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 + trick70,000.The 7 is in the ten-thousands column. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 1B | Place value Similar practice |
What is the value of 4 in 8,643,219? | Show answer + formula + trick40,000.The 4 is in the ten-thousands column. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 2A | Rounding | Round 4,567,891 to the nearest 10,000. | Show answer + formula + trick4,570,000.Look at the thousands digit, 7. Since it is 5 or more, round the ten-thousands digit up. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 2B | Rounding Similar practice |
Round 7,842,615 to the nearest 100,000. | Show answer + formula + trick7,800,000.Look at the ten-thousands digit, 4. Since it is less than 5, keep the hundred-thousands digit unchanged. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 3A | Highest common factor | Find the HCF of 36 and 48. | Show answer + formula + trick12.Factors common to both include 1, 2, 3, 4, 6 and 12. The greatest is 12. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 3B | Highest common factor Similar practice |
Find the HCF of 54 and 72. | Show answer + formula + trick18.Common factors include 1, 2, 3, 6, 9 and 18. The greatest is 18. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 4A | Lowest common multiple | Find the LCM of 12 and 18. | Show answer + formula + trick36.Multiples of 12 are 12, 24, 36... and multiples of 18 are 18, 36... Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 4B | Lowest common multiple Similar practice |
Find the LCM of 15 and 20. | Show answer + formula + trick60.Multiples of 15 are 15, 30, 45, 60... and multiples of 20 are 20, 40, 60... Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 5A | Prime numbers | List all prime numbers between 40 and 50. | Show answer + formula + trick41, 43, 47.A prime has exactly two factors: 1 and itself. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 5B | Prime numbers Similar practice |
List all prime numbers between 50 and 60. | Show answer + formula + trick53, 59.53 and 59 have exactly two factors: 1 and themselves. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 6A | Odd and even | Is 135 × 27 odd or even? | Show answer + formula + trickOdd.Odd × odd = odd. You do not need to calculate the product. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 6B | Odd and even Similar practice |
Is 246 × 19 odd or even? | Show answer + formula + trickEven.Even × odd = even. You do not need to calculate the product. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 7A | Negative numbers | Calculate -7 + 12 - 9. | Show answer + formula + trick-4.-7 + 12 = 5, then 5 - 9 = -4. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 7B | Negative numbers Similar practice |
Calculate -12 + 7 + 9. | Show answer + formula + trick4.-12 + 7 = -5, then -5 + 9 = 4. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 8A | Roman numerals | Write 49 in Roman numerals. | Show answer + formula + trickXLIX.40 = XL and 9 = IX, so 49 = XLIX. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 8B | Roman numerals Similar practice |
Write 74 in Roman numerals. | Show answer + formula + trickLXXIV.70 = LXX and 4 = IV, so 74 = LXXIV. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 9A | Divisibility | Without dividing, decide whether 7,362 is divisible by 9. | Show answer + formula + trickYes.Add the digits: 7 + 3 + 6 + 2 = 18. Since 18 is divisible by 9, so is 7,362. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 9B | Divisibility Similar practice |
Without dividing, decide whether 8,127 is divisible by 3. | Show answer + formula + trickYes.Add the digits: 8 + 1 + 2 + 7 = 18. Since 18 is divisible by 3, so is 8,127. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 10A | Square numbers | 144 is the square of which whole number? | Show answer + formula + trick12.12 × 12 = 144. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
| 10B | Square numbers Similar practice |
169 is the square of which whole number? | Show answer + formula + trick13.13 × 13 = 169. Formula / rule Rule / formula: Use place value, factor pairs, multiples and divisibility tests. HCF = greatest common factor; LCM = smallest common multiple. Tip / trick Quick trick: For divisibility, test the digits first before doing long division. For rounding, circle the target digit and look only at the digit immediately to its right. Remember Memory cue: 5 or more, raise the score; 4 or less, let it rest. |
2. Arithmetic Operations
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Mental multiplication | Calculate 48 × 25 mentally. | Show answer + formula + trick1,200.25 is one quarter of 100. So 48 × 25 = 48 × 100 ÷ 4 = 1,200. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 1B | Mental multiplication Similar practice |
Calculate 64 × 25 mentally. | Show answer + formula + trick1,600.25 is one quarter of 100. So 64 × 25 = 64 × 100 ÷ 4 = 1,600. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 2A | Long multiplication | Calculate 307 × 46. | Show answer + formula + trick14,122.307 × 40 = 12,280 and 307 × 6 = 1,842. Total = 14,122. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 2B | Long multiplication Similar practice |
Calculate 428 × 37. | Show answer + formula + trick15,836.428 × 30 = 12,840 and 428 × 7 = 2,996. Total = 15,836. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 3A | Division with remainder | Calculate 987 ÷ 23. | Show answer + formula + trick42 remainder 21.23 × 42 = 966, and 987 - 966 = 21. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 3B | Division with remainder Similar practice |
Calculate 1,075 ÷ 24. | Show answer + formula + trick44 remainder 19.24 × 44 = 1,056, and 1,075 - 1,056 = 19. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 4A | BODMAS | Calculate 18 - 3 × (4 + 2) + 24 ÷ 6. | Show answer + formula + trick4.Brackets first: 6. Then 3 × 6 = 18 and 24 ÷ 6 = 4. So 18 - 18 + 4 = 4. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 4B | BODMAS Similar practice |
Calculate 30 - 4 × (5 + 1) + 18 ÷ 3. | Show answer + formula + trick12.Brackets first: 6. Then 4 × 6 = 24 and 18 ÷ 3 = 6. So 30 - 24 + 6 = 12. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 5A | Estimation | Estimate 398 × 51. | Show answer + formula + trickAbout 20,000.Round to 400 × 50 = 20,000. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 5B | Estimation Similar practice |
Estimate 612 × 39. | Show answer + formula + trickAbout 24,000.Round to 600 × 40 = 24,000. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 6A | Inverse operations | A number divided by 8 gives 37. What is the number? | Show answer + formula + trick296.Reverse the division: 37 × 8 = 296. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 6B | Inverse operations Similar practice |
A number multiplied by 7 gives 364. What is the number? | Show answer + formula + trick52.Reverse the multiplication: 364 ÷ 7 = 52. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 7A | Missing number | □ × 16 = 1,248. Find □. | Show answer + formula + trick78.1,248 ÷ 16 = 78. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 7B | Missing number Similar practice |
□ × 18 = 1,674. Find □. | Show answer + formula + trick93.1,674 ÷ 18 = 93. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 8A | Efficient subtraction | Calculate 10,000 - 3,998. | Show answer + formula + trick6,002.Subtract 4,000 to get 6,000, then add back 2. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 8B | Efficient subtraction Similar practice |
Calculate 8,000 - 2,997. | Show answer + formula + trick5,003.Subtract 3,000 to get 5,000, then add back 3. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 9A | Remainder interpretation | 253 pupils travel in coaches holding 48 each. What is the minimum number of coaches? | Show answer + formula + trick6.253 ÷ 48 = 5 remainder 13. Five coaches are not enough, so 6 are needed. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 9B | Remainder interpretation Similar practice |
327 pupils travel in minibuses holding 15 each. What is the minimum number of minibuses? | Show answer + formula + trick22.327 ÷ 15 = 21 remainder 12. Twenty-one are not enough, so 22 are needed. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 10A | Multi-step arithmetic | A shop has 24 boxes of 36 pencils. It sells 275 pencils. How many remain? | Show answer + formula + trick589.24 × 36 = 864. Then 864 - 275 = 589. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
| 10B | Multi-step arithmetic Similar practice |
A warehouse has 18 crates of 42 bottles. It ships 289 bottles. How many remain? | Show answer + formula + trick467.18 × 42 = 756. Then 756 - 289 = 467. Formula / rule Rule / formula: BIDMAS/BODMAS = Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction. Tip / trick Quick trick: Estimate first. Break awkward numbers into friendly parts, for example ×25 = ×100 ÷4 and ×50 = ×100 ÷2. Remember Memory cue: Brackets first, then × and ÷, then + and −. |
3. Fractions
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Equivalent fractions | Complete: 3/5 = ?/40. | Show answer + formula + trick24/40.Multiply numerator and denominator by 8. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 1B | Equivalent fractions Similar practice |
Complete: 7/9 = ?/63. | Show answer + formula + trick49/63.Multiply numerator and denominator by 7. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 2A | Simplifying | Simplify 42/56. | Show answer + formula + trick3/4.Divide numerator and denominator by 14. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 2B | Simplifying Similar practice |
Simplify 63/84. | Show answer + formula + trick3/4.Divide numerator and denominator by 21. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 3A | Comparing | Which is larger: 5/8 or 7/12? | Show answer + formula + trick5/8.Cross-multiply: 5 × 12 = 60 and 7 × 8 = 56. Since 60 > 56, 5/8 is larger. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 3B | Comparing Similar practice |
Which is larger: 7/10 or 5/8? | Show answer + formula + trick7/10.Cross-multiply: 7 × 8 = 56 and 5 × 10 = 50. Since 56 > 50, 7/10 is larger. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 4A | Mixed to improper | Convert 3 2/5 to an improper fraction. | Show answer + formula + trick17/5.3 × 5 + 2 = 17. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 4B | Mixed to improper Similar practice |
Convert 4 3/7 to an improper fraction. | Show answer + formula + trick31/7.4 × 7 + 3 = 31. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 5A | Adding unlike fractions | Calculate 3/4 + 5/6. | Show answer + formula + trick1 7/12.Common denominator 12: 9/12 + 10/12 = 19/12 = 1 7/12. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 5B | Adding unlike fractions Similar practice |
Calculate 5/6 + 3/8. | Show answer + formula + trick1 5/24.Common denominator 24: 20/24 + 9/24 = 29/24 = 1 5/24. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 6A | Subtracting fractions | Calculate 7/8 - 1/3. | Show answer + formula + trick13/24.Common denominator 24: 21/24 - 8/24 = 13/24. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 6B | Subtracting fractions Similar practice |
Calculate 11/12 - 2/5. | Show answer + formula + trick31/60.Common denominator 60: 55/60 - 24/60 = 31/60. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 7A | Multiplying fractions | Calculate 3/5 × 10/21. | Show answer + formula + trick2/7.Cancel 10 with 5 to get 2; cancel 3 with 21 to get 1/7. Answer 2/7. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 7B | Multiplying fractions Similar practice |
Calculate 4/7 × 21/10. | Show answer + formula + trick6/5 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 Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 8A | Dividing fractions | Calculate 4/9 ÷ 2/3. | Show answer + formula + trick2/3.Keep 4/9, change ÷ to ×, flip 2/3 to 3/2: 4/9 × 3/2 = 2/3. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 8B | Dividing fractions Similar practice |
Calculate 5/8 ÷ 15/16. | Show answer + formula + trick2/3.Keep 5/8, change ÷ to ×, flip 15/16 to 16/15: 5/8 × 16/15 = 2/3. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 9A | Fraction of an amount | Find 7/12 of 360. | Show answer + formula + trick210.360 ÷ 12 = 30, then 30 × 7 = 210. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 9B | Fraction of an amount Similar practice |
Find 5/9 of 324. | Show answer + formula + trick180.324 ÷ 9 = 36, then 36 × 5 = 180. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 10A | Missing fraction | 2/3 + ? = 11/12. Find ?. | Show answer + formula + trick1/4.2/3 = 8/12. So ? = 11/12 - 8/12 = 3/12 = 1/4. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
| 10B | Missing fraction Similar practice |
3/4 + ? = 11/12. Find ?. | Show answer + formula + trick1/6.3/4 = 9/12. So ? = 2/12 = 1/6. Formula / rule Rule / formula: Add/subtract using a common denominator. Multiply tops × tops and bottoms × bottoms. Divide by keep-change-flip. Tip / trick Quick trick: Cancel common factors before multiplying. For a fraction of an amount, divide by the denominator first, then multiply by the numerator. Remember Memory cue: Add/subtract = same bottom. Multiply = straight across. Divide = keep, change, flip. |
4. Decimals
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Place value | What is the value of 6 in 14.063? | Show answer + formula + trick0.06.It is in the hundredths column. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 1B | Place value Similar practice |
What is the value of 8 in 72.184? | Show answer + formula + trick0.08.It is in the hundredths column. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 2A | Comparing decimals | Which is greater: 4.507 or 4.57? | Show answer + formula + trick4.57.Write 4.57 as 4.570. Then 4.570 > 4.507. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 2B | Comparing decimals Similar practice |
Which is greater: 6.305 or 6.35? | Show answer + formula + trick6.35.Write 6.35 as 6.350. Then 6.350 > 6.305. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 3A | Addition | Calculate 18.75 + 6.089. | Show answer + formula + trick24.839.Align decimal points: 18.750 + 6.089 = 24.839. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 3B | Addition Similar practice |
Calculate 27.406 + 8.95. | Show answer + formula + trick36.356.Align decimal points: 27.406 + 8.950 = 36.356. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 4A | Subtraction | Calculate 20 - 7.386. | Show answer + formula + trick12.614.Write 20 as 20.000, then subtract. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 4B | Subtraction Similar practice |
Calculate 50 - 18.725. | Show answer + formula + trick31.275.Write 50 as 50.000, then subtract. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 5A | Multiply by whole number | Calculate 3.75 × 24. | Show answer + formula + trick90.3.75 × 6 = 22.5, then ×4 gives 90, or use 3.75 × (20 + 4). Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 5B | Multiply by whole number Similar practice |
Calculate 4.25 × 16. | Show answer + formula + trick68.4.25 × 8 = 34, then double to get 68. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 6A | Multiply decimals | Calculate 0.6 × 0.35. | Show answer + formula + trick0.21.6 × 35 = 210. There are three decimal places in total, so 0.210 = 0.21. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 6B | Multiply decimals Similar practice |
Calculate 0.8 × 0.45. | Show answer + formula + trick0.36.8 × 45 = 360. There are three decimal places in total, so 0.360 = 0.36. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 7A | Divide decimals | Calculate 7.2 ÷ 0.3. | Show answer + formula + trick24.Multiply both numbers by 10: 72 ÷ 3 = 24. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 7B | Divide decimals Similar practice |
Calculate 8.4 ÷ 0.7. | Show answer + formula + trick12.Multiply both numbers by 10: 84 ÷ 7 = 12. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 8A | Fraction to decimal | Convert 7/20 to a decimal. | Show answer + formula + trick0.35.Multiply by 5 to get 35/100 = 0.35. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 8B | Fraction to decimal Similar practice |
Convert 9/25 to a decimal. | Show answer + formula + trick0.36.Multiply numerator and denominator by 4 to get 36/100 = 0.36. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 9A | Decimal to fraction | Convert 0.375 to a fraction in simplest form. | Show answer + formula + trick3/8.0.375 = 375/1000. Divide by 125 to get 3/8. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 9B | Decimal to fraction Similar practice |
Convert 0.625 to a fraction in simplest form. | Show answer + formula + trick5/8.0.625 = 625/1000. Divide by 125 to get 5/8. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 10A | Rounding decimals | Round 8.746 to 2 decimal places. | Show answer + formula + trick8.75.Look at the third decimal digit, 6, so 8.74 rounds up to 8.75. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
| 10B | Rounding decimals Similar practice |
Round 13.485 to 2 decimal places. | Show answer + formula + trick13.49.Look at the third decimal digit, 5, so 13.48 rounds up to 13.49. Formula / rule Rule / formula: Line up decimal points for addition/subtraction. When multiplying decimals, multiply as whole numbers then replace the total number of decimal places. Tip / trick Quick trick: Add zeros to the right when comparing, for example 4.57 = 4.570. Remember Memory cue: Decimal points must stand in a straight column. |
5. Percentages
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Fraction to percentage | Convert 3/5 to a percentage. | Show answer + formula + trick60%.3 ÷ 5 = 0.6 = 60%. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 1B | Fraction to percentage Similar practice |
Convert 7/20 to a percentage. | Show answer + formula + trick35%.7 ÷ 20 = 0.35 = 35%. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 2A | Decimal to percentage | Convert 0.275 to a percentage. | Show answer + formula + trick27.5%.Multiply by 100. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 2B | Decimal to percentage Similar practice |
