2−4+7=5; y:5−6=−1. Formula / rule Combine horizontal movement in x and vertical movement in y; opposite directions cancel. Applied to 7B: the worked result is (5,−1). Why it works here x:2−4+7=5; y:5−6=−1. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 7B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Grid displacement. Checkpoint for 7B: (5,−1).
8A	Route and turn	Walk north 4, east 3, south 4. Where are you relative to start?	Show answer + formula + trick 3 units east. North and south cancel. Formula / rule Combine horizontal movement in x and vertical movement in y; opposite directions cancel. Applied to 8A: the worked result is 3 units east. Why it works here North and south cancel. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Route and turn. Checkpoint for 8A: 3 units east.
8B	Route and turn Similar practice	Walk west 5, north 2, east 8. Relative position?	Show answer + formula + trick 3 east and 2 north. Horizontal net=3 east; vertical=2 north. Formula / rule Combine horizontal movement in x and vertical movement in y; opposite directions cancel. Applied to 8B: the worked result is 3 east and 2 north. Why it works here Horizontal net=3 east; vertical=2 north. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Route and turn. Checkpoint for 8B: 3 east and 2 north.
9A	Symmetry count	How many lines of symmetry does a regular hexagon have?	Show answer + formula + trick 6. A regular n-gon has n lines of symmetry. Formula / rule A line of symmetry divides a shape into mirror-image halves. Applied to 9A: the worked result is 6. Why it works here A regular n-gon has n lines of symmetry. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Symmetry count. Checkpoint for 9A: 6.
9B	Symmetry count Similar practice	How many lines of symmetry does a non-square rectangle have?	Show answer + formula + trick 2. Horizontal and vertical lines through the centre. Formula / rule A line of symmetry divides a shape into mirror-image halves. Applied to 9B: the worked result is 2. Why it works here Horizontal and vertical lines through the centre. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Symmetry count. Checkpoint for 9B: 2.
10A	Rotation order	What is the order of rotational symmetry of a square?	Show answer + formula + trick 4. It matches at 90°,180°,270°,360°. Formula / rule Rotational symmetry order = number of matches in one full 360° turn. Applied to 10A: the worked result is 4. Why it works here It matches at 90°,180°,270°,360°. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 10A, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 10A: 4.
10B	Rotation order Similar practice	What is the order of rotational symmetry of an equilateral triangle?	Show answer + formula + trick 3. It matches at 120°,240°,360°. Formula / rule Rotational symmetry order = number of matches in one full 360° turn. Applied to 10B: the worked result is 3. Why it works here It matches at 120°,240°,360°. Tip / trick Track one feature or coordinate at a time. For transformations, keep the centre/axis fixed and check distance from it. For 10B, re-check the exact numbers in this row before entering the answer. Remember A rotation keeps equal distance from the centre. Checkpoint for 10B: 3.

29. Tiffin Focus: Statistics, Tables & Graph Interpretation

#	Question type	Example	Worked solution
1A	Mean from table	Values 2,3,4 have frequencies 3,4,3. Find mean.	Show answer + formula + trick 3. Weighted total=2×3+3×4+4×3=30; frequency total=10; mean=3. Formula / rule Mean from frequency table = Σ(value×frequency) ÷ Σfrequency. Applied to 1A: the worked result is 3. Why it works here Weighted total=2×3+3×4+4×3=30; frequency total=10; mean=3. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 1A, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 1A: 3.
1B	Mean from table Similar practice	Values 1,2,5 have frequencies 2,5,3. Find mean.	Show answer + formula + trick 2.7. Weighted total=2+10+15=27; total=10. Formula / rule Mean from frequency table = Σ(value×frequency) ÷ Σfrequency. Applied to 1B: the worked result is 2.7. Why it works here Weighted total=2+10+15=27; total=10. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 1B, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 1B: 2.7.
2A	Missing frequency	Scores 1,2,3 have frequencies 4, x, 5. Total frequency is 15. Find x.	Show answer + formula + trick 6. 4+x+5=15, so x=6. Formula / rule Missing amount = required total − sum of known amounts. Applied to 2A: the worked result is 6. Why it works here 4+x+5=15, so x=6. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing frequency. Checkpoint for 2A: 6.
