What you'll be able to do by the end
- Simplify a ratio and write it in the form 1 : n
- Divide a quantity in a given ratio
- Solve problems where you know one part and need the total, or the difference
- Convert between