Convert 0.462 to a percentage. | Show answer + formula + trick46.2%.Multiply by 100. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 3A | Percentage of amount | Find 35% of £240. | Show answer + formula + trick£84.10% = £24, 30% = £72, 5% = £12. Total £84. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 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 Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 4A | Quick 15% | Find 15% of 360. | Show answer + formula + trick54.10% = 36 and 5% = 18, total 54. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 4B | Quick 15% Similar practice |
Find 15% of 480. | Show answer + formula + trick72.10% = 48 and 5% = 24, total 72. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 5A | Percentage increase | Increase £80 by 25%. | Show answer + formula + trick£100.25% of 80 = 20. Add: 80 + 20 = 100. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷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 Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 6A | Percentage decrease | Decrease 350 by 12%. | Show answer + formula + trick308.10% = 35, 2% = 7, so 12% = 42. Then 350 - 42 = 308. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 6B | Percentage decrease Similar practice |
Decrease 500 by 18%. | Show answer + formula + trick410.10% = 50, 5% = 25, 3% = 15, so 18% = 90. Then 500 - 90 = 410. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 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 Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 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 Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 8A | Percentage change | A value rises from 50 to 65. What is the percentage increase? | Show answer + formula + trick30%.Increase = 15. 15 ÷ 50 = 0.3 = 30%. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 8B | Percentage change Similar practice |
A value falls from 80 to 68. What is the percentage decrease? | Show answer + formula + trick15%.Decrease = 12. 12 ÷ 80 = 0.15 = 15%. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 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 Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 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 Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 10A | Find the whole | 42 is 35% of a number. Find the number. | Show answer + formula + trick120.1% = 42 ÷ 35 = 1.2, so 100% = 120. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
| 10B | Find the whole Similar practice |
54 is 45% of a number. Find the number. | Show answer + formula + trick120.1% = 54 ÷ 45 = 1.2, so 100% = 120. Formula / rule Rule / formula: Percentage of amount = percentage ÷ 100 × amount. Percentage change = change ÷ original × 100. Tip / trick Quick trick: Build from 10%, 5%, 1%, 25%, 50% and 75%. For reverse percentages, divide by the percentage that remains. Remember Memory cue: 10% = ÷10, 5% = half of 10%, 1% = ÷100. |
6. Ratios and Proportions
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Simplify ratio | Simplify 42:56. | Show answer + formula + trick3:4.Divide both parts by 14. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 1B | Simplify ratio Similar practice |
Simplify 54:72. | Show answer + formula + trick3:4.Divide both parts by 18. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 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 Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 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 Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 3A | Find missing ratio value | 3:5 = 21:x. Find x. | Show answer + formula + trick35.3 has been multiplied by 7, so 5 × 7 = 35. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 3B | Find missing ratio value Similar practice |
4:7 = 28:x. Find x. | Show answer + formula + trick49.4 has been multiplied by 7, so 7 × 7 = 49. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 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 Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 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 Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 5A | Recipe proportion | A recipe for 4 uses 300 g flour. How much for 10? | Show answer + formula + trick750 g.300 ÷ 4 = 75 g per person. 75 × 10 = 750 g. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 5B | Recipe proportion Similar practice |
A recipe for 6 uses 450 g flour. How much for 14? | Show answer + formula + trick1,050 g.450 ÷ 6 = 75 g per person. 75 × 14 = 1,050 g. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 6A | Scale factor | A shape side of 6 cm is enlarged by scale factor 2.5. New length? | Show answer + formula + trick15 cm.6 × 2.5 = 15. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 6B | Scale factor Similar practice |
A side of 8 cm is enlarged by scale factor 1.75. New length? | Show answer + formula + trick14 cm.8 × 1.75 = 14. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 7A | Map scale | On a 1:50,000 map, two places are 6 cm apart. What is the real distance? | Show answer + formula + trick3 km.6 × 50,000 = 300,000 cm = 3,000 m = 3 km. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 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 + trick2 km.8 × 25,000 = 200,000 cm = 2,000 m = 2 km. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 8A | Ratio from total | Red:blue = 7:5 and there are 96 counters. How many are red? | Show answer + formula + trick56.12 parts in total. 96 ÷ 12 = 8 per part. Red = 7 × 8 = 56. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 8B | Ratio from total Similar practice |
Green:yellow = 5:3 and there are 64 counters. How many are green? | Show answer + formula + trick40.8 parts in total. 64 ÷ 8 = 8 per part. Green = 5 × 8 = 40. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 9A | Part-to-whole | Boys:girls = 3:7. What fraction are girls? | Show answer + formula + trick7/10.There are 10 equal parts in total; 7 are girls. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 9B | Part-to-whole Similar practice |
Cats:dogs = 2:5. What fraction are dogs? | Show answer + formula + trick5/7.There are 7 equal parts in total; 5 are dogs. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 10A | Ratio after change | Red:blue = 3:2 with 18 red counters. How many blue counters? | Show answer + formula + trick12.3 parts = 18, so 1 part = 6. Blue = 2 parts = 12. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
| 10B | Ratio after change Similar practice |
Red:blue = 4:3 with 28 red counters. How many blue counters? | Show answer + formula + trick21.4 parts = 28, so 1 part = 7. Blue = 3 parts = 21. Formula / rule Rule / formula: To share in ratio a:b, total parts = a+b, one part = total ÷ total parts. Tip / trick Quick trick: Always find one part or one unit first. For maps, keep units consistent before converting. Remember Memory cue: Add parts → divide total → multiply needed parts. |
7. Basic Algebra
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Simplify like terms | Simplify 4x + 3x - 5. | Show answer + formula + trick7x - 5.Combine only like terms. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 1B | Simplify like terms Similar practice |
Simplify 6y + 5y - 4. | Show answer + formula + trick11y - 4.Combine only like terms. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 2A | Substitution | Find 3a + 2 when a = 7. | Show answer + formula + trick23.3 × 7 + 2 = 23. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 2B | Substitution Similar practice |
Find 4b - 3 when b = 6. | Show answer + formula + trick21.4 × 6 - 3 = 21. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 3A | One-step equation | Solve x + 17 = 43. | Show answer + formula + trickx = 26.Subtract 17 from both sides. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 3B | One-step equation Similar practice |
Solve x - 14 = 29. | Show answer + formula + trickx = 43.Add 14 to both sides. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 4A | Multiplication equation | Solve 6x = 78. | Show answer + formula + trickx = 13.Divide both sides by 6. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 4B | Multiplication equation Similar practice |
Solve 9x = 117. | Show answer + formula + trickx = 13.Divide both sides by 9. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 5A | Two-step equation | Solve 4x + 7 = 31. | Show answer + formula + trickx = 6.Subtract 7 to get 4x = 24, then divide by 4. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 5B | Two-step equation Similar practice |
Solve 5x - 6 = 29. | Show answer + formula + trickx = 7.Add 6 to get 5x = 35, then divide by 5. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 6A | Brackets | Simplify 3(x + 4). | Show answer + formula + trick3x + 12.Multiply every term inside the bracket by 3. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 6B | Brackets Similar practice |
Simplify 4(x - 3). | Show answer + formula + trick4x - 12.Multiply every term inside the bracket by 4. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 7A | Formula use | Use A = l × w. Find A when l = 8.5 and w = 6. | Show answer + formula + trick51.8.5 × 6 = 51. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 7B | Formula use Similar practice |
Use P = 2(l + w). Find P when l = 9 and w = 4.5. | Show answer + formula + trick27.P = 2(9 + 4.5) = 2 × 13.5 = 27. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 8A | Rearranging simple formula | If y = x + 9 and y = 25, find x. | Show answer + formula + trick16.x = 25 - 9 = 16. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 8B | Rearranging simple formula Similar practice |
If y = x - 8 and y = 17, find x. | Show answer + formula + trick25.x = 17 + 8 = 25. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 9A | Word equation | A number doubled then increased by 5 equals 27. Find the number. | Show answer + formula + trick11.2x + 5 = 27, so 2x = 22 and x = 11. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 9B | Word equation Similar practice |
A number tripled then decreased by 4 equals 29. Find the number. | Show answer + formula + trick11.3x - 4 = 29, so 3x = 33 and x = 11. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 10A | Function machine | Input x is multiplied by 4 then 3 is subtracted. Output is 29. Find x. | Show answer + formula + trick8.4x - 3 = 29, so 4x = 32 and x = 8. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
| 10B | Function machine Similar practice |
Input x is multiplied by 5 then 2 is added. Output is 42. Find x. | Show answer + formula + trick8.5x + 2 = 42, so 5x = 40 and x = 8. Formula / rule Rule / formula: Do the same operation to both sides of an equation. Combine only like terms. Tip / trick Quick trick: Work backwards using inverse operations. Check by substituting your answer back into the original equation. Remember Memory cue: Undo operations in reverse order. |
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 + trick12.3x + 8 = 44, so 3x = 36 and x = 12. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 1B | Word problem equation Similar practice |
Four times a number minus 6 is 34. Find the number. | Show answer + formula + trick10.4x - 6 = 34, so 4x = 40 and x = 10. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 2A | Inequality | Solve x + 7 < 15. | Show answer + formula + trickx < 8.Subtract 7 from both sides. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 2B | Inequality Similar practice |
Solve x - 5 > 12. | Show answer + formula + trickx > 17.Add 5 to both sides. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 3A | Arithmetic sequence | Find the next two terms: 7, 12, 17, 22, ... | Show answer + formula + trick27, 32.The common difference is +5. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 3B | Arithmetic sequence Similar practice |
Find the next two terms: 11, 15, 19, 23, ... | Show answer + formula + trick27, 31.The common difference is +4. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 4A | Nth term | Find the nth term of 4, 7, 10, 13, ... | Show answer + formula + trick3n + 1.The common difference is 3, so start with 3n. At n=1, 3n gives 3, so add 1. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 4B | Nth term Similar practice |
Find the nth term of 6, 10, 14, 18, ... | Show answer + formula + trick4n + 2.The common difference is 4, so start with 4n. At n=1, 4n gives 4, so add 2. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 5A | Use nth term | For the sequence 5n - 2, find the 12th term. | Show answer + formula + trick58.5 × 12 - 2 = 58. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 5B | Use nth term Similar practice |
For the sequence 7n + 1, find the 9th term. | Show answer + formula + trick64.7 × 9 + 1 = 64. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 6A | Geometric sequence | Find the next term: 3, 6, 12, 24, ... | Show answer + formula + trick48.Each term is multiplied by 2. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 6B | Geometric sequence Similar practice |
Find the next term: 5, 15, 45, 135, ... | Show answer + formula + trick405.Each term is multiplied by 3. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 7A | Missing term | Find x: 2, 6, 18, x, 162. | Show answer + formula + trick54.The rule is ×3 each time. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 7B | Missing term Similar practice |
Find x: 4, 12, 36, x, 324. | Show answer + formula + trick108.The rule is ×3 each time. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 8A | Consecutive numbers | Three consecutive integers sum to 72. Find them. | Show answer + formula + trick23, 24, 25.Middle number is 72 ÷ 3 = 24, so the neighbours are 23 and 25. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 8B | Consecutive numbers Similar practice |
Three consecutive integers sum to 96. Find them. | Show answer + formula + trick31, 32, 33.Middle number is 96 ÷ 3 = 32, so the neighbours are 31 and 33. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 9A | Age problem | Mia is 4 years older than Ava. Together they are 26. How old is Ava? | Show answer + formula + trick11.Let Ava = x. Mia = x + 4. Then 2x + 4 = 26, so x = 11. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 9B | Age problem Similar practice |
Noah is 6 years older than Leo. Together they are 34. How old is Leo? | Show answer + formula + trick14.Let Leo = x. Noah = x + 6. Then 2x + 6 = 34, so x = 14. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 10A | Pattern formula | A matchstick pattern uses 4, 7, 10, 13 sticks. How many in figure 20? | Show answer + formula + trick61.Nth term is 3n + 1. At n=20, 60 + 1 = 61. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
| 10B | Pattern formula Similar practice |
A pattern uses 6, 10, 14, 18 tiles. How many in figure 25? | Show answer + formula + trick102.Nth term is 4n + 2. At n=25, 100 + 2 = 102. Formula / rule Rule / formula: Arithmetic sequence nth term starts with common difference × n, then adjust by a constant. Tip / trick Quick trick: For word problems, define the unknown first. For inequalities, solve like an equation unless multiplying/dividing by a negative number. Remember Memory cue: Sequence = difference first, then adjustment. |
9. Number Patterns and Sequences
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Constant difference | Continue: 81, 74, 67, 60, ... | Show answer + formula + trick53.Subtract 7 each time. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 1B | Constant difference Similar practice |
Continue: 95, 87, 79, 71, ... | Show answer + formula + trick63.Subtract 8 each time. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 2A | Increasing difference | Continue: 2, 5, 9, 14, 20, ... | Show answer + formula + trick27.Differences are +3, +4, +5, +6, so next is +7. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 2B | Increasing difference Similar practice |
Continue: 3, 7, 12, 18, 25, ... | Show answer + formula + trick33.Differences are +4, +5, +6, +7, so next is +8. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 3A | Geometric | Continue: 160, 80, 40, 20, ... | Show answer + formula + trick10.Divide by 2 each time. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 3B | Geometric Similar practice |
Continue: 243, 81, 27, 9, ... | Show answer + formula + trick3.Divide by 3 each time. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 4A | Fibonacci style | Continue: 4, 7, 11, 18, 29, ... | Show answer + formula + trick47.Add the previous two terms. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 4B | Fibonacci style Similar practice |
Continue: 5, 8, 13, 21, 34, ... | Show answer + formula + trick55.Add the previous two terms. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 5A | Square numbers | What comes next: 1, 4, 9, 16, 25, ...? | Show answer + formula + trick36.These are 1², 2², 3², 4², 5², so next is 6² = 36. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 5B | Square numbers Similar practice |
What comes next: 4, 9, 16, 25, 36, ...? | Show answer + formula + trick49.These are 2², 3², 4², 5², 6², so next is 7² = 49. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 6A | Triangular numbers | Continue: 1, 3, 6, 10, 15, ... | Show answer + formula + trick21.Add 2, 3, 4, 5, then 6. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 6B | Triangular numbers Similar practice |
Continue: 3, 6, 10, 15, 21, ... | Show answer + formula + trick28.Add 3, 4, 5, 6, then 7. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 7A | Alternating rule | Continue: 3, 8, 6, 11, 9, 14, ... | Show answer + formula + trick12.Alternate +5, -2. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 7B | Alternating rule Similar practice |
Continue: 10, 14, 11, 15, 12, 16, ... | Show answer + formula + trick13.Alternate +4, -3. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 8A | Decimal sequence | Continue: 0.4, 0.7, 1.0, 1.3, ... | Show answer + formula + trick1.6.Add 0.3. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 8B | Decimal sequence Similar practice |
Continue: 1.2, 1.6, 2.0, 2.4, ... | Show answer + formula + trick2.8.Add 0.4. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 9A | Find missing term | 5, 11, 23, ?, 95. | Show answer + formula + trick47.Multiply by 2 and add 1 each time: 23 × 2 + 1 = 47. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 9B | Find missing term Similar practice |