2B	Missing frequency Similar practice	Categories A,B,C have frequencies 7,x,9. Total=24. Find x.	Show answer + formula + trick 8. 7+x+9=24, so x=8. Formula / rule Missing amount = required total − sum of known amounts. Applied to 2B: the worked result is 8. Why it works here 7+x+9=24, so x=8. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Missing frequency. Checkpoint for 2B: 8.
3A	Mean reverse	The mean of 8 values is 12.5. What is their total?	Show answer + formula + trick 100. 12.5×8=100. Formula / rule Total = mean × number of values. Applied to 3A: the worked result is 100. Why it works here 12.5×8=100. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 3A, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 3A: 100.
3B	Mean reverse Similar practice	Mean of 6 values is 14. What is total?	Show answer + formula + trick 84. 14×6=84. Formula / rule Total = mean × number of values. Applied to 3B: the worked result is 84. Why it works here 14×6=84. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 3B, re-check the exact numbers in this row before entering the answer. Remember Mean needs the total. Checkpoint for 3B: 84.
4A	Pie chart count	A 90° sector represents 18 pupils. How many pupils in the whole group?	Show answer + formula + trick 72. 90° is 1/4 of 360°, so whole=18×4=72. Formula / rule Whole count = sector count ÷ (sector angle/360). Applied to 4A: the worked result is 72. Why it works here 90° is 1/4 of 360°, so whole=18×4=72. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Pie chart count. Checkpoint for 4A: 72.
4B	Pie chart count Similar practice	A 72° sector represents 10 pupils. Whole group?	Show answer + formula + trick 50. 72° is 1/5 of 360°, so total=50. Formula / rule Whole count = sector count ÷ (sector angle/360). Applied to 4B: the worked result is 50. Why it works here 72° is 1/5 of 360°, so total=50. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Pie chart count. Checkpoint for 4B: 50.
5A	Pie chart sector from count	12 of 48 pupils choose art. What sector angle?	Show answer + formula + trick 90°. 12/48=1/4; 1/4×360=90°. Formula / rule Sector angle = part/whole ×360°. Applied to 5A: the worked result is 90°. Why it works here 12/48=1/4; 1/4×360=90°. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 5A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Pie chart sector from count. Checkpoint for 5A: 90°.
5B	Pie chart sector from count Similar practice	15 of 60 choose music. Sector angle?	Show answer + formula + trick 90°. 15/60=1/4; ×360=90°. Formula / rule Sector angle = part/whole ×360°. Applied to 5B: the worked result is 90°. Why it works here 15/60=1/4; ×360=90°. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 5B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Pie chart sector from count. Checkpoint for 5B: 90°.
6A	Bar chart scale trap	A bar reaches the third small interval after 40. Each interval is 5. What value?	Show answer + formula + trick 55. 40+3×5=55. Formula / rule Minor interval value = difference between labelled values ÷ number of equal intervals. Applied to 6A: the worked result is 55. Why it works here 40+3×5=55. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 6A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Bar chart scale trap. Checkpoint for 6A: 55.
6B	Bar chart scale trap Similar practice	A bar reaches two small intervals after 70. Each interval is 4. Value?	Show answer + formula + trick 78. 70+2×4=78. Formula / rule Minor interval value = difference between labelled values ÷ number of equal intervals. Applied to 6B: the worked result is 78. Why it works here 70+2×4=78. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 6B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Bar chart scale trap. Checkpoint for 6B: 78.
7A	Percentage from table	Out of 80 pupils, 28 choose option A. What percentage?	Show answer + formula + trick 35%. 28/80×100=35%. Formula / rule Percentage = part ÷ whole ×100%. Applied to 7A: the worked result is 35%. Why it works here 28/80×100=35%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 7A: 35%.
7B	Percentage from table Similar practice	Out of 120 people, 42 choose B. Percentage?	Show answer + formula + trick 35%. 42/120×100=35%. Formula / rule Percentage = part ÷ whole ×100%. Applied to 7B: the worked result is 35%. Why it works here 42/120×100=35%. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 7B: 35%.
8A	Median with frequency	Values 1,2,3 occur 2,5,3 times. What is median?	Show answer + formula + trick 2. There are 10 values; 5th and 6th values are both 2. Formula / rule Use cumulative frequencies to locate the middle position(s). Applied to 8A: the worked result is 2. Why it works here There are 10 values; 5th and 6th values are both 2. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 8A, re-check the exact numbers in this row before entering the answer. Remember Order the data first. Checkpoint for 8A: 2.
