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. 190 question varieties with worked solutions
Try each example before opening the solution. If a type is already effortless, move on. Spend time on the types that cause errors.
1. Number Concepts
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
2. Arithmetic Operations
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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 −. |
| 2 | 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 −. |
| 3 | 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 −. |
| 4 | 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 −. |
| 5 | 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 −. |
| 6 | 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 −. |
| 7 | 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 −. |
| 8 | 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 −. |
| 9 | 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 −. |
| 10 | 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 −. |
3. Fractions
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
4. Decimals
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
5. Percentages
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
6. Ratios and Proportions
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
7. Basic Algebra
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
8. Advanced Algebra
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
9. Number Patterns and Sequences
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
10. Basic Geometry
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
11. Shapes and Symmetry
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
12. 3D Shapes and Volume
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
13. Measurement and Units
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
14. Data Handling
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
15. Advanced Data Handling
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
16. Venn Diagrams and Probability
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
17. Word Problems and Applied Maths
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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? |
| 2 | 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? |
| 3 | 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? |
| 4 | 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? |
| 5 | 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? |
| 6 | 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? |
| 7 | 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? |
| 8 | 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? |
| 9 | 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? |
| 10 | 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? |
18. Spatial Reasoning
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
19. Financial Mathematics
| # | Question type | Example | Worked solution |
|---|---|---|---|
| 1 | 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. |
| 2 | 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. |
| 3 | 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. |
| 4 | 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. |
| 5 | 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. |
| 6 | 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. |
| 7 | 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. |
| 8 | 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. |
| 9 | 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. |
| 10 | 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. |
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.