7, 15, 31, ?, 127. | Show answer + formula + trick63.Multiply by 2 and add 1 each time: 31 × 2 + 1 = 63. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 10A | Backward sequence | 100, 91, 83, 76, 70, ... | Show answer + formula + trick65.Subtract 9, 8, 7, 6, then 5. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
| 10B | Backward sequence Similar practice |
120, 109, 99, 90, 82, ... | Show answer + formula + trick75.Subtract 11, 10, 9, 8, then 7. Formula / rule Rule / formula: Check first differences, second differences, multiplication/division, then alternating rules. Tip / trick Quick trick: Write the change between terms above the sequence. If changes are not constant, inspect how the changes themselves vary. Remember Memory cue: Difference → second difference → ×/÷ → alternating. |
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 + trick52°.Angles on a straight line total 180°. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 1B | Straight line Similar practice |
One angle on a straight line is 143°. Find the other. | Show answer + formula + trick37°.Angles on a straight line total 180°. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 2A | Around a point | Angles around a point are 95°, 110°, 80° and x. Find x. | Show answer + formula + trick75°.Angles around a point total 360°. 360 - 285 = 75. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 2B | Around a point Similar practice |
Angles around a point are 120°, 85°, 64° and x. Find x. | Show answer + formula + trick91°.Angles around a point total 360°. 360 - 269 = 91. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 3A | Triangle | A triangle has angles 48° and 67°. Find the third. | Show answer + formula + trick65°.Angles in a triangle total 180°. 180 - 115 = 65. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 3B | Triangle Similar practice |
A triangle has angles 56° and 79°. Find the third. | Show answer + formula + trick45°.Angles in a triangle total 180°. 180 - 135 = 45. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 4A | Isosceles triangle | An isosceles triangle has vertex angle 38°. Find each base angle. | Show answer + formula + trick71°.Remaining total is 142°, split equally: 71° each. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 4B | Isosceles triangle Similar practice |
An isosceles triangle has vertex angle 44°. Find each base angle. | Show answer + formula + trick68°.Remaining total is 136°, split equally: 68° each. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 5A | Quadrilateral | Three angles of a quadrilateral are 90°, 105° and 88°. Find the fourth. | Show answer + formula + trick77°.Quadrilateral angles total 360°. 360 - 283 = 77. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 5B | Quadrilateral Similar practice |
Three angles of a quadrilateral are 96°, 82° and 103°. Find the fourth. | Show answer + formula + trick79°.Quadrilateral angles total 360°. 360 - 281 = 79. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 6A | Vertically opposite | Two straight lines cross. One angle is 63°. Find the vertically opposite angle. | Show answer + formula + trick63°.Vertically opposite angles are equal. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 6B | Vertically opposite Similar practice |
Two straight lines cross. One angle is 117°. Find the vertically opposite angle. | Show answer + formula + trick117°.Vertically opposite angles are equal. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 7A | Corresponding angles | Two parallel lines are crossed by a transversal. One corresponding angle is 112°. Find the matching corresponding angle. | Show answer + formula + trick112°.Corresponding angles on parallel lines are equal. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 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 + trick74°.Corresponding angles on parallel lines are equal. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 8A | Alternate angles | A pair of alternate interior angles lie on parallel lines. One is 68°. Find the other. | Show answer + formula + trick68°.Alternate interior angles are equal. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 8B | Alternate angles Similar practice |
A pair of alternate interior angles lie on parallel lines. One is 121°. Find the other. | Show answer + formula + trick121°.Alternate interior angles are equal. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 9A | Exterior triangle angle | The two opposite interior angles of a triangle are 42° and 73°. Find the exterior angle. | Show answer + formula + trick115°.An exterior angle equals the sum of the two opposite interior angles: 42 + 73 = 115. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 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 + trick105°.An exterior angle equals the sum of the two opposite interior angles: 58 + 47 = 105. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 10A | Angle type | Classify 235°. | Show answer + formula + trickReflex angle.A reflex angle is greater than 180° and less than 360°. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
| 10B | Angle type Similar practice |
Classify 152°. | Show answer + formula + trickObtuse angle.An obtuse angle is greater than 90° and less than 180°. Formula / rule Rule / formula: Straight line = 180°. Around a point = 360°. Triangle = 180°. Quadrilateral = 360°. Vertically opposite angles are equal. Tip / trick Quick trick: Write the known total beside the diagram before calculating. Never trust how an angle looks. Remember Memory cue: Line 180, point 360, triangle 180, quad 360. |
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 + trick38 cm.P = 2(l + w) = 2(12 + 7) = 38. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 1B | Rectangle perimeter Similar practice |
A rectangle is 15 cm by 9 cm. Find its perimeter. | Show answer + formula + trick48 cm.P = 2(l + w) = 2(15 + 9) = 48. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 2A | Rectangle area | A rectangle is 13 cm by 8 cm. Find its area. | Show answer + formula + trick104 cm².A = l × w = 13 × 8. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 2B | Rectangle area Similar practice |
A rectangle is 17 cm by 6 cm. Find its area. | Show answer + formula + trick102 cm².A = l × w = 17 × 6. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 3A | Triangle area | A triangle has base 14 cm and perpendicular height 9 cm. Find its area. | Show answer + formula + trick63 cm².A = 1/2 × base × height = 1/2 × 14 × 9. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 3B | Triangle area Similar practice |
A triangle has base 18 cm and perpendicular height 7 cm. Find its area. | Show answer + formula + trick63 cm².A = 1/2 × 18 × 7 = 63. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 4A | Parallelogram area | Base 11 cm, perpendicular height 6 cm. Area? | Show answer + formula + trick66 cm².A = base × perpendicular height. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 4B | Parallelogram area Similar practice |
Base 13 cm, perpendicular height 8 cm. Area? | Show answer + formula + trick104 cm².A = base × perpendicular height. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 5A | Trapezium area | Parallel sides are 8 cm and 14 cm, height 5 cm. Area? | Show answer + formula + trick55 cm².A = 1/2 × (8 + 14) × 5 = 55. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 5B | Trapezium area Similar practice |
Parallel sides are 10 cm and 16 cm, height 6 cm. Area? | Show answer + formula + trick78 cm².A = 1/2 × (10 + 16) × 6 = 78. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 6A | Regular polygon perimeter | A regular octagon has side 6.5 cm. Perimeter? | Show answer + formula + trick52 cm.8 × 6.5 = 52. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 6B | Regular polygon perimeter Similar practice |
A regular decagon has side 4.2 cm. Perimeter? | Show answer + formula + trick42 cm.10 × 4.2 = 42. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 7A | Line symmetry | How many lines of symmetry does a square have? | Show answer + formula + trick4.Two diagonals and two lines through opposite side midpoints. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 7B | Line symmetry Similar practice |
How many lines of symmetry does an equilateral triangle have? | Show answer + formula + trick3.Each line goes from a vertex to the midpoint of the opposite side. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 8A | Rotational symmetry | What is the order of rotational symmetry of a regular hexagon? | Show answer + formula + trick6.It matches itself 6 times in a full turn. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 8B | Rotational symmetry Similar practice |
What is the order of rotational symmetry of a rectangle that is not a square? | Show answer + formula + trick2.It matches itself after 180° and 360°. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 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 Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 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 Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 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 + trick68 cm².80 - 12 = 68. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
| 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 + trick98 cm².108 - 10 = 98. Formula / rule Rule / formula: Rectangle A=lw, P=2(l+w). Triangle A=½bh. Parallelogram A=bh. Trapezium A=½(a+b)h. Tip / trick Quick trick: For composite shapes, split into rectangles/triangles or find a large shape and subtract the missing part. Remember Memory cue: Area uses square units; perimeter uses ordinary length units. |
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 + trick120 cm³.V = l × w × h. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 1B | Cuboid volume Similar practice |
Find the volume of a cuboid 9 cm × 4 cm × 5 cm. | Show answer + formula + trick180 cm³.V = l × w × h. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 2A | Cube volume | A cube has side 6 cm. Find its volume. | Show answer + formula + trick216 cm³.V = 6³ = 216. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 2B | Cube volume Similar practice |
A cube has side 7 cm. Find its volume. | Show answer + formula + trick343 cm³.V = 7³ = 343. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 3A | Cube surface area | A cube has side 4 cm. Find its surface area. | Show answer + formula + trick96 cm².Six faces, each 4 × 4 = 16. Total 6 × 16 = 96. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 3B | Cube surface area Similar practice |
A cube has side 5 cm. Find its surface area. | Show answer + formula + trick150 cm².Six faces, each 5 × 5 = 25. Total 6 × 25 = 150. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 4A | Cuboid surface area | Find the surface area of a 3 cm × 4 cm × 5 cm cuboid. | Show answer + formula + trick94 cm².2(3×4 + 3×5 + 4×5) = 2(12+15+20) = 94. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 4B | Cuboid surface area Similar practice |
Find the surface area of a 2 cm × 6 cm × 7 cm cuboid. | Show answer + formula + trick136 cm².2(2×6 + 2×7 + 6×7) = 2(12+14+42) = 136. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 5A | Faces edges vertices | How many faces, edges and vertices does a cube have? | Show answer + formula + trick6 faces, 12 edges, 8 vertices.Memorise this core fact. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 5B | Faces edges vertices Similar practice |
How many faces, edges and vertices does a cuboid have? | Show answer + formula + trick6 faces, 12 edges, 8 vertices.A cuboid has the same counts as a cube. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 6A | Triangular prism | How many faces does a triangular prism have? | Show answer + formula + trick5.Two triangular ends and three rectangular faces. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 6B | Triangular prism Similar practice |
How many vertices does a triangular prism have? | Show answer + formula + trick6.Three vertices on each triangular end. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 7A | Pyramid recognition | A square-based pyramid has how many vertices? | Show answer + formula + trick5.Four on the base plus the apex. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 7B | Pyramid recognition Similar practice |
A triangular-based pyramid has how many faces? | Show answer + formula + trick4.One triangular base plus three triangular side faces. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 8A | Cylinder properties | How many flat circular faces does a cylinder have? | Show answer + formula + trick2.Top and bottom are flat circles; the side is curved. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 8B | Cylinder properties Similar practice |
How many curved surfaces does a cylinder have? | Show answer + formula + trick1.The side wraps around as one curved surface. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 9A | Net reasoning | A valid cube net must contain how many equal squares? | Show answer + formula + trick6.A cube has six square faces. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 9B | Net reasoning Similar practice |
A valid triangular prism net must contain how many faces altogether? | Show answer + formula + trick5.Two triangles and three rectangles. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 10A | Volume reverse problem | A cuboid has volume 240 cm³, length 10 cm and width 6 cm. Find height. | Show answer + formula + trick4 cm.h = 240 ÷ (10 × 6) = 4. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
| 10B | Volume reverse problem Similar practice |
A cuboid has volume 360 cm³, length 12 cm and width 5 cm. Find height. | Show answer + formula + trick6 cm.h = 360 ÷ (12 × 5) = 6. Formula / rule Rule / formula: Cuboid volume = lwh. Cube volume = s³. Cube surface area = 6s². Cuboid surface area = 2(lw+lh+wh). Tip / trick Quick trick: Volume counts cubes, so use cubic units. Surface area counts outside faces, so use square units. Remember Memory cue: Volume = space inside; surface area = skin outside. |
13. Measurement and Units
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Length conversion | Convert 3.45 km to metres. | Show answer + formula + trick3,450 m.1 km = 1,000 m. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 1B | Length conversion Similar practice |
Convert 5.08 km to metres. | Show answer + formula + trick5,080 m.1 km = 1,000 m. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 2A | Mass conversion | Convert 2.75 kg to grams. | Show answer + formula + trick2,750 g.1 kg = 1,000 g. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 2B | Mass conversion Similar practice |
Convert 3.6 kg to grams. | Show answer + formula + trick3,600 g.1 kg = 1,000 g. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 3A | Capacity conversion | Convert 4.2 litres to millilitres. | Show answer + formula + trick4,200 ml.1 litre = 1,000 ml. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 3B | Capacity conversion Similar practice |
Convert 2.35 litres to millilitres. | Show answer + formula + trick2,350 ml.1 litre = 1,000 ml. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 4A | Time conversion | Convert 2 hours 35 minutes to minutes. | Show answer + formula + trick155 minutes.2 hours = 120 minutes; 120 + 35 = 155. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 4B | Time conversion Similar practice |
Convert 3 hours 42 minutes to minutes. | Show answer + formula + trick222 minutes.3 hours = 180 minutes; 180 + 42 = 222. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 5A | 24-hour time | Write 7:45 pm in 24-hour time. | Show answer + formula + trick19:45.For pm times after noon, add 12 to the hour. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 5B | 24-hour time Similar practice |
Write 6:20 am in 24-hour time. | Show answer + formula + trick06:20.For am times before noon, keep the hour and use two digits. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 6A | Duration | A train leaves at 13:48 and arrives at 16:17. Journey time? | Show answer + formula + trick2 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 Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 6B | Duration Similar practice |
A bus leaves at 09:37 and arrives at 12:05. Journey time? | Show answer + formula + trick2 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 Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 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 Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 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 Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 8A | Reading scales | A scale has 5 equal intervals between 2 kg and 3 kg. What is each interval? | Show answer + formula + trick0.2 kg.Difference is 1 kg. 1 ÷ 5 = 0.2 kg. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 8B | Reading scales Similar practice |
A scale has 8 equal intervals between 4 kg and 6 kg. What is each interval? | Show answer + formula + trick0.25 kg.Difference is 2 kg. 2 ÷ 8 = 0.25 kg. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 9A | Area units | Convert 3.2 m² to cm². | Show answer + formula + trick32,000 cm².1 m² = 10,000 cm², so 3.2 × 10,000 = 32,000. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 9B | Area units Similar practice |
Convert 1.7 m² to cm². | Show answer + formula + trick17,000 cm².1 m² = 10,000 cm², so 1.7 × 10,000 = 17,000. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 10A | Speed | A car travels 150 km in 2.5 hours. Average speed? | Show answer + formula + trick60 km/h.Speed = distance ÷ time = 150 ÷ 2.5 = 60. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
| 10B | Speed Similar practice |
A cyclist travels 84 km in 3.5 hours. Average speed? | Show answer + formula + trick24 km/h.Speed = distance ÷ time = 84 ÷ 3.5 = 24. Formula / rule Rule / formula: 1 km=1000 m, 1 m=100 cm=1000 mm, 1 kg=1000 g, 1 L=1000 ml. Speed = distance ÷ time. Tip / trick Quick trick: Convert everything to the same unit before calculating. For time, bridge to the next hour first. Remember Memory cue: Big unit to small unit = multiply; small to big = divide. |
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 + trick90.18 + 24 + 17 + 31 = 90. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 1B | Read a table Similar practice |