8B	Median with frequency Similar practice	Values 4,5,6 occur 3,4,3 times. Median?	Show answer + formula + trick 5. 10 values; 5th and 6th are 5. Formula / rule Use cumulative frequencies to locate the middle position(s). Applied to 8B: the worked result is 5. Why it works here 10 values; 5th and 6th are 5. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 8B, re-check the exact numbers in this row before entering the answer. Remember Order the data first. Checkpoint for 8B: 5.
9A	Range after change	A data set has minimum 12 and maximum 47. If maximum increases to 52, new range?	Show answer + formula + trick 40. 52−12=40. Formula / rule Range = maximum − minimum. Applied to 9A: the worked result is 40. Why it works here 52−12=40. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 9A, re-check the exact numbers in this row before entering the answer. Remember Range = max − min. Checkpoint for 9A: 40.
9B	Range after change Similar practice	Minimum is 8, maximum 33. Minimum changes to 5. New range?	Show answer + formula + trick 28. 33−5=28. Formula / rule Range = maximum − minimum. Applied to 9B: the worked result is 28. Why it works here 33−5=28. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 9B, re-check the exact numbers in this row before entering the answer. Remember Range = max − min. Checkpoint for 9B: 28.
10A	Interpreting misleading graph	Two bars are 52 and 55, but the axis starts at 50. Is the absolute difference 3 or 5?	Show answer + formula + trick 3. Axis truncation can exaggerate visual difference, but arithmetic difference is 55−52=3. Formula / rule Use labelled numerical values, not apparent bar height, especially when the axis is truncated. Applied to 10A: the worked result is 3. Why it works here Axis truncation can exaggerate visual difference, but arithmetic difference is 55−52=3. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Interpreting misleading graph. Checkpoint for 10A: 3.
10B	Interpreting misleading graph Similar practice	Bars are 101 and 104 with axis starting at 100. Absolute difference?	Show answer + formula + trick 3. 104−101=3. Formula / rule Use labelled numerical values, not apparent bar height, especially when the axis is truncated. Applied to 10B: the worked result is 3. Why it works here 104−101=3. Tip / trick Read the scale and identify the statistic requested before calculating; do not use visual appearance as the value. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Interpreting misleading graph. Checkpoint for 10B: 3.

30. Tiffin Focus: Algebra, Sequences & Function Machines

#	Question type	Example	Worked solution
1A	Two-step function	A machine multiplies by 3 then adds 5. Input 7. Output?	Show answer + formula + trick 26. 7×3+5=26. Formula / rule Apply function-machine operations in the stated order. Applied to 1A: the worked result is 26. Why it works here 7×3+5=26. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 1A, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 1A: 26.
1B	Two-step function Similar practice	A machine multiplies by 4 then subtracts 6. Input 9. Output?	Show answer + formula + trick 30. 9×4−6=30. Formula / rule Apply function-machine operations in the stated order. Applied to 1B: the worked result is 30. Why it works here 9×4−6=30. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 1B, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 1B: 30.
2A	Reverse function	A machine multiplies by 6 then adds 4. Output is 58. Input?	Show answer + formula + trick 9. 58−4=54; 54÷6=9. Formula / rule Start from the output and undo each operation in reverse order. Applied to 2A: the worked result is 9. Why it works here 58−4=54; 54÷6=9. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 2A, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 2A: 9.
2B	Reverse function Similar practice	A machine doubles then subtracts 7. Output 19. Input?	Show answer + formula + trick 13. 19+7=26; 26÷2=13. Formula / rule Start from the output and undo each operation in reverse order. Applied to 2B: the worked result is 13. Why it works here 19+7=26; 26÷2=13. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 2B, re-check the exact numbers in this row before entering the answer. Remember Undo in reverse order and check by substitution. Checkpoint for 2B: 13.
3A	Two unknown-style relationship	x+y=20 and x is 4 more than y. Find x.	Show answer + formula + trick 12. Let y=8, x=12; or 2y+4=20. Formula / rule Use the total relationship and the stated difference to form and solve a simple equation. Applied to 3A: the worked result is 12. Why it works here Let y=8, x=12; or 2y+4=20. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Two unknown-style relationship. Checkpoint for 3A: 12.