A table shows 26, 19, 34 and 21 visitors over four days. Total? | Show answer + formula + trick100.26 + 19 + 34 + 21 = 100. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 2A | Difference from bar chart | One bar shows 48 and another 35. What is the difference? | Show answer + formula + trick13.48 - 35 = 13. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 2B | Difference from bar chart Similar practice |
One bar shows 63 and another 47. What is the difference? | Show answer + formula + trick16.63 - 47 = 16. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 3A | Line graph change | A value rises from 22 to 37. By how much? | Show answer + formula + trick15.37 - 22 = 15. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 3B | Line graph change Similar practice |
A value falls from 54 to 38. By how much? | Show answer + formula + trick16.54 - 38 = 16. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 4A | Pie chart angle | A category is 25% of a pie chart. What angle represents it? | Show answer + formula + trick90°.25% of 360° = 90°. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 4B | Pie chart angle Similar practice |
A category is 35% of a pie chart. What angle represents it? | Show answer + formula + trick126°.35% of 360° = 0.35 × 360 = 126°. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 5A | Pie chart from angle | A sector is 72°. What fraction of the whole is this? | Show answer + formula + trick1/5.72/360 = 1/5. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 5B | Pie chart from angle Similar practice |
A sector is 45°. What fraction of the whole is this? | Show answer + formula + trick1/8.45/360 = 1/8. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 6A | Pictogram | One symbol represents 8 pupils. What do 3.5 symbols represent? | Show answer + formula + trick28 pupils.3.5 × 8 = 28. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 6B | Pictogram Similar practice |
One symbol represents 6 pupils. What do 4.5 symbols represent? | Show answer + formula + trick27 pupils.4.5 × 6 = 27. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 7A | Missing table value | A total is 120. Three categories are 35, 28 and 41. Find the fourth. | Show answer + formula + trick16.120 - (35 + 28 + 41) = 16. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 7B | Missing table value Similar practice |
A total is 150. Three categories are 42, 37 and 51. Find the fourth. | Show answer + formula + trick20.150 - (42 + 37 + 51) = 20. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 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 + trick60.18 + 14 + 11 + 17 = 60. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 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 + trick70.22 + 16 + 13 + 19 = 70. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 9A | Compare proportions | Class A: 12 of 30 choose chess. Class B: 15 of 50. Which has the higher proportion? | Show answer + formula + trickClass A.A = 40%; B = 30%. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 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 + trickThey are equal.18/45 = 40%; 14/35 = 40%. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 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 + trick5.20 ÷ 4 = 5. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
| 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 + trick5.30 ÷ 6 = 5. Formula / rule Rule / formula: Pie chart angle = fraction × 360°. Proportion = part ÷ whole. Tip / trick Quick trick: Read the axis scale before reading bars or points. Check whether minor intervals are 1, 2, 5, 10, etc. Remember Memory cue: Read scale first, data second. |
15. Advanced Data Handling
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1A | Mean | Find the mean of 7, 9, 12, 16, 21. | Show answer + formula + trick13.Total = 65; 65 ÷ 5 = 13. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 1B | Mean Similar practice |
Find the mean of 8, 10, 15, 17, 20. | Show answer + formula + trick14.Total = 70; 70 ÷ 5 = 14. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 2A | Median odd set | Find the median of 3, 11, 7, 8, 20. | Show answer + formula + trick8.Order: 3, 7, 8, 11, 20. Middle is 8. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 2B | Median odd set Similar practice |
Find the median of 14, 5, 9, 21, 11. | Show answer + formula + trick11.Order: 5, 9, 11, 14, 21. Middle is 11. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 3A | Median even set | Find the median of 4, 7, 10, 15, 21, 30. | Show answer + formula + trick12.5.Middle pair is 10 and 15. Mean = 12.5. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 3B | Median even set Similar practice |
Find the median of 6, 8, 13, 17, 22, 28. | Show answer + formula + trick15.Middle pair is 13 and 17. Mean = 15. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 4A | Mode | Find the mode: 4, 6, 6, 7, 8, 6, 9. | Show answer + formula + trick6.6 occurs most often. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 4B | Mode Similar practice |
Find the mode: 3, 5, 7, 5, 8, 5, 9. | Show answer + formula + trick5.5 occurs most often. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 5A | Range | Find the range of 18, 4, 27, 11, 9. | Show answer + formula + trick23.27 - 4 = 23. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 5B | Range Similar practice |
Find the range of 31, 12, 45, 28, 19. | Show answer + formula + trick33.45 - 12 = 33. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 6A | Missing value from mean | The mean of 5 numbers is 12. Four are 9, 11, 13, 15. Find the fifth. | Show answer + formula + trick12.Required total = 5 × 12 = 60. Known total = 48. Missing = 12. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 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 + trick16.Required total = 6 × 15 = 90. Known total = 74. Missing = 16. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 7A | Frequency mean | Scores 1, 2, 3 occur with frequencies 2, 3, 5. Find the mean. | Show answer + formula + trick2.3.Weighted total = 1×2 + 2×3 + 3×5 = 23; total frequency = 10; mean = 2.3. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 7B | Frequency mean Similar practice |
Scores 2, 3, 4 occur with frequencies 3, 2, 5. Find the mean. | Show answer + formula + trick3.2.Weighted total = 2×3 + 3×2 + 4×5 = 32; total frequency = 10; mean = 3.2. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 8A | Simple probability | A bag has 3 red, 5 blue and 2 green counters. Probability of blue? | Show answer + formula + trick1/2.5 blue out of 10 total = 5/10 = 1/2. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 8B | Simple probability Similar practice |
A bag has 4 red, 3 blue and 5 green counters. Probability of red? | Show answer + formula + trick1/3.4 red out of 12 total = 4/12 = 1/3. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 9A | Scatter graph interpretation | As x increases, y generally decreases. What type of correlation? | Show answer + formula + trickNegative correlation.The variables move in opposite directions. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 9B | Scatter graph interpretation Similar practice |
As x increases, y generally increases. What type of correlation? | Show answer + formula + trickPositive correlation.The variables tend to increase together. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 10A | Frequency reasoning | The modal class in a grouped frequency table is the class with what property? | Show answer + formula + trickThe greatest frequency.Mode means most frequent. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
| 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 + trick10-20.It has the greatest frequency, 9. Formula / rule Rule / formula: Mean = total ÷ number. Median = middle after ordering. Mode = most frequent. Range = highest − lowest. Tip / trick Quick trick: For a missing value from a mean, first find the required total: mean × number of values. Remember Memory cue: Mean needs a total; median needs order; mode means most. |
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 + trick13.18 - 5 = 13. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 1B | Intersection Similar practice |
24 pupils like Art, 15 like Music and 7 like both. How many like Art only? | Show answer + formula + trick17.24 - 7 = 17. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 2A | Union | Using the same data, how many like Maths or Science? | Show answer + formula + trick25.18 + 12 - 5 = 25. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 2B | Union Similar practice |
Using the same data, how many like Art or Music? | Show answer + formula + trick32.24 + 15 - 7 = 32. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 3A | Neither | In a class of 30, 25 like Maths or Science. How many like neither? | Show answer + formula + trick5.30 - 25 = 5. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 3B | Neither Similar practice |
In a class of 36, 32 like Art or Music. How many like neither? | Show answer + formula + trick4.36 - 32 = 4. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 4A | Set total | Maths only 9, Science only 7, both 6, neither 3. Class size? | Show answer + formula + trick25.9 + 7 + 6 + 3 = 25. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 4B | Set total Similar practice |
Art only 11, Music only 8, both 5, neither 4. Class size? | Show answer + formula + trick28.11 + 8 + 5 + 4 = 28. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 5A | Probability complement | Probability of rain is 0.35. Probability of no rain? | Show answer + formula + trick0.65.Probabilities of complementary events sum to 1. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 5B | Probability complement Similar practice |
Probability of winning is 0.28. Probability of not winning? | Show answer + formula + trick0.72.Complementary probabilities sum to 1. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 6A | Probability scale | What does probability 0 mean? | Show answer + formula + trickImpossible.0 = impossible, 1 = certain. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 6B | Probability scale Similar practice |
What does probability 1 mean? | Show answer + formula + trickCertain.1 = certain, 0 = impossible. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 7A | Two independent events | A fair coin is tossed twice. Probability of two heads? | Show answer + formula + trick1/4.1/2 × 1/2 = 1/4. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 7B | Two independent events Similar practice |
A fair coin is tossed three times. Probability of three heads? | Show answer + formula + trick1/8.1/2 × 1/2 × 1/2 = 1/8. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 8A | Dice compound event | A fair die is rolled. Probability of an even number greater than 3? | Show answer + formula + trick1/3.Favourable outcomes are 4 and 6: 2/6 = 1/3. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 8B | Dice compound event Similar practice |
A fair die is rolled. Probability of a prime number greater than 2? | Show answer + formula + trick1/3.Favourable outcomes are 3 and 5: 2/6 = 1/3. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 9A | Without replacement | A bag has 3 red and 2 blue counters. Two are drawn without replacement. Probability both are red? | Show answer + formula + trick3/10.3/5 × 2/4 = 6/20 = 3/10. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 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 + trick2/7.4/7 × 3/6 = 12/42 = 2/7. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 10A | At least one | Two fair coins are tossed. Probability of at least one head? | Show answer + formula + trick3/4.Use complement: only no head is TT = 1/4. So 1 - 1/4 = 3/4. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
| 10B | At least one Similar practice |
Two fair coins are tossed. Probability of at least one tail? | Show answer + formula + trick3/4.Use complement: only no tail is HH = 1/4. So 1 - 1/4 = 3/4. Formula / rule Rule / formula: P(event)=favourable outcomes ÷ total outcomes. Complementary probabilities add to 1. Tip / trick Quick trick: In Venn diagrams, fill the overlap first. For 'at least one', often use 1 − P(none). Remember Memory cue: Overlap first; neither last. |
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 Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 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 Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 2A | Travel | A train covers 168 miles in 3 hours. Average speed? | Show answer + formula + trick56 mph.168 ÷ 3 = 56. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 2B | Travel Similar practice |
A car covers 210 miles in 3.5 hours. Average speed? | Show answer + formula + trick60 mph.210 ÷ 3.5 = 60. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 3A | Consecutive numbers | Two consecutive numbers sum to 71. Find them. | Show answer + formula + trick35 and 36.Half of 71 is 35.5, so the consecutive integers are 35 and 36. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 3B | Consecutive numbers Similar practice |
Two consecutive numbers sum to 93. Find them. | Show answer + formula + trick46 and 47.Half of 93 is 46.5, so the consecutive integers are 46 and 47. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 4A | Ages | A mother is 28 years older than her daughter. Together they are 54. Ages? | Show answer + formula + trickDaughter 13, mother 41.Let daughter = x. Then x + x + 28 = 54, so 2x = 26 and x = 13. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 4B | Ages Similar practice |
A father is 30 years older than his son. Together they are 58. Ages? | Show answer + formula + trickSon 14, father 44.Let son = x. Then x + x + 30 = 58, so 2x = 28 and x = 14. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 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 + trick102 m².Garden = 108 m², shed = 6 m², difference = 102 m². Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 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 + trick138 m².Patio = 150 m², pond = 12 m², difference = 138 m². Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 6A | Time | A film starts at 18:35 and lasts 1 h 48 min. Finish time? | Show answer + formula + trick20:23.18:35 + 1 h = 19:35, +48 min = 20:23. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 6B | Time Similar practice |
A concert starts at 19:20 and lasts 2 h 15 min. Finish time? | Show answer + formula + trick21:35.19:20 + 2 h = 21:20, +15 min = 21:35. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 7A | Packing | 487 sweets are packed equally into bags of 24. How many full bags and how many left? | Show answer + formula + trick20 full bags, 7 left.24 × 20 = 480, remainder 7. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 7B | Packing Similar practice |
562 biscuits are packed equally into boxes of 25. How many full boxes and how many left? | Show answer + formula + trick22 full boxes, 12 left.25 × 22 = 550, remainder 12. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 8A | Mixture ratio | Orange:apple juice = 3:2. A jug contains 1.5 litres. How much orange juice? | Show answer + formula + trick0.9 litres.5 parts total. 1.5 ÷ 5 = 0.3 L per part. Orange = 3 × 0.3 = 0.9 L. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 8B | Mixture ratio Similar practice |
Red:yellow paint = 2:3. A tin contains 2 litres. How much yellow paint? | Show answer + formula + trick1.2 litres.5 parts total. 2 ÷ 5 = 0.4 L per part. Yellow = 3 × 0.4 = 1.2 L. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 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 + trick20.Total 60; 60 ÷ 3 = 20. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 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 + trick20.Total 80; 80 ÷ 4 = 20. Formula / rule Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 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 Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
| 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 Rule / formula: No single formula. Translate the story into operations and keep units consistent. Tip / trick Quick trick: Underline what is being asked, circle useful numbers, estimate the answer, then calculate. Ask whether the answer must be rounded up. Remember Memory cue: What do I know? What do I need? Which operation connects them? |
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 Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 + trick3 units above.A quarter-turn anticlockwise sends right to up. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 + trick4 units to the right.A quarter-turn clockwise sends up to right. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 5A | Half-turn rotation | What happens to a point 2 right and 5 up after a 180° rotation about the centre? | Show answer + formula + trick2 left and 5 down.A half-turn reverses both directions. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 + trick3 right and 2 up.A half-turn reverses both directions. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 6A | Tessellation | Can regular hexagons tessellate on their own? | Show answer + formula + trickYes.Their 120° interior angles fit three around a point: 3 × 120° = 360°. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 6B | Tessellation Similar practice |
Can regular pentagons tessellate on their own? | Show answer + formula + trickNo.Their interior angle is 108°, and 360 ÷ 108 is not a whole number. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 7A | Cube net | How many square faces must a cube net contain? | Show answer + formula + trick6.One for each face of the cube. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 7B | Cube net Similar practice |
How many edges does a cube have? | Show answer + formula + trick12.A cube has 12 edges. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 + trick1.The rear cube can be hidden directly behind the front cube. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 + trick1.The two rear cubes can be completely hidden behind the front cube. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 9A | Symmetry after reflection | A shape is reflected twice in the same mirror line. What is the final position? | Show answer + formula + trickThe original position.Two identical reflections cancel each other. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 + trickThe original position.Two identical reflections cancel each other. Formula / rule Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
| 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 Rule / formula: Translation adds a vector. Reflection changes position across a mirror line. Rotation keeps distance from the centre fixed. Tip / trick Quick trick: For coordinate reflections: in x-axis change y sign; in y-axis change x sign. Remember Memory cue: x-axis flips y; y-axis flips x. |
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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 2A | Loss | An item costs £85 and sells for £72. Loss? | Show answer + formula + trick£13.Cost price - selling price = £13. Formula / rule Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 3A | Percentage profit | A toy costs £40 and sells for £50. Profit percentage? | Show answer + formula + trick25%.Profit = £10. £10 ÷ £40 = 25%. Formula / rule Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 3B | Percentage profit Similar practice |
A book costs £24 and sells for £30. Profit percentage? | Show answer + formula + trick25%.Profit = £6. £6 ÷ £24 = 25%. Formula / rule Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 7A | Best buy | 750 g cereal costs £3.00; 1 kg costs £3.80. Which is cheaper per kg? | Show answer + formula + trick1 kg pack.750 g pack costs £4.00 per kg. The 1 kg pack costs £3.80 per kg. Formula / rule Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 7B | Best buy Similar practice |
600 g cereal costs £2.70; 1 kg costs £4.20. Which is cheaper per kg? | Show answer + formula + trick1 kg pack.600 g pack costs £4.50 per kg. The 1 kg pack costs £4.20 per kg. Formula / rule Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
| 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 Rule / formula: Profit = selling price − cost price. Loss = cost price − selling price. Profit % = profit ÷ cost price × 100. Tip / trick Quick trick: For best-buy questions, compare the same unit, usually cost per item, per 100 g or per kg. Remember Memory cue: Profit compares with cost price, not selling price. |
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 + trick4,572.Digit sum 4+5+7+2=18, which is divisible by 9. The other digit sum is 19. Formula / rule Divisibility by 3 or 9 is decided by the sum of the digits. Tip / trick Add the digits before attempting any division. Remember Digit sum first. |
| 1B | Divisibility by 3 and 9 Similar practice | Which of 6,381 and 6,382 is divisible by 3? | Show answer + formula + trick6,381.Digit sum 6+3+8+1=18, which is divisible by 3. Formula / rule Divisibility by 3 or 9 is decided by the sum of the digits. Tip / trick Add the digits before attempting any division. Remember Digit sum first. |
| 2A | Divisibility by 4 | Is 7,316 divisible by 4? | Show answer + formula + trickYes.Only the last two digits matter. 16 is divisible by 4. Formula / rule A whole number is divisible by 4 if its last two digits form a number divisible by 4. Tip / trick Ignore every digit except the final two. Remember Last two for ÷4. |
| 2B | Divisibility by 4 Similar practice | Is 9,542 divisible by 4? | Show answer + formula + trickNo.Only the last two digits matter. 42 is not divisible by 4. Formula / rule A whole number is divisible by 4 if its last two digits form a number divisible by 4. Tip / trick Ignore every digit except the final two. Remember Last two for ÷4. |
| 3A | Divisibility by 6 | Is 2,154 divisible by 6? | Show answer + formula + trickYes.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 if it is divisible by both 2 and 3. Tip / trick Check evenness and digit sum. Remember ÷6 = ÷2 and ÷3. |
| 3B | Divisibility by 6 Similar practice | Is 3,428 divisible by 6? | Show answer + formula + trickNo.It is even, but its digit sum is 17, not divisible by 3. Formula / rule A number is divisible by 6 if it is divisible by both 2 and 3. Tip / trick Check evenness and digit sum. Remember ÷6 = ÷2 and ÷3. |
| 4A | Prime factor reasoning | Which is prime: 51, 53, 55, 57? | Show answer + formula + trick53.51 and 57 are divisible by 3, and 55 is divisible by 5. Formula / rule A prime number has exactly two factors, 1 and itself. Tip / trick For numbers under 100, test divisibility by 2, 3, 5 and 7 where needed. Remember Prime means exactly two factors. |
| 4B | Prime factor reasoning Similar practice | Which is prime: 61, 63, 65, 69? | Show answer + formula + trick61.63 and 69 are divisible by 3; 65 is divisible by 5. Formula / rule A prime number has exactly two factors, 1 and itself. Tip / trick For numbers under 100, test divisibility by 2, 3, 5 and 7 where needed. Remember Prime means exactly two factors. |
| 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 + trick24.The HCF of 48 and 72 is 24. Formula / rule HCF is the greatest factor shared by both numbers. Tip / trick List factor pairs or use prime factors if helpful. Remember Shared factor, choose greatest. |
| 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 + trick12.The HCF of 60 and 84 is 12. Formula / rule HCF is the greatest factor shared by both numbers. Tip / trick List factor pairs or use prime factors if helpful. Remember Shared factor, choose greatest. |
| 6A | Common multiple puzzle | What is the smallest number greater than 20 that is a multiple of both 6 and 8? | Show answer + formula + trick24.The LCM of 6 and 8 is 24. Formula / rule LCM is the smallest positive multiple common to both numbers. Tip / trick Write a short multiples list for the larger number first. Remember Common multiple, choose smallest. |
| 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 + trick36.The LCM of 9 and 12 is 36. Formula / rule LCM is the smallest positive multiple common to both numbers. Tip / trick Write a short multiples list for the larger number first. Remember Common multiple, choose smallest. |
| 7A | Square and cube recognition | Which number is both a square number and less than 50: 18, 25, 32, 45? | Show answer + formula + trick25.25=5². Formula / rule Square numbers are n². Cube numbers are n³. Tip / trick Know squares to 15² and cubes to at least 10³. Remember Squares: n×n. Cubes: n×n×n. |
| 7B | Square and cube recognition Similar practice | Which number is a cube number: 24, 27, 30, 36? | Show answer + formula + trick27.27=3³. Formula / rule Square numbers are n². Cube numbers are n³. Tip / trick Know squares to 15² and cubes to at least 10³. Remember Squares: n×n. Cubes: n×n×n. |
| 8A | Remainder pattern | What remainder is left when 1,003 is divided by 10? | Show answer + formula + trick3.Division by 10 leaves the units digit as the remainder. Formula / rule When dividing by 10, 100 or 1000, the trailing digits determine the remainder. Tip / trick Do not perform long division for powers of 10. Remember Remainder lives at the end. |
| 8B | Remainder pattern Similar practice | What remainder is left when 2,047 is divided by 100? | Show answer + formula + trick47.Division by 100 leaves the final two digits as the remainder. Formula / rule When dividing by 10, 100 or 1000, the trailing digits determine the remainder. Tip / trick Do not perform long division for powers of 10. Remember Remainder lives at the end. |
| 9A | Consecutive integer properties | The sum of three consecutive integers is 75. What is the largest? | Show answer + formula + trick26.The middle is 75÷3=25, so the integers are 24,25,26. Formula / rule For an odd number of consecutive integers, their average is the middle integer. Tip / trick Divide the total by the number of terms. Remember Odd count: average = middle. |
| 9B | Consecutive integer properties Similar practice | The sum of five consecutive integers is 110. What is the middle integer? | Show answer + formula + trick22.The middle is the average: 110÷5=22. Formula / rule For an odd number of consecutive integers, their average is the middle integer. Tip / trick Divide the total by the number of terms. Remember Odd count: average = middle. |
| 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 + trick74.Let units=u, tens=u+3. Then 2u+3=11, so u=4, tens=7. Formula / rule A two-digit number is 10×tens + units. Tip / trick Translate digit conditions into a tiny equation. Remember Tens and units are separate values. |
| 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 + trick94.Let units=u, tens=u+5. Then 2u+5=13, so u=4, tens=9. Formula / rule A two-digit number is 10×tens + units. Tip / trick Translate digit conditions into a tiny equation. Remember Tens and units are separate values. |
21. Tiffin Focus: Missing Numbers, Digits & Arithmetic Puzzles
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 | Missing digit in addition | Fill the digit: 4□7 + 268 = 735. | Show answer + formula + trick6.735-268=467, so the missing digit is 6. Formula / rule Use inverse operations to reconstruct the missing number. Tip / trick Subtract the known addend from the total. Remember Addition mystery? Subtract. |
| 1B | Missing digit in addition Similar practice | Fill the digit: 5□4 + 379 = 913. | Show answer + formula + trick3.913-379=534, so the missing digit is 3. Formula / rule Use inverse operations to reconstruct the missing number. Tip / trick Subtract the known addend from the total. Remember Addition mystery? Subtract. |
| 2A | Missing digit in subtraction | Fill the digit: 8□2 - 347 = 475. | Show answer + formula + trick2.475+347=822, so the missing digit is 2. Formula / rule For A-B=C, A=B+C. Tip / trick Reverse subtraction with addition. Remember Missing start? Add. |
| 2B | Missing digit in subtraction Similar practice | Fill the digit: 9□5 - 468 = 457. | Show answer + formula + trick2.457+468=925, so the missing digit is 2. Formula / rule For A-B=C, A=B+C. Tip / trick Reverse subtraction with addition. Remember Missing start? Add. |
| 3A | Missing digit in multiplication | Fill the digit: 3□ × 7 = 238. | Show answer + formula + trick4.238÷7=34. Formula / rule Use division to reverse multiplication. Tip / trick Divide the product by the known factor. Remember Product mystery? Divide. |
| 3B | Missing digit in multiplication Similar practice | Fill the digit: 4□ × 6 = 288. | Show answer + formula + trick8.288÷6=48. Formula / rule Use division to reverse multiplication. Tip / trick Divide the product by the known factor. Remember Product mystery? Divide. |
| 4A | Missing divisor | 432 ÷ □ = 18. Find □. | Show answer + formula + trick24.432÷18=24. Formula / rule If A÷B=C, then B=A÷C. Tip / trick Check by multiplying your divisor by the quotient. Remember Division triangle: A=B×C. |
| 4B | Missing divisor Similar practice | 588 ÷ □ = 21. Find □. | Show answer + formula + trick28.588÷21=28. Formula / rule If A÷B=C, then B=A÷C. Tip / trick Check by multiplying your divisor by the quotient. Remember Division triangle: A=B×C. |
| 5A | Digit rearrangement | Using digits 2, 5 and 8 once each, make the largest even number. | Show answer + formula + trick852.An even number must end in 2 or 8. To maximise it, put 8 in hundreds and 2 in units: 852. Formula / rule Even numbers end in 0,2,4,6 or 8. Tip / trick Choose the final digit for the condition first, then maximise/minimise the front digits. Remember Condition first, order second. |
| 5B | Digit rearrangement Similar practice | Using digits 3, 6 and 9 once each, make the smallest even number. | Show answer + formula + trick396.The units digit must be 6, then use the smallest remaining digit first. Formula / rule Even numbers end in 0,2,4,6 or 8. Tip / trick Choose the final digit for the condition first, then maximise/minimise the front digits. Remember Condition first, order second. |
| 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 + trick9.Work backwards: 52-7=45, then 45÷5=9. Formula / rule Reverse a function by undoing operations in reverse order. Tip / trick Write the forward steps, then reverse the arrows. Remember Reverse order, inverse operation. |
| 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 + trick44.Work backwards: 8+3=11, then 11×4=44. Formula / rule Reverse a function by undoing operations in reverse order. Tip / trick Write the forward steps, then reverse the arrows. Remember Reverse order, inverse operation. |
| 7A | Balance puzzle | 3 identical boxes weigh 27 kg altogether. What do 5 boxes weigh? | Show answer + formula + trick45 kg.One box=27÷3=9 kg. Five=45 kg. Formula / rule Unitary method: find one, then scale. Tip / trick Always find the value of one identical object. Remember One first. |
| 7B | Balance puzzle Similar practice | 4 identical parcels weigh 30 kg altogether. What do 10 parcels weigh? | Show answer + formula + trick75 kg.One parcel=30÷4=7.5 kg. Ten=75 kg. Formula / rule Unitary method: find one, then scale. Tip / trick Always find the value of one identical object. Remember One first. |
| 8A | Operation symbol puzzle | Which operation makes 48 □ 6 = 8 true? | Show answer + formula + trick÷.48÷6=8. Formula / rule Test the four basic operations systematically. Tip / trick Estimate the size of the answer to eliminate operations quickly. Remember Ask: bigger, smaller or same scale? |
| 8B | Operation symbol puzzle Similar practice | Which operation makes 15 □ 4 = 60 true? | Show answer + formula + trick×.15×4=60. Formula / rule Test the four basic operations systematically. Tip / trick Estimate the size of the answer to eliminate operations quickly. Remember Ask: bigger, smaller or same scale? |
| 9A | Target-number puzzle | Use 8, 4 and 2 once each with +, −, × or ÷ to make 20. | Show answer + formula + trick8×2+4=20.8×2=16, then +4=20. Formula / rule Use brackets if needed and check every number is used once. Tip / trick Try multiplication first because targets often need a large jump. Remember Use every number exactly once. |
| 9B | Target-number puzzle Similar practice | Use 9, 3 and 2 once each to make 21. | Show answer + formula + trick9×2+3=21.9×2=18, then +3=21. Formula / rule Use brackets if needed and check every number is used once. Tip / trick Try multiplication first because targets often need a large jump. Remember Use every number exactly once. |
| 10A | Largest possible result | Choose + or × in 7 □ 6 □ 2 to make the largest result, using normal order of operations. | Show answer + formula + trick84.7×6×2=84. Formula / rule Multiplication often produces the largest positive result for whole numbers greater than 1. Tip / trick Still check BIDMAS and compare plausible alternatives. Remember Estimate before choosing. |
| 10B | Largest possible result Similar practice | Choose + or × in 9 □ 4 □ 3 to make the largest result. | Show answer + formula + trick108.9×4×3=108. Formula / rule Multiplication often produces the largest positive result for whole numbers greater than 1. Tip / trick Still check BIDMAS and compare plausible alternatives. Remember Estimate before choosing. |
22. Tiffin Focus: Fractions, Decimals & Percentages Challenge
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 | Ordering mixed forms | Order from smallest to largest: 0.6, 5/8, 62%. | Show answer + formula + trick0.6, 62%, 5/8.Convert to percentages: 60%, 62%, 62.5%. Formula / rule Convert all quantities to the same form before comparing. Tip / trick Percentages are often fastest for comparison. Remember Same form, then order. |
| 1B | Ordering mixed forms Similar practice | Order from smallest to largest: 3/4, 0.72, 73%. | Show answer + formula + trick0.72, 73%, 3/4.Convert to percentages: 75%,72%,73%. Formula / rule Convert all quantities to the same form before comparing. Tip / trick Percentages are often fastest for comparison. Remember Same form, then order. |
| 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 + trick9/20.Used=1/4 of 3/5=3/20. Remaining=12/20-3/20=9/20. Formula / rule When a fraction of a fraction is taken, multiply. Tip / trick Distinguish 'fraction of the container' from 'fraction of what is there'. Remember Of means multiply. |
| 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 + trick5/8.Removed=2/7×7/8=1/4. Remaining=7/8-2/8=5/8. Formula / rule When a fraction of a fraction is taken, multiply. Tip / trick Distinguish 'fraction of the container' from 'fraction of what is there'. Remember Of means multiply. |
| 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 Successive percentage changes apply to the new amount, not the original. Tip / trick Equal percentage up and down do not cancel. Remember New base each time. |
| 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 + trick198.200×0.9=180; 180×1.1=198. Formula / rule Successive percentage changes apply to the new amount, not the original. Tip / trick Equal percentage up and down do not cancel. Remember New base each time. |
| 4A | Reverse fraction | 3/7 of a number is 24. Find the number. | Show answer + formula + trick56.One seventh=24÷3=8; seven sevenths=56. Formula / rule If a/b of N = x, then N=x÷a×b. Tip / trick Find one fractional part first. Remember Part first, whole next. |
| 4B | Reverse fraction Similar practice | 5/8 of a number is 45. Find the number. | Show answer + formula + trick72.One eighth=45÷5=9; eight eighths=72. Formula / rule If a/b of N = x, then N=x÷a×b. Tip / trick Find one fractional part first. Remember Part first, whole next. |
| 5A | Recurring relationship | Which is greater: 2/3 or 0.66? | Show answer + formula + trick2/3.2/3≈0.666..., which is greater than 0.66. Formula / rule Convert fractions to decimals or compare by multiplication. Tip / trick Know 1/3≈0.333 and 2/3≈0.667. Remember Thirds recur. |
| 5B | Recurring relationship Similar practice | Which is greater: 5/6 or 0.83? | Show answer + formula + trick5/6.5/6≈0.833..., which is greater than 0.83. Formula / rule Convert fractions to decimals or compare by multiplication. Tip / trick Know 1/3≈0.333 and 2/3≈0.667. Remember Thirds recur. |
| 6A | Percentage points vs percentage | A score rises from 60% to 75%. By how many percentage points? | Show answer + formula + trick15 percentage points.75-60=15. Formula / rule Percentage points are found by direct subtraction of percentages. Tip / trick Do not use percentage-change formula when asked for percentage points. Remember Points = subtract. |