3B	Two unknown-style relationship Similar practice	a+b=34 and a is 6 more than b. Find b.	Show answer + formula + trick 14. 2b+6=34, so b=14. Formula / rule Use the total relationship and the stated difference to form and solve a simple equation. Applied to 3B: the worked result is 14. Why it works here 2b+6=34, so b=14. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Two unknown-style relationship. Checkpoint for 3B: 14.
4A	Nth term use	Sequence has nth term 4n−1. What is 15th term?	Show answer + formula + trick 59. 4×15−1=59. Formula / rule For a linear sequence, nth term = common difference × n + adjustment. Applied to 4A: the worked result is 59. Why it works here 4×15−1=59. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 4A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 4A: 59.
4B	Nth term use Similar practice	Nth term 6n+2. What is 12th term?	Show answer + formula + trick 74. 6×12+2=74. Formula / rule For a linear sequence, nth term = common difference × n + adjustment. Applied to 4B: the worked result is 74. Why it works here 6×12+2=74. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 4B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 4B: 74.
5A	Find position from nth term	In sequence 5n+1, which term equals 46?	Show answer + formula + trick 9th. 5n+1=46, so 5n=45, n=9. Formula / rule Set the nth-term formula equal to the target value and solve for n. Applied to 5A: the worked result is 9th. Why it works here 5n+1=46, so 5n=45, n=9. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 5A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 5A: 9th.
5B	Find position from nth term Similar practice	In sequence 3n+4, which term equals 31?	Show answer + formula + trick 9th. 3n+4=31; n=9. Formula / rule Set the nth-term formula equal to the target value and solve for n. Applied to 5B: the worked result is 9th. Why it works here 3n+4=31; n=9. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 5B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 5B: 9th.
6A	Second-difference pattern	Continue 2,5,10,17,26,...	Show answer + formula + trick 37. Differences 3,5,7,9; next difference 11; 26+11=37. Formula / rule Write first differences; if they change regularly, inspect the second differences. Applied to 6A: the worked result is 37. Why it works here Differences 3,5,7,9; next difference 11; 26+11=37. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 6A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 6A: 37.
6B	Second-difference pattern Similar practice	Continue 1,4,9,16,25,...	Show answer + formula + trick 36. Differences 3,5,7,9; next 11; 25+11=36. Formula / rule Write first differences; if they change regularly, inspect the second differences. Applied to 6B: the worked result is 36. Why it works here Differences 3,5,7,9; next 11; 25+11=36. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 6B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 6B: 36.
7A	Alternating sequence	Continue 4,12,9,27,24,72,...	Show answer + formula + trick 69. Pattern ×3,−3 repeated. Formula / rule Look for two repeating operations or separate patterns in odd/even positions. Applied to 7A: the worked result is 69. Why it works here Pattern ×3,−3 repeated. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 7A, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 7A: 69.
7B	Alternating sequence Similar practice	Continue 5,10,8,16,14,28,...	Show answer + formula + trick 26. Pattern ×2,−2 repeated. Formula / rule Look for two repeating operations or separate patterns in odd/even positions. Applied to 7B: the worked result is 26. Why it works here Pattern ×2,−2 repeated. Tip / trick Keep each algebraic step on a separate line and substitute your final value back to check it. For 7B, re-check the exact numbers in this row before entering the answer. Remember Write the differences/rule before guessing. Checkpoint for 7B: 26.
8A	Fractional sequence	Continue 1/2, 3/4, 1, 1 1/4, ...	Show answer + formula + trick 1 1/2. Add 1/4 each time. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 8A: the worked result is 1 1/2. Why it works here Add 1/4 each time. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8A, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 8A: 1 1/2.
8B	Fractional sequence Similar practice	Continue 2/3, 1, 1 1/3, 1 2/3,...	Show answer + formula + trick 2. Add 1/3 each time. Formula / rule Identify the step rule between terms before extending or reversing the sequence. Applied to 8B: the worked result is 2. Why it works here Add 1/3 each time. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 8B, re-check the exact numbers in this row before entering the answer. Remember Keep track of the whole and the denominator. Checkpoint for 8B: 2.