| 6B | Percentage points vs percentage Similar practice | A score falls from 82% to 70%. By how many percentage points? | Show answer + formula + trick12 percentage points.82-70=12. Formula / rule Percentage points are found by direct subtraction of percentages. Tip / trick Do not use percentage-change formula when asked for percentage points. Remember Points = subtract. |
| 7A | Mixed-number proportion | Calculate 1 1/2 × 2 2/3. | Show answer + formula + trick4.3/2×8/3=4. Formula / rule Convert mixed numbers to improper fractions before multiplying. Tip / trick Cancel before multiplying. Remember Mixed → improper → cancel. |
| 7B | Mixed-number proportion Similar practice | Calculate 2 1/4 × 1 1/3. | Show answer + formula + trick3.9/4×4/3=3. Formula / rule Convert mixed numbers to improper fractions before multiplying. Tip / trick Cancel before multiplying. Remember Mixed → improper → cancel. |
| 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 rate = total cost ÷ quantity. Tip / trick Scale decimals by 100 if that makes division easier. Remember Per one = divide. |
| 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 rate = total cost ÷ quantity. Tip / trick Scale decimals by 100 if that makes division easier. Remember Per one = divide. |
| 9A | Fraction difference | How much greater is 7/8 than 2/3? | Show answer + formula + trick5/24.7/8=21/24 and 2/3=16/24; difference=5/24. Formula / rule Subtract fractions using a common denominator. Tip / trick LCM of denominators often keeps numbers smaller. Remember Same bottom before subtracting. |
| 9B | Fraction difference Similar practice | How much greater is 5/6 than 3/5? | Show answer + formula + trick7/30.25/30-18/30=7/30. Formula / rule Subtract fractions using a common denominator. Tip / trick LCM of denominators often keeps numbers smaller. Remember Same bottom before subtracting. |
| 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 + trick10%.40%×25%=0.4×0.25=0.10=10%. Formula / rule A percentage of a percentage means multiply the decimal forms. Tip / trick Convert to fractions if easier, e.g. 25%=1/4. Remember Of means multiply. |
| 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 + trick18%.0.6×0.3=0.18=18%. Formula / rule A percentage of a percentage means multiply the decimal forms. Tip / trick Convert to fractions if easier, e.g. 25%=1/4. Remember Of means multiply. |
23. Tiffin Focus: Ratio, Proportion & Scale Problems
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 | Ratio chain | A:B=2:3 and B:C=6:5. Find A:B:C. | Show answer + formula + trick4:6:5.Scale 2:3 by 2 to make B=6. Formula / rule For linked ratios, make the shared part equal. Tip / trick Use the LCM of the shared ratio numbers. Remember Match the middle. |
| 1B | Ratio chain Similar practice | P:Q=3:4 and Q:R=8:7. Find P:Q:R. | Show answer + formula + trick6:8:7.Scale 3:4 by 2 to make Q=8. Formula / rule For linked ratios, make the shared part equal. Tip / trick Use the LCM of the shared ratio numbers. Remember Match the middle. |
| 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 + trick12.Let one part=4 because blue needs one extra part to equal red. Red=3×4=12. Formula / rule Use the difference in ratio parts to match the number added. Tip / trick Translate 'become equal' into the gap between ratio parts. Remember Difference in parts = added amount. |
| 2B | Changing a ratio Similar practice | Cats:dogs=5:3. After adding 8 dogs, numbers are equal. How many cats originally? | Show answer + formula + trick20.The difference is 2 parts=8, so 1 part=4; cats=20. Formula / rule Use the difference in ratio parts to match the number added. Tip / trick Translate 'become equal' into the gap between ratio parts. Remember Difference in parts = added amount. |
| 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 + trick48 worker-days.8×6=48. Formula / rule For simple work-rate questions, total work can be represented by workers×time. Tip / trick Only use inverse proportion if the question states the same total task/rate conditions. Remember More workers, less time for same work. |
| 3B | Proportion with inverse thinking Similar practice | 5 machines run for 12 hours. How many machine-hours? | Show answer + formula + trick60 machine-hours.5×12=60. Formula / rule For simple work-rate questions, total work can be represented by workers×time. Tip / trick Only use inverse proportion if the question states the same total task/rate conditions. Remember More workers, less time for same work. |
| 4A | Recipe scaling non-integer | A recipe for 8 uses 600 g flour. How much for 14? | Show answer + formula + trick1,050 g.600÷8=75 g each; 75×14=1,050. Formula / rule Direct proportion: multiply by new amount ÷ original amount. Tip / trick Unitary method avoids fraction mistakes. Remember Find one serving. |
| 4B | Recipe scaling non-integer Similar practice | A recipe for 5 uses 350 ml milk. How much for 12? | Show answer + formula + trick840 ml.350÷5=70 ml each; 70×12=840. Formula / rule Direct proportion: multiply by new amount ÷ original amount. Tip / trick Unitary method avoids fraction mistakes. Remember Find one serving. |
| 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 + trick8 cm.4 m=400 cm; 400÷50=8 cm. Formula / rule For scale 1:n, drawing length = real length ÷ n, after matching units. Tip / trick Convert units before applying the scale. Remember Same units first. |
| 5B | Scale drawing reverse Similar practice | A plan uses scale 1:200. A real wall is 12 m. Length on plan? | Show answer + formula + trick6 cm.12 m=1,200 cm; 1,200÷200=6 cm. Formula / rule For scale 1:n, drawing length = real length ÷ n, after matching units. Tip / trick Convert units before applying the scale. Remember Same units first. |
| 6A | Ratio and fraction conversion | In a bag red:green=4:7. What fraction are red? | Show answer + formula + trick4/11.Total parts=11; red=4 parts. Formula / rule Fraction of a ratio part = chosen parts ÷ total parts. Tip / trick Add ratio numbers for the denominator. Remember Part over total parts. |
| 6B | Ratio and fraction conversion Similar practice | Boys:girls=5:8. What fraction are boys? | Show answer + formula + trick5/13.Total parts=13; boys=5 parts. Formula / rule Fraction of a ratio part = chosen parts ÷ total parts. Tip / trick Add ratio numbers for the denominator. Remember Part over total parts. |
| 7A | Ratio from difference | A:B=7:4 and A is 18 more than B. Find A. | Show answer + formula + trick42.Difference=3 parts=18, so 1 part=6; A=7×6=42. Formula / rule Difference in values corresponds to difference in ratio parts. Tip / trick Find one part from the difference. Remember Gap in ratio ↔ gap in value. |
| 7B | Ratio from difference Similar practice | P:Q=9:5 and P is 28 more than Q. Find Q. | Show answer + formula + trick35.Difference=4 parts=28, so 1 part=7; Q=5×7=35. Formula / rule Difference in values corresponds to difference in ratio parts. Tip / trick Find one part from the difference. Remember Gap in ratio ↔ gap in value. |
| 8A | Ratio from one amount | Apples:pears=5:3. There are 45 apples. How many fruit altogether? | Show answer + formula + trick72.5 parts=45 so 1 part=9; total 8 parts=72. Formula / rule Use known amount to find one part. Tip / trick Do not stop after finding the other category if the question asks total. Remember Known part → one part → required. |
| 8B | Ratio from one amount Similar practice | Red:blue=4:7. There are 28 red. How many counters altogether? | Show answer + formula + trick77.4 parts=28 so 1 part=7; total 11 parts=77. Formula / rule Use known amount to find one part. Tip / trick Do not stop after finding the other category if the question asks total. Remember Known part → one part → required. |
| 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 Cost is directly proportional to quantity when unit price is fixed. Tip / trick Find cost per one unit first. Remember Unit price first. |
| 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 Cost is directly proportional to quantity when unit price is fixed. Tip / trick Find cost per one unit first. Remember Unit price first. |
| 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 + trick30 km.7.5×4=30. Formula / rule Apply the scale as a multiplicative relationship. Tip / trick For unusual scales, first find what 1 cm represents. Remember One centimetre first. |
| 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 + trick20 km.8÷2=4 scale units; 4×5=20 km. Formula / rule Apply the scale as a multiplicative relationship. Tip / trick For unusual scales, first find what 1 cm represents. Remember One centimetre first. |
24. Tiffin Focus: Time, Timetables & Time Zones
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 | Elapsed time across hour | A lesson starts at 09:47 and ends at 11:13. How long is it? | Show answer + formula + trick1 h 26 min.09:47→10:00=13 min, →11:00=1 h, →11:13=13 min. Formula / rule Duration = finish time - start time. Tip / trick Bridge to the next hour instead of subtracting minutes with borrowing. Remember Bridge through whole hours. |
| 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 + trick1 h 27 min.14:38→15:00=22 min, →16:00=1 h, →16:05=5 min. Formula / rule Duration = finish time - start time. Tip / trick Bridge to the next hour instead of subtracting minutes with borrowing. Remember Bridge through whole hours. |
| 2A | Find start time | A film ends at 21:35 and lasts 1 h 48 min. When did it start? | Show answer + formula + trick19:47.21:35-1 h=20:35; subtract 48 min=19:47. Formula / rule Start = finish - duration. Tip / trick Subtract hours first, then minutes carefully. Remember Work backwards. |
| 2B | Find start time Similar practice | A train arrives at 17:12 after 2 h 37 min. When did it leave? | Show answer + formula + trick14:35.17:12-2 h=15:12; subtract 37 min=14:35. Formula / rule Start = finish - duration. Tip / trick Subtract hours first, then minutes carefully. Remember Work backwards. |
| 3A | Find finish time | A test starts at 10:25 and lasts 1 h 55 min. When does it finish? | Show answer + formula + trick12:20.10:25+1 h=11:25; +55 min=12:20. Formula / rule Finish = start + duration. Tip / trick When minutes pass 60, exchange 60 minutes for 1 hour. Remember 60 minutes = 1 hour. |
| 3B | Find finish time Similar practice | A match starts at 13:48 and lasts 2 h 36 min. Finish? | Show answer + formula + trick16:24.13:48+2 h=15:48; +36 min=16:24. Formula / rule Finish = start + duration. Tip / trick When minutes pass 60, exchange 60 minutes for 1 hour. Remember 60 minutes = 1 hour. |
| 4A | Timetable wait | A bus arrives at 08:42. The next train leaves at 09:07. Waiting time? | Show answer + formula + trick25 min.08:42→09:00=18 min, plus 7=25 min. Formula / rule Waiting time = departure of next service - arrival. Tip / trick Bridge to the hour. Remember Arrival to next departure. |
| 4B | Timetable wait Similar practice | A train arrives at 15:53. A connection leaves at 16:21. Waiting time? | Show answer + formula + trick28 min.15:53→16:00=7 min, plus 21=28 min. Formula / rule Waiting time = departure of next service - arrival. Tip / trick Bridge to the hour. Remember Arrival to next departure. |
| 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 + trick10:06.The fourth is 3 intervals later: 3×18=54 min; 09:12+54=10:06. Formula / rule Nth departure is (n-1) intervals after the first. Tip / trick Do not multiply by n; the first service is already at the starting time. Remember Number of gaps = number of services minus one. |
| 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 + trick15:50.Four intervals later: 4×25=100 min; 14:10+1 h 40=15:50. Formula / rule Nth departure is (n-1) intervals after the first. Tip / trick Do not multiply by n; the first service is already at the starting time. Remember Number of gaps = number of services minus one. |
| 6A | 12-hour to 24-hour | Write 12:35 am in 24-hour time. | Show answer + formula + trick00:35.After midnight, 12:xx am becomes 00:xx. Formula / rule Midnight=00:00; noon=12:00. Tip / trick 12 am and 12 pm are the common traps. Remember 12 am → 00; 12 pm → 12. |
| 6B | 12-hour to 24-hour Similar practice | Write 12:20 pm in 24-hour time. | Show answer + formula + trick12:20.Noon is 12:00, so 12:20 pm stays 12:20. Formula / rule Midnight=00:00; noon=12:00. Tip / trick 12 am and 12 pm are the common traps. Remember 12 am → 00; 12 pm → 12. |
| 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 + trick16:35.Athens is ahead, so add 2 hours. Formula / rule If destination is ahead, add the time difference. Tip / trick Write an arrow showing which city is ahead. Remember Ahead = add. |
| 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 + trick13:20.Dubai is ahead, so add 5 hours. Formula / rule If destination is ahead, add the time difference. Tip / trick Write an arrow showing which city is ahead. Remember Ahead = add. |
| 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 + trick03:30 next day.20:30+7 h=27:30, which is 03:30 next day. Formula / rule When total hours reach 24, subtract 24 and move to the next day. Tip / trick Mark 'next day' whenever you cross midnight. Remember Past 24 → 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 + trick03:15 next day.18:15+9 h=27:15=03:15 next day. Formula / rule When total hours reach 24, subtract 24 and move to the next day. Tip / trick Mark 'next day' whenever you cross midnight. Remember Past 24 → next day. |
| 9A | Minutes as fraction of hour | What fraction of an hour is 45 minutes? | Show answer + formula + trick3/4.45/60=3/4. Formula / rule Fraction of hour = minutes/60, then simplify. Tip / trick Memorise 15=1/4, 20=1/3, 30=1/2, 45=3/4. Remember Minutes over 60. |
| 9B | Minutes as fraction of hour Similar practice | What fraction of an hour is 20 minutes? | Show answer + formula + trick1/3.20/60=1/3. Formula / rule Fraction of hour = minutes/60, then simplify. Tip / trick Memorise 15=1/4, 20=1/3, 30=1/2, 45=3/4. Remember Minutes over 60. |
| 10A | Decimal hours | Convert 2.5 hours to hours and minutes. | Show answer + formula + trick2 h 30 min.0.5 hour=30 min. Formula / rule Decimal part of an hour ×60 = minutes. Tip / trick Do not read 1.75 h as 1 h 75 min. Remember Decimal ×60. |
| 10B | Decimal hours Similar practice | Convert 1.75 hours to hours and minutes. | Show answer + formula + trick1 h 45 min.0.75 hour=45 min. Formula / rule Decimal part of an hour ×60 = minutes. Tip / trick Do not read 1.75 h as 1 h 75 min. Remember Decimal ×60. |
25. Tiffin Focus: Speed, Distance, Rates & Unitary Method
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 | Speed | A car travels 216 km in 3 hours. What is its average speed? | Show answer + formula + trick72 km/h.216÷3=72. Formula / rule Speed = distance ÷ time. Tip / trick Cover the quantity you want in the speed-distance-time triangle. Remember S = D/T. |
| 1B | Speed Similar practice | A train travels 315 km in 4.5 hours. Average speed? | Show answer + formula + trick70 km/h.315÷4.5=70. Formula / rule Speed = distance ÷ time. Tip / trick Cover the quantity you want in the speed-distance-time triangle. Remember S = D/T. |
| 2A | Distance | A cyclist travels at 18 km/h for 2.5 hours. Distance? | Show answer + formula + trick45 km.18×2.5=45. Formula / rule Distance = speed × time. Tip / trick Convert minutes to hours before multiplying when needed. Remember D = S×T. |
| 2B | Distance Similar practice | A boat travels at 24 km/h for 3.25 hours. Distance? | Show answer + formula + trick78 km.24×3.25=78. Formula / rule Distance = speed × time. Tip / trick Convert minutes to hours before multiplying when needed. Remember D = S×T. |
| 3A | Time | How long does it take to travel 150 km at 60 km/h? | Show answer + formula + trick2.5 h.150÷60=2.5. Formula / rule Time = distance ÷ speed. Tip / trick State whether answer is decimal hours or hours/minutes. Remember T = D/S. |
| 3B | Time Similar practice | How long to travel 96 km at 32 km/h? | Show answer + formula + trick3 h.96÷32=3. Formula / rule Time = distance ÷ speed. Tip / trick State whether answer is decimal hours or hours/minutes. Remember T = D/S. |
| 4A | Minutes in speed problem | A car travels 50 km/h for 36 minutes. How far? | Show answer + formula + trick30 km.36 min=0.6 h; 50×0.6=30. Formula / rule Convert minutes to hours by dividing by 60. Tip / trick Common fractions: 30 min=1/2 h, 45=3/4 h, 15=1/4 h. Remember Minutes ÷60. |
| 4B | Minutes in speed problem Similar practice | A cyclist travels 24 km/h for 45 minutes. How far? | Show answer + formula + trick18 km.45 min=0.75 h; 24×0.75=18. Formula / rule Convert minutes to hours by dividing by 60. Tip / trick Common fractions: 30 min=1/2 h, 45=3/4 h, 15=1/4 h. Remember Minutes ÷60. |
| 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 ÷ number of units. Tip / trick Find one before scaling up. Remember Per one = divide. |
| 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 ÷ number of units. Tip / trick Find one before scaling up. Remember Per one = divide. |
| 6A | Rate comparison | Machine A makes 120 items in 4 h. Machine B makes 140 in 5 h. Which is faster? | Show answer + formula + trickMachine A.A=30/h, B=28/h. Formula / rule Compare rates using the same unit. Tip / trick Never compare totals when times differ. Remember Convert to per hour. |
| 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 + trickRunner A.A=9 km/h; B≈8.33 km/h. Formula / rule Compare rates using the same unit. Tip / trick Never compare totals when times differ. Remember Convert to per hour. |
| 7A | Filling rate | A tap fills 15 litres in 3 minutes. How much in 8 minutes at the same rate? | Show answer + formula + trick40 L.5 L/min ×8=40. Formula / rule Direct proportion: amount = rate × time. Tip / trick Find one-minute rate first. Remember One minute first. |
| 7B | Filling rate Similar practice | A machine packs 42 boxes in 6 minutes. How many in 15 minutes? | Show answer + formula + trick105.7 boxes/min ×15=105. Formula / rule Direct proportion: amount = rate × time. Tip / trick Find one-minute rate first. Remember One minute first. |
| 8A | Combined rate interpretation | Two taps each fill 6 L/min. Together, how much in 7 min? | Show answer + formula + trick84 L.Combined rate=12 L/min; 12×7=84. Formula / rule For identical simultaneous rates, add the rates. Tip / trick Find total per minute before multiplying by time. Remember Add rates, then × time. |
| 8B | Combined rate interpretation Similar practice | Three identical machines each make 8 parts/min. How many parts in 5 min? | Show answer + formula + trick120.Combined rate=24/min; ×5=120. Formula / rule For identical simultaneous rates, add the rates. Tip / trick Find total per minute before multiplying by time. Remember Add rates, then × time. |
| 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 + trick3 h.First leg 2 h, second 1 h; total 3 h. Formula / rule For multi-leg journeys, calculate time for each leg separately. Tip / trick Do not average the speeds unless the question justifies it. Remember Find each leg. |
| 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 + trick3 h.2 h +1 h =3 h. Formula / rule For multi-leg journeys, calculate time for each leg separately. Tip / trick Do not average the speeds unless the question justifies it. Remember Find each leg. |
| 10A | Rate word problem | A printer produces 18 pages per minute. How long for 450 pages? | Show answer + formula + trick25 min.450÷18=25. Formula / rule Time = total amount ÷ rate. Tip / trick Check units match. Remember Total ÷ per-minute. |
| 10B | Rate word problem Similar practice | A pump moves 24 L/min. How long for 600 L? | Show answer + formula + trick25 min.600÷24=25. Formula / rule Time = total amount ÷ rate. Tip / trick Check units match. Remember Total ÷ per-minute. |
26. Tiffin Focus: Geometry, Angles & Shape Properties
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 | 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 + trick116°.Adjacent angles on a straight line total 180°: 180-64=116. Formula / rule Corresponding/alternate angles are equal; adjacent angles on a straight line total 180°. Tip / trick Mark equal angles first, then use 180°. Remember Equal or supplementary. |
| 1B | Angles in parallel lines Similar practice | An acute angle formed by parallel lines is 73°. Adjacent obtuse angle? | Show answer + formula + trick107°.180-73=107. Formula / rule Corresponding/alternate angles are equal; adjacent angles on a straight line total 180°. Tip / trick Mark equal angles first, then use 180°. Remember Equal or supplementary. |
| 2A | Triangle angle ratio | Angles in a triangle are in ratio 2:3:4. Find the largest angle. | Show answer + formula + trick80°.Total parts=9; one part=180÷9=20; largest=4×20=80. Formula / rule Triangle angle total is 180°. Tip / trick Share 180 in the given ratio. Remember Triangle ratio shares 180. |
| 2B | Triangle angle ratio Similar practice | Triangle angles are in ratio 3:4:5. Find the smallest. | Show answer + formula + trick45°.12 parts; one=15°; smallest=45°. Formula / rule Triangle angle total is 180°. Tip / trick Share 180 in the given ratio. Remember Triangle ratio shares 180. |
| 3A | Quadrilateral angle ratio | Angles of a quadrilateral are in ratio 2:3:4:3. Find the largest. | Show answer + formula + trick120°.12 parts; one=30°; largest=4×30=120°. Formula / rule Quadrilateral angle total is 360°. Tip / trick Share 360 in the ratio. Remember Quad ratio shares 360. |
| 3B | Quadrilateral angle ratio Similar practice | Angles are in ratio 1:2:3:4. Find the second largest. | Show answer + formula + trick108°.10 parts; one=36°; second largest=3×36=108°. Formula / rule Quadrilateral angle total is 360°. Tip / trick Share 360 in the ratio. Remember Quad ratio shares 360. |
| 4A | Regular polygon exterior angle | What is each exterior angle of a regular hexagon? | Show answer + formula + trick60°.360÷6=60°. Formula / rule Exterior angles of any polygon total 360°; regular polygon exterior angle=360÷number of sides. Tip / trick Exterior angle is often quicker than interior angle. Remember Outside total = 360. |
| 4B | Regular polygon exterior angle Similar practice | What is each exterior angle of a regular decagon? | Show answer + formula + trick36°.360÷10=36°. Formula / rule Exterior angles of any polygon total 360°; regular polygon exterior angle=360÷number of sides. Tip / trick Exterior angle is often quicker than interior angle. Remember Outside total = 360. |
| 5A | Regular polygon interior angle | Find each interior angle of a regular pentagon. | Show answer + formula + trick108°.Exterior=360÷5=72; interior=180-72=108. Formula / rule Interior angle of regular polygon = 180° - exterior angle. Tip / trick Find exterior first using 360÷n. Remember Interior + exterior =180. |
| 5B | Regular polygon interior angle Similar practice | Find each interior angle of a regular octagon. | Show answer + formula + trick135°.Exterior=360÷8=45; interior=180-45=135. Formula / rule Interior angle of regular polygon = 180° - exterior angle. Tip / trick Find exterior first using 360÷n. Remember Interior + exterior =180. |
| 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 + trick65°.Interior vertex=50°. Remaining 130° shared equally=65°. Formula / rule Exterior and adjacent interior total 180°; triangle total 180°. Tip / trick Convert exterior to interior first. Remember Exterior → interior → share. |
| 6B | Isosceles exterior Similar practice | Vertex exterior angle is 100°. Find each base angle. | Show answer + formula + trick50°.Interior vertex=80°, remaining 100°, each base=50°. Formula / rule Exterior and adjacent interior total 180°; triangle total 180°. Tip / trick Convert exterior to interior first. Remember Exterior → interior → share. |
| 7A | Shape property identification | Which quadrilateral always has exactly one pair of parallel sides in the usual UK definition? | Show answer + formula + trickTrapezium.A trapezium has one pair of parallel sides. Formula / rule Know defining properties of square, rectangle, rhombus, parallelogram, kite and trapezium. Tip / trick Look for the word 'always'. Remember Properties define the shape. |
| 7B | Shape property identification Similar practice | Which quadrilateral has four equal sides but need not have four right angles? | Show answer + formula + trickRhombus.A rhombus has four equal sides. Formula / rule Know defining properties of square, rectangle, rhombus, parallelogram, kite and trapezium. Tip / trick Look for the word 'always'. Remember Properties define the shape. |
| 8A | Diagonal properties | Which quadrilateral has diagonals that are equal and bisect each other but need not meet at right angles? | Show answer + formula + trickRectangle.Rectangle diagonals are equal and bisect each other. Formula / rule Diagonal properties help distinguish quadrilaterals. Tip / trick Square satisfies both rectangle and rhombus diagonal properties. Remember Equal? Perpendicular? Bisect? |
| 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 + trickRhombus.Rhombus diagonals are perpendicular bisectors. Formula / rule Diagonal properties help distinguish quadrilaterals. Tip / trick Square satisfies both rectangle and rhombus diagonal properties. Remember Equal? Perpendicular? Bisect? |
| 9A | Symmetry properties | A regular pentagon has how many lines of symmetry? | Show answer + formula + trick5.A regular n-gon has n lines of symmetry. Formula / rule Regular n-gon: n lines of symmetry and rotational order n. Tip / trick Regular shapes are highly symmetric. Remember Regular n → n. |
| 9B | Symmetry properties Similar practice | A regular octagon has rotational symmetry of what order? | Show answer + formula + trick8.A regular n-gon has rotational symmetry order n. Formula / rule Regular n-gon: n lines of symmetry and rotational order n. Tip / trick Regular shapes are highly symmetric. Remember Regular n → n. |
| 10A | Angle chase multi-step | A triangle has one angle 52°. The other two are equal. Find each equal angle. | Show answer + formula + trick64°.180-52=128; 128÷2=64. Formula / rule Use the total angle sum then divide the remaining amount equally where stated. Tip / trick Subtract known angles before sharing. Remember Total, subtract, share. |
| 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 + trick90°.360-180=180; 180÷2=90. Formula / rule Use the total angle sum then divide the remaining amount equally where stated. Tip / trick Subtract known angles before sharing. Remember Total, subtract, share. |
27. Tiffin Focus: Perimeter, Area & Volume Reverse Problems
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 | Rectangle missing side from perimeter | A rectangle has perimeter 46 cm and length 15 cm. Find width. | Show answer + formula + trick8 cm.2(L+W)=46, so L+W=23; W=8. Formula / rule Rectangle perimeter P=2(L+W). Tip / trick Halve the perimeter first. Remember Half P = L+W. |
| 1B | Rectangle missing side from perimeter Similar practice | Perimeter 62 cm, length 19 cm. Width? | Show answer + formula + trick12 cm.L+W=31, so W=12. Formula / rule Rectangle perimeter P=2(L+W). Tip / trick Halve the perimeter first. Remember Half P = L+W. |
| 2A | Rectangle missing side from area | A rectangle has area 96 cm² and length 12 cm. Find width. | Show answer + formula + trick8 cm.96÷12=8. Formula / rule Rectangle area A=L×W. Tip / trick Missing side = area ÷ known side. Remember Area divided by side. |
| 2B | Rectangle missing side from area Similar practice | Area 135 cm², length 15 cm. Width? | Show answer + formula + trick9 cm.135÷15=9. Formula / rule Rectangle area A=L×W. Tip / trick Missing side = area ÷ known side. Remember Area divided by side. |
| 3A | Triangle missing height | A triangle has area 54 cm² and base 12 cm. Find perpendicular height. | Show answer + formula + trick9 cm.54=1/2×12×h=6h, so h=9. Formula / rule Triangle area A=1/2 bh. Tip / trick Double the area, then divide by base. Remember Height = 2A÷b. |
| 3B | Triangle missing height Similar practice | Area 70 cm², base 14 cm. Height? | Show answer + formula + trick10 cm.70=7h, so h=10. Formula / rule Triangle area A=1/2 bh. Tip / trick Double the area, then divide by base. Remember Height = 2A÷b. |
| 4A | Composite L-shape area | A 12×10 rectangle has a 4×3 rectangle removed. Remaining area? | Show answer + formula + trick108 cm².120-12=108. Formula / rule Composite area = add useful parts or subtract missing parts. Tip / trick Use the biggest easy rectangle then subtract the cut-out. Remember Big minus missing. |
| 4B | Composite L-shape area Similar practice | A 15×8 rectangle has a 5×2 rectangle removed. Remaining area? | Show answer + formula + trick110 cm².120-10=110. Formula / rule Composite area = add useful parts or subtract missing parts. Tip / trick Use the biggest easy rectangle then subtract the cut-out. Remember Big minus missing. |
| 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 + trick32 cm.The removed corner replaces two outer lengths with equal inner lengths, so perimeter remains 2(10+6)=32. Formula / rule A rectangular corner notch can leave total perimeter unchanged because removed lengths are replaced by equal inner lengths. Tip / trick Trace every boundary segment once. Remember Trace, do not guess. |
| 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 + trick46 cm.Perimeter remains 2(14+9)=46. Formula / rule A rectangular corner notch can leave total perimeter unchanged because removed lengths are replaced by equal inner lengths. Tip / trick Trace every boundary segment once. Remember Trace, do not guess. |
| 6A | Cuboid missing dimension | Volume=360 cm³, length=12 cm, width=5 cm. Height? | Show answer + formula + trick6 cm.h=360÷(12×5)=6. Formula / rule Cuboid volume V=LWH. Tip / trick Missing dimension = volume ÷ product of other two. Remember Volume divided by face area. |
| 6B | Cuboid missing dimension Similar practice | Volume=504 cm³, length=9 cm, width=7 cm. Height? | Show answer + formula + trick8 cm.504÷63=8. Formula / rule Cuboid volume V=LWH. Tip / trick Missing dimension = volume ÷ product of other two. Remember Volume divided by face area. |
| 7A | Cube side from volume | A cube has volume 512 cm³. Side length? | Show answer + formula + trick8 cm.8³=512. Formula / rule Cube volume=s³. Tip / trick Know common cubes: 1,8,27,64,125,216,343,512,729,1000. Remember Cube root asks: what number cubed? |
| 7B | Cube side from volume Similar practice | A cube has volume 729 cm³. Side length? | Show answer + formula + trick9 cm.9³=729. Formula / rule Cube volume=s³. Tip / trick Know common cubes: 1,8,27,64,125,216,343,512,729,1000. Remember Cube root asks: what number cubed? |
| 8A | Area scaling | A rectangle's length and width are both doubled. By what factor does its area change? | Show answer + formula + trick4.Area scale factor=2×2=4. Formula / rule If all lengths scale by k, area scales by k². Tip / trick Do not say 'double length means double area'. Remember Area squares the scale. |
| 8B | Area scaling Similar practice | A square's side is tripled. By what factor does area change? | Show answer + formula + trick9.3²=9. Formula / rule If all lengths scale by k, area scales by k². Tip / trick Do not say 'double length means double area'. Remember Area squares the scale. |
| 9A | Volume scaling | A cube's side length doubles. By what factor does volume change? | Show answer + formula + trick8.2³=8. Formula / rule If all lengths scale by k, volume scales by k³. Tip / trick Volume changes faster than length. Remember Volume cubes the scale. |
| 9B | Volume scaling Similar practice | All dimensions of a cuboid are tripled. Volume factor? | Show answer + formula + trick27.3³=27. Formula / rule If all lengths scale by k, volume scales by k³. Tip / trick Volume changes faster than length. Remember Volume cubes the scale. |
| 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 + trick6 cm.L+W=20, so W=6. Formula / rule Equal perimeter means set the perimeter expressions equal. Tip / trick Find the shared perimeter first. Remember Same perimeter, different shape. |
| 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 + trick8 cm.Square perimeter=48; rectangle L+W=24; W=8. Formula / rule Equal perimeter means set the perimeter expressions equal. Tip / trick Find the shared perimeter first. Remember Same perimeter, different shape. |
28. Tiffin Focus: Position, Direction, Reflection & Rotation
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 | 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: new point=(x+a,y+b). Tip / trick Handle x and y separately. Remember Add the vector. |
| 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: new point=(x+a,y+b). Tip / trick Handle x and y separately. Remember Add the vector. |
| 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 In x-axis: (x,y)→(x,−y). Tip / trick x-axis flips the y-coordinate. Remember x-axis: y changes. |
| 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 In x-axis: (x,y)→(x,−y). Tip / trick x-axis flips the y-coordinate. Remember x-axis: y changes. |
| 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 In y-axis: (x,y)→(−x,y). Tip / trick y-axis flips the x-coordinate. Remember y-axis: x changes. |
| 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 In y-axis: (x,y)→(−x,y). Tip / trick y-axis flips the x-coordinate. Remember y-axis: x changes. |
| 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° about origin: (x,y)→(−x,−y). Tip / trick A half-turn sends each point directly opposite. Remember 180°: change both signs. |
| 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° about origin: (x,y)→(−x,−y). Tip / trick A half-turn sends each point directly opposite. Remember 180°: change both signs. |
| 5A | 90° direction reasoning | You face north and turn 90° clockwise. Which direction now? | Show answer + formula + trickEast.Clockwise quarter-turn from north points east. Formula / rule A quarter-turn is 90°. Tip / trick Draw N-E-S-W as a tiny compass if needed. Remember Clockwise goes N→E→S→W. |
| 5B | 90° direction reasoning Similar practice | You face west and turn 90° anticlockwise. Which direction now? | Show answer + formula + trickSouth.Anticlockwise from west points south. Formula / rule A quarter-turn is 90°. Tip / trick Draw N-E-S-W as a tiny compass if needed. Remember Clockwise goes N→E→S→W. |
| 6A | Three-quarter turn | You face south and turn 270° clockwise. Which direction now? | Show answer + formula + trickEast.270° clockwise equals 90° anticlockwise; south→east. Formula / rule 270° one way equals 90° the opposite way. Tip / trick Convert three-quarter turns into easier quarter-turns. Remember 270° = opposite 90°. |
| 6B | Three-quarter turn Similar practice | You face east and turn 270° anticlockwise. Direction? | Show answer + formula + trickSouth.270° anticlockwise equals 90° clockwise; east→south. Formula / rule 270° one way equals 90° the opposite way. Tip / trick Convert three-quarter turns into easier quarter-turns. Remember 270° = opposite 90°. |
| 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 Track horizontal and vertical movement separately. Tip / trick Combine all x-movements and all y-movements. Remember Horizontal with x, vertical with y. |
| 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 Track horizontal and vertical movement separately. Tip / trick Combine all x-movements and all y-movements. Remember Horizontal with x, vertical with y. |
| 8A | Route and turn | Walk north 4, east 3, south 4. Where are you relative to start? | Show answer + formula + trick3 units east.North and south cancel. Formula / rule Net displacement combines signed horizontal and vertical movement. Tip / trick Cancel opposite movements first. Remember Opposites cancel. |
| 8B | Route and turn Similar practice | Walk west 5, north 2, east 8. Relative position? | Show answer + formula + trick3 east and 2 north.Horizontal net=3 east; vertical=2 north. Formula / rule Net displacement combines signed horizontal and vertical movement. Tip / trick Cancel opposite movements first. Remember Opposites cancel. |
| 9A | Symmetry count | How many lines of symmetry does a regular hexagon have? | Show answer + formula + trick6.A regular n-gon has n lines of symmetry. Formula / rule Use exact shape properties. Tip / trick Regular polygons have symmetry equal to number of sides. Remember Regular n → n lines. |
| 9B | Symmetry count Similar practice | How many lines of symmetry does a non-square rectangle have? | Show answer + formula + trick2.Horizontal and vertical lines through the centre. Formula / rule Use exact shape properties. Tip / trick Regular polygons have symmetry equal to number of sides. Remember Regular n → n lines. |