9A	Expression from words	Write an expression for 'five more than three times n'.	Show answer + formula + trick 3n+5. Three times n is 3n; five more gives +5. Formula / rule Translate the wording into algebra in the same logical order. Applied to 9A: the worked result is 3n+5. Why it works here Three times n is 3n; five more gives +5. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Expression from words. Checkpoint for 9A: 3n+5.
9B	Expression from words Similar practice	Write an expression for 'seven less than four times x'.	Show answer + formula + trick 4x−7. Four times x is 4x; seven less gives −7. Formula / rule Translate the wording into algebra in the same logical order. Applied to 9B: the worked result is 4x−7. Why it works here Four times x is 4x; seven less gives −7. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Expression from words. Checkpoint for 9B: 4x−7.
10A	Equation from perimeter	A rectangle has length x+2 and width x. Perimeter is 28. Find x.	Show answer + formula + trick 6. 2[(x+2)+x]=28; 4x+4=28; x=6. Formula / rule Substitute algebraic side lengths into P=2(L+W), then solve the resulting equation. Applied to 10A: the worked result is 6. Why it works here 2[(x+2)+x]=28; 4x+4=28; x=6. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 10A, re-check the exact numbers in this row before entering the answer. Remember Perimeter traces the outside boundary. Checkpoint for 10A: 6.
10B	Equation from perimeter Similar practice	A rectangle has length x+3 and width x. Perimeter 34. Find x.	Show answer + formula + trick 7. 2[(x+3)+x]=34; 4x+6=34; x=7. Formula / rule Substitute algebraic side lengths into P=2(L+W), then solve the resulting equation. Applied to 10B: the worked result is 7. Why it works here 2[(x+3)+x]=34; 4x+6=34; x=7. Tip / trick Write the formula with units before substituting values. Check whether the answer needs linear, square or cubic units. For 10B, re-check the exact numbers in this row before entering the answer. Remember Perimeter traces the outside boundary. Checkpoint for 10B: 7.

31. Tiffin Focus: Multi-Step Problem Solving & Answer-Entry Traps

#	Question type	Example	Worked solution
1A	Round up containers	A hall needs seats for 286 people. Each row holds 24. Minimum number of rows?	Show answer + formula + trick 12. 286÷24=11 remainder 22, so 12 rows are required. Formula / rule If every item/person must fit and division leaves a remainder, required containers = quotient + 1. Applied to 1A: the worked result is 12. Why it works here 286÷24=11 remainder 22, so 12 rows are required. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Round up containers. Checkpoint for 1A: 12.
1B	Round up containers Similar practice	523 books fit 40 per box. Minimum boxes?	Show answer + formula + trick 14. 523÷40=13 remainder 3, so 14 boxes. Formula / rule If every item/person must fit and division leaves a remainder, required containers = quotient + 1. Applied to 1B: the worked result is 14. Why it works here 523÷40=13 remainder 3, so 14 boxes. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 1B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Round up containers. Checkpoint for 1B: 14.
2A	Ignore unused information	A shop opens at 9 am. A pen costs £1.80 and a notebook £3.20. Mia buys 4 pens. Cost?	Show answer + formula + trick £7.20. Only pen price and quantity matter: 4×1.80. Formula / rule Use only quantities that are mathematically linked to the question being asked. Applied to 2A: the worked result is £7.20. Why it works here Only pen price and quantity matter: 4×1.80. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Ignore unused information. Checkpoint for 2A: £7.20.
2B	Ignore unused information Similar practice	A train has 8 carriages. A ticket costs £6.50. Ben buys 3 tickets. Cost?	Show answer + formula + trick £19.50. 3×£6.50. Formula / rule Use only quantities that are mathematically linked to the question being asked. Applied to 2B: the worked result is £19.50. Why it works here 3×£6.50. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 2B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Ignore unused information. Checkpoint for 2B: £19.50.
3A	Two-stage discount and change	A £50 game is reduced by 20%. You pay with £50. Change?	Show answer + formula + trick £10. 20% of £50=£10 discount; sale price £40; change £10. Formula / rule Sale price = original×(1−discount rate); then apply any payment/change step. Applied to 3A: the worked result is £10. Why it works here 20% of £50=£10 discount; sale price £40; change £10. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 3A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Two-stage discount and change. Checkpoint for 3A: £10.
3B	Two-stage discount and change Similar practice	A £80 coat is reduced by 25%. You pay £70. Change?	Show answer + formula + trick £10. Sale price £60; change £10. Formula / rule Sale price = original×(1−discount rate); then apply any payment/change step. Applied to 3B: the worked result is £10. Why it works here Sale price £60; change £10. Tip / trick Label each amount as cost, selling price, discount, total or change so you do not subtract from the wrong quantity. For 3B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Two-stage discount and change. Checkpoint for 3B: £10.
4A	Average after adding value	Mean of 4 numbers is 12. A fifth number 22 is added. New mean?	Show answer + formula + trick 14. Original total=48; new total=70; 70÷5=14. Formula / rule Recover the old total with mean×count, add the new value, then divide by the new count. Applied to 4A: the worked result is 14. Why it works here Original total=48; new total=70; 70÷5=14. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Average after adding value. Checkpoint for 4A: 14.
4B	Average after adding value Similar practice	Mean of 5 numbers is 18. Add 30. New mean?	Show answer + formula + trick 20. Original total=90; new total=120; ÷6=20. Formula / rule Recover the old total with mean×count, add the new value, then divide by the new count. Applied to 4B: the worked result is 20. Why it works here Original total=90; new total=120; ÷6=20. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 4B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Average after adding value. Checkpoint for 4B: 20.
5A	Fraction then percentage	A box has 240 sweets. 3/8 are red. 20% of the red sweets are strawberry. How many strawberry red sweets?	Show answer + formula + trick 18. 3/8 of 240=90; 20% of 90=18. Formula / rule Complete the fraction-of-total stage first, then take the required percentage of that result. Applied to 5A: the worked result is 18. Why it works here 3/8 of 240=90; 20% of 90=18. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 5A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 5A: 18.
5B	Fraction then percentage Similar practice	A box has 320 sweets. 3/8 are red. 25% of red are strawberry. How many?	Show answer + formula + trick 30. 3/8 of 320=120; 25% of 120=30. Formula / rule Complete the fraction-of-total stage first, then take the required percentage of that result. Applied to 5B: the worked result is 30. Why it works here 3/8 of 320=120; 25% of 120=30. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 5B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 5B: 30.
6A	Unit conversion mid-problem	A ribbon 3.5 m long is cut into 25 cm pieces. How many full pieces?	Show answer + formula + trick 14. 3.5 m=350 cm; 350÷25=14. Formula / rule Convert all quantities to a common unit before dividing or multiplying. Applied to 6A: the worked result is 14. Why it works here 3.5 m=350 cm; 350÷25=14. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 6A, re-check the exact numbers in this row before entering the answer. Remember Convert before calculating. Checkpoint for 6A: 14.
6B	Unit conversion mid-problem Similar practice	A 2.4 kg bag is packed into 150 g portions. How many full portions?	Show answer + formula + trick 16. 2.4 kg=2400 g; 2400÷150=16. Formula / rule Convert all quantities to a common unit before dividing or multiplying. Applied to 6B: the worked result is 16. Why it works here 2.4 kg=2400 g; 2400÷150=16. Tip / trick Write the conversion factor first and decide whether you are moving from a larger unit to a smaller one or vice versa. For 6B, re-check the exact numbers in this row before entering the answer. Remember Convert before calculating. Checkpoint for 6B: 16.
7A	Reverse percentage multi-step	After a 20% discount, a price is £96. A £4 delivery fee is then added. Final paid amount?	Show answer + formula + trick £100. Discounted price is already £96; add £4=£100. Formula / rule Identify which stage the stated amount belongs to before applying any reverse percentage calculation. Applied to 7A: the worked result is £100. Why it works here Discounted price is already £96; add £4=£100. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7A, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 7A: £100.
7B	Reverse percentage multi-step Similar practice	After a 25% discount, a price is £75. Then £6 delivery. Final amount?	Show answer + formula + trick £81. £75+£6=£81. Formula / rule Identify which stage the stated amount belongs to before applying any reverse percentage calculation. Applied to 7B: the worked result is £81. Why it works here £75+£6=£81. Tip / trick Write one clean intermediate line before the final answer; most errors here come from applying the correct rule to the wrong base amount or denominator. For 7B, re-check the exact numbers in this row before entering the answer. Remember Percent means 'out of 100'. Checkpoint for 7B: £81.
8A	Comparison by unit price	Pack A: 6 bottles for £4.50. Pack B: 10 bottles for £7.20. Which is cheaper per bottle?	Show answer + formula + trick Pack B. A=75p each; B=72p each. Formula / rule Unit price = pack price ÷ pack quantity; compare using the same unit. Applied to 8A: the worked result is Pack B. Why it works here A=75p each; B=72p each. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Comparison by unit price. Checkpoint for 8A: Pack B.
8B	Comparison by unit price Similar practice	Pack A: 8 bars for £5.20. Pack B: 12 bars for £7.50. Cheaper per bar?	Show answer + formula + trick Pack B. A=65p; B=62.5p. Formula / rule Unit price = pack price ÷ pack quantity; compare using the same unit. Applied to 8B: the worked result is Pack B. Why it works here A=65p; B=62.5p. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 8B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Comparison by unit price. Checkpoint for 8B: Pack B.
9A	Estimate to catch error	Which is a sensible value for 49×203: 994, 9,947, 99,470 or 994,700?	Show answer + formula + trick 9,947. Estimate 50×200=10,000, so 9,947 is plausible. Formula / rule Estimate by rounding to friendly numbers, then compare the exact or proposed answer with the expected size. Applied to 9A: the worked result is 9,947. Why it works here Estimate 50×200=10,000, so 9,947 is plausible. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 9A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Estimate to catch error. Checkpoint for 9A: 9,947.
9B	Estimate to catch error Similar practice	Which is sensible for 398×21: 836, 8,358, 83,580 or 835,800?	Show answer + formula + trick 8,358. Estimate 400×20=8,000. Formula / rule Estimate by rounding to friendly numbers, then compare the exact or proposed answer with the expected size. Applied to 9B: the worked result is 8,358. Why it works here Estimate 400×20=8,000. Tip / trick Use the quick property test before doing full arithmetic; these questions are often designed to reward recognition. For 9B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Estimate to catch error. Checkpoint for 9B: 8,358.
10A	Numeric answer-sheet care	A calculation gives 7. What should be entered as the numerical answer if the question asks for the number only?	Show answer + formula + trick 7. Enter only the required number, not units or working. Formula / rule Enter the answer in exactly the requested numerical format, without adding unwanted symbols or units. Applied to 10A: the worked result is 7. Why it works here Enter only the required number, not units or working. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10A, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Numeric answer-sheet care. Checkpoint for 10A: 7.
10B	Numeric answer-sheet care Similar practice	A time answer is 7:05 am. In a four-digit 24-hour format, what should be entered?	Show answer + formula + trick 0705. 07:05 becomes digits 0,7,0,5. Formula / rule Enter the answer in exactly the requested numerical format, without adding unwanted symbols or units. Applied to 10B: the worked result is 0705. Why it works here 07:05 becomes digits 0,7,0,5. Tip / trick Estimate or reverse-check the result before moving on, especially if the answer sheet only records the final number. For 10B, re-check the exact numbers in this row before entering the answer. Remember Identify the question type first: Numeric answer-sheet care. Checkpoint for 10B: 0705.

6. Do not ignore English

The official Stage One test includes English as well as Maths, and the English paper is multiple choice. The school does not publish past papers, so these are general high-value techniques rather than claims about exact question types.

Stage One English is officially multiple choice. Focus on precise reading rather than speed alone.
For inference, choose the answer best supported by the text, not the answer that could possibly be true.
Separate the narrator's opinion from objective events.
For vocabulary in context, substitute each option into the sentence and test tone as well as dictionary meaning.
For punctuation and grammar, read the whole sentence aloud silently and identify the grammatical job of the missing word or punctuation.
If two answers look plausible, find the exact word or phrase in the passage that distinguishes them.

7. Final checklist

I know fraction, percentage and ratio conversions without hesitation.
I know all angle totals and area/volume formulas.
I can convert units without mixing linear and square units.
I check whether a word problem asks for a remainder, a rounded answer or an extra container/person/coach.
I can use estimation to reject impossible multiple-choice answers.
I move on from a stuck question and return later.
I enter answers carefully on the answer sheet.