| 10A | Rotation order | What is the order of rotational symmetry of a square? | Show answer + formula + trick4.It matches at 90°,180°,270°,360°. Formula / rule Order = number of times shape matches itself in a full 360° turn. Tip / trick Angle of smallest rotation=360÷order. Remember Order counts matches in 360°. |
| 10B | Rotation order Similar practice | What is the order of rotational symmetry of an equilateral triangle? | Show answer + formula + trick3.It matches at 120°,240°,360°. Formula / rule Order = number of times shape matches itself in a full 360° turn. Tip / trick Angle of smallest rotation=360÷order. Remember Order counts matches in 360°. |
29. Tiffin Focus: Statistics, Tables & Graph Interpretation
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 | Mean from table | Values 2,3,4 have frequencies 3,4,3. Find mean. | Show answer + formula + trick3.Weighted total=2×3+3×4+4×3=30; frequency total=10; mean=3. Formula / rule Frequency mean=sum(value×frequency) ÷ total frequency. Tip / trick Build a weighted total before dividing. Remember Multiply, add, divide. |
| 1B | Mean from table Similar practice | Values 1,2,5 have frequencies 2,5,3. Find mean. | Show answer + formula + trick2.7.Weighted total=2+10+15=27; total=10. Formula / rule Frequency mean=sum(value×frequency) ÷ total frequency. Tip / trick Build a weighted total before dividing. Remember Multiply, add, divide. |
| 2A | Missing frequency | Scores 1,2,3 have frequencies 4, x, 5. Total frequency is 15. Find x. | Show answer + formula + trick6.4+x+5=15, so x=6. Formula / rule Frequencies add to the total number of observations. Tip / trick Subtract known frequencies from total. Remember Total minus known. |
| 2B | Missing frequency Similar practice | Categories A,B,C have frequencies 7,x,9. Total=24. Find x. | Show answer + formula + trick8.7+x+9=24, so x=8. Formula / rule Frequencies add to the total number of observations. Tip / trick Subtract known frequencies from total. Remember Total minus known. |
| 3A | Mean reverse | The mean of 8 values is 12.5. What is their total? | Show answer + formula + trick100.12.5×8=100. Formula / rule Total = mean × number of values. Tip / trick This is the reverse of finding mean. Remember Mean × count = total. |
| 3B | Mean reverse Similar practice | Mean of 6 values is 14. What is total? | Show answer + formula + trick84.14×6=84. Formula / rule Total = mean × number of values. Tip / trick This is the reverse of finding mean. Remember Mean × count = total. |
| 4A | Pie chart count | A 90° sector represents 18 pupils. How many pupils in the whole group? | Show answer + formula + trick72.90° is 1/4 of 360°, so whole=18×4=72. Formula / rule Count represented by sector is proportional to sector angle. Tip / trick Find what fraction of 360° the sector is. Remember Sector fraction → scale up. |
| 4B | Pie chart count Similar practice | A 72° sector represents 10 pupils. Whole group? | Show answer + formula + trick50.72° is 1/5 of 360°, so total=50. Formula / rule Count represented by sector is proportional to sector angle. Tip / trick Find what fraction of 360° the sector is. Remember Sector fraction → scale up. |
| 5A | Pie chart sector from count | 12 of 48 pupils choose art. What sector angle? | Show answer + formula + trick90°.12/48=1/4; 1/4×360=90°. Formula / rule Sector angle = part/whole ×360°. Tip / trick Simplify the fraction first. Remember Part over whole ×360. |
| 5B | Pie chart sector from count Similar practice | 15 of 60 choose music. Sector angle? | Show answer + formula + trick90°.15/60=1/4; ×360=90°. Formula / rule Sector angle = part/whole ×360°. Tip / trick Simplify the fraction first. Remember Part over whole ×360. |
| 6A | Bar chart scale trap | A bar reaches the third small interval after 40. Each interval is 5. What value? | Show answer + formula + trick55.40+3×5=55. Formula / rule Read minor interval value before reading the bar. Tip / trick Count spaces, not just grid lines. Remember Scale first. |
| 6B | Bar chart scale trap Similar practice | A bar reaches two small intervals after 70. Each interval is 4. Value? | Show answer + formula + trick78.70+2×4=78. Formula / rule Read minor interval value before reading the bar. Tip / trick Count spaces, not just grid lines. Remember Scale first. |
| 7A | Percentage from table | Out of 80 pupils, 28 choose option A. What percentage? | Show answer + formula + trick35%.28/80×100=35%. Formula / rule Percentage = part ÷ whole ×100. Tip / trick Simplify fraction if possible. Remember Part/whole ×100. |
| 7B | Percentage from table Similar practice | Out of 120 people, 42 choose B. Percentage? | Show answer + formula + trick35%.42/120×100=35%. Formula / rule Percentage = part ÷ whole ×100. Tip / trick Simplify fraction if possible. Remember Part/whole ×100. |
| 8A | Median with frequency | Values 1,2,3 occur 2,5,3 times. What is median? | Show answer + formula + trick2.There are 10 values; 5th and 6th values are both 2. Formula / rule Median is the middle value(s) after ordering all data. Tip / trick Use cumulative frequency positions rather than writing every value. Remember Find middle positions. |
| 8B | Median with frequency Similar practice | Values 4,5,6 occur 3,4,3 times. Median? | Show answer + formula + trick5.10 values; 5th and 6th are 5. Formula / rule Median is the middle value(s) after ordering all data. Tip / trick Use cumulative frequency positions rather than writing every value. Remember Find middle positions. |
| 9A | Range after change | A data set has minimum 12 and maximum 47. If maximum increases to 52, new range? | Show answer + formula + trick40.52−12=40. Formula / rule Range=max−min. Tip / trick Only minimum and maximum matter. Remember Ignore middle values. |
| 9B | Range after change Similar practice | Minimum is 8, maximum 33. Minimum changes to 5. New range? | Show answer + formula + trick28.33−5=28. Formula / rule Range=max−min. Tip / trick Only minimum and maximum matter. Remember Ignore middle values. |
| 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 + trick3.Axis truncation can exaggerate visual difference, but arithmetic difference is 55−52=3. Formula / rule Always calculate from the labelled values rather than visual bar height. Tip / trick Truncated axes can make small differences look huge. Remember Read numbers, not appearance. |
| 10B | Interpreting misleading graph Similar practice | Bars are 101 and 104 with axis starting at 100. Absolute difference? | Show answer + formula + trick3.104−101=3. Formula / rule Always calculate from the labelled values rather than visual bar height. Tip / trick Truncated axes can make small differences look huge. Remember Read numbers, not appearance. |
30. Tiffin Focus: Algebra, Sequences & Function Machines
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 | Two-step function | A machine multiplies by 3 then adds 5. Input 7. Output? | Show answer + formula + trick26.7×3+5=26. Formula / rule Apply operations in the stated order. Tip / trick Write each intermediate result. Remember Follow arrows in order. |
| 1B | Two-step function Similar practice | A machine multiplies by 4 then subtracts 6. Input 9. Output? | Show answer + formula + trick30.9×4−6=30. Formula / rule Apply operations in the stated order. Tip / trick Write each intermediate result. Remember Follow arrows in order. |
| 2A | Reverse function | A machine multiplies by 6 then adds 4. Output is 58. Input? | Show answer + formula + trick9.58−4=54; 54÷6=9. Formula / rule Reverse operations in reverse order. Tip / trick Start at the output and undo. Remember Backwards and inverse. |
| 2B | Reverse function Similar practice | A machine doubles then subtracts 7. Output 19. Input? | Show answer + formula + trick13.19+7=26; 26÷2=13. Formula / rule Reverse operations in reverse order. Tip / trick Start at the output and undo. Remember Backwards and inverse. |
| 3A | Two unknown-style relationship | x+y=20 and x is 4 more than y. Find x. | Show answer + formula + trick12.Let y=8, x=12; or 2y+4=20. Formula / rule Translate comparative language into equations. Tip / trick Use the total and the difference together. Remember Total + difference. |
| 3B | Two unknown-style relationship Similar practice | a+b=34 and a is 6 more than b. Find b. | Show answer + formula + trick14.2b+6=34, so b=14. Formula / rule Translate comparative language into equations. Tip / trick Use the total and the difference together. Remember Total + difference. |
| 4A | Nth term use | Sequence has nth term 4n−1. What is 15th term? | Show answer + formula + trick59.4×15−1=59. Formula / rule Substitute n with the term position. Tip / trick Do not continue the sequence term by term for large n. Remember Plug in n. |
| 4B | Nth term use Similar practice | Nth term 6n+2. What is 12th term? | Show answer + formula + trick74.6×12+2=74. Formula / rule Substitute n with the term position. Tip / trick Do not continue the sequence term by term for large n. Remember Plug in n. |
| 5A | Find position from nth term | In sequence 5n+1, which term equals 46? | Show answer + formula + trick9th.5n+1=46, so 5n=45, n=9. Formula / rule Set nth-term formula equal to target and solve for n. Tip / trick Reverse the formula. Remember Target = formula. |
| 5B | Find position from nth term Similar practice | In sequence 3n+4, which term equals 31? | Show answer + formula + trick9th.3n+4=31; n=9. Formula / rule Set nth-term formula equal to target and solve for n. Tip / trick Reverse the formula. Remember Target = formula. |
| 6A | Second-difference pattern | Continue 2,5,10,17,26,... | Show answer + formula + trick37.Differences 3,5,7,9; next difference 11; 26+11=37. Formula / rule If first differences change regularly, inspect second differences. Tip / trick Write difference rows beneath the sequence. Remember Difference of differences. |
| 6B | Second-difference pattern Similar practice | Continue 1,4,9,16,25,... | Show answer + formula + trick36.Differences 3,5,7,9; next 11; 25+11=36. Formula / rule If first differences change regularly, inspect second differences. Tip / trick Write difference rows beneath the sequence. Remember Difference of differences. |
| 7A | Alternating sequence | Continue 4,12,9,27,24,72,... | Show answer + formula + trick69.Pattern ×3,−3 repeated. Formula / rule Some sequences use two alternating operations. Tip / trick Test odd-position and even-position terms separately. Remember Two rules can alternate. |
| 7B | Alternating sequence Similar practice | Continue 5,10,8,16,14,28,... | Show answer + formula + trick26.Pattern ×2,−2 repeated. Formula / rule Some sequences use two alternating operations. Tip / trick Test odd-position and even-position terms separately. Remember Two rules can alternate. |
| 8A | Fractional sequence | Continue 1/2, 3/4, 1, 1 1/4, ... | Show answer + formula + trick1 1/2.Add 1/4 each time. Formula / rule Convert mixed fractions to a consistent form if needed. Tip / trick Look for a constant fractional difference. Remember Same denominator helps. |
| 8B | Fractional sequence Similar practice | Continue 2/3, 1, 1 1/3, 1 2/3,... | Show answer + formula + trick2.Add 1/3 each time. Formula / rule Convert mixed fractions to a consistent form if needed. Tip / trick Look for a constant fractional difference. Remember Same denominator helps. |
| 9A | Expression from words | Write an expression for 'five more than three times n'. | Show answer + formula + trick3n+5.Three times n is 3n; five more gives +5. Formula / rule Translate key phrases carefully: 'more than' adds; 'less than' subtracts. Tip / trick Order matters in 'less than'. Remember Words → algebra. |
| 9B | Expression from words Similar practice | Write an expression for 'seven less than four times x'. | Show answer + formula + trick4x−7.Four times x is 4x; seven less gives −7. Formula / rule Translate key phrases carefully: 'more than' adds; 'less than' subtracts. Tip / trick Order matters in 'less than'. Remember Words → algebra. |
| 10A | Equation from perimeter | A rectangle has length x+2 and width x. Perimeter is 28. Find x. | Show answer + formula + trick6.2[(x+2)+x]=28; 4x+4=28; x=6. Formula / rule Rectangle perimeter=2(L+W). Tip / trick Substitute algebraic sides into the perimeter formula. Remember Formula first, solve second. |
| 10B | Equation from perimeter Similar practice | A rectangle has length x+3 and width x. Perimeter 34. Find x. | Show answer + formula + trick7.2[(x+3)+x]=34; 4x+6=34; x=7. Formula / rule Rectangle perimeter=2(L+W). Tip / trick Substitute algebraic sides into the perimeter formula. Remember Formula first, solve second. |
31. Tiffin Focus: Multi-Step Problem Solving & Answer-Entry Traps
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 | Round up containers | A hall needs seats for 286 people. Each row holds 24. Minimum number of rows? | Show answer + formula + trick12.286÷24=11 remainder 22, so 12 rows are required. Formula / rule If every item/person must fit, any remainder means round up. Tip / trick Read whether the question asks full containers or containers needed. Remember Remainder + 'minimum needed' → round up. |
| 1B | Round up containers Similar practice | 523 books fit 40 per box. Minimum boxes? | Show answer + formula + trick14.523÷40=13 remainder 3, so 14 boxes. Formula / rule If every item/person must fit, any remainder means round up. Tip / trick Read whether the question asks full containers or containers needed. Remember Remainder + 'minimum needed' → round up. |
| 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 Not every number in a word problem is needed. Tip / trick Underline only quantities linked to the question. Remember Relevant numbers only. |
| 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 Not every number in a word problem is needed. Tip / trick Underline only quantities linked to the question. Remember Relevant numbers only. |
| 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 Apply discount to original price, then perform payment calculation. Tip / trick Separate stages and label each result. Remember Discount first, change second. |
| 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 Apply discount to original price, then perform payment calculation. Tip / trick Separate stages and label each result. Remember Discount first, change second. |
| 4A | Average after adding value | Mean of 4 numbers is 12. A fifth number 22 is added. New mean? | Show answer + formula + trick14.Original total=48; new total=70; 70÷5=14. Formula / rule Mean problems are easiest through totals. Tip / trick Mean×count gives old total. Remember Total first. |
| 4B | Average after adding value Similar practice | Mean of 5 numbers is 18. Add 30. New mean? | Show answer + formula + trick20.Original total=90; new total=120; ÷6=20. Formula / rule Mean problems are easiest through totals. Tip / trick Mean×count gives old total. Remember Total first. |
| 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 + trick18.3/8 of 240=90; 20% of 90=18. Formula / rule Apply operations in sequence: fraction of total, then percentage of result. Tip / trick Calculate one stage at a time and sense-check the final answer. Remember Stage by stage. |
| 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 + trick30.3/8 of 320=120; 25% of 120=30. Formula / rule Apply operations in sequence: fraction of total, then percentage of result. Tip / trick Calculate one stage at a time and sense-check the final answer. Remember Stage by stage. |
| 6A | Unit conversion mid-problem | A ribbon 3.5 m long is cut into 25 cm pieces. How many full pieces? | Show answer + formula + trick14.3.5 m=350 cm; 350÷25=14. Formula / rule Convert to the same unit before dividing. Tip / trick Use the smaller unit to avoid decimals. Remember Same units first. |
| 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 + trick16.2.4 kg=2400 g; 2400÷150=16. Formula / rule Convert to the same unit before dividing. Tip / trick Use the smaller unit to avoid decimals. Remember Same units first. |
| 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 Read carefully whether the given amount is before or after discount. Tip / trick Do not reverse-calculate unless asked for original price. Remember Identify the stage of the given number. |
| 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 Read carefully whether the given amount is before or after discount. Tip / trick Do not reverse-calculate unless asked for original price. Remember Identify the stage of the given number. |
| 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 + trickPack B.A=75p each; B=72p each. Formula / rule Unit price = total price ÷ quantity. Tip / trick Compare like-for-like, not pack totals. Remember Price per one. |
| 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 + trickPack B.A=65p; B=62.5p. Formula / rule Unit price = total price ÷ quantity. Tip / trick Compare like-for-like, not pack totals. Remember Price per one. |
| 9A | Estimate to catch error | Which is a sensible value for 49×203: 994, 9,947, 99,470 or 994,700? | Show answer + formula + trick9,947.Estimate 50×200=10,000, so 9,947 is plausible. Formula / rule Estimate by rounding before exact calculation. Tip / trick Use order of magnitude to eliminate impossible options. Remember Estimate first. |
| 9B | Estimate to catch error Similar practice | Which is sensible for 398×21: 836, 8,358, 83,580 or 835,800? | Show answer + formula + trick8,358.Estimate 400×20=8,000. Formula / rule Estimate by rounding before exact calculation. Tip / trick Use order of magnitude to eliminate impossible options. Remember Estimate first. |
| 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 + trick7.Enter only the required number, not units or working. Formula / rule Follow the answer-sheet format exactly. Tip / trick Separate solving from recording, then perform a final transfer check. Remember Correct answer + correct entry. |
| 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 + trick0705.07:05 becomes digits 0,7,0,5. Formula / rule Follow the answer-sheet format exactly. Tip / trick Separate solving from recording, then perform a final transfer check. Remember Correct answer + correct entry. |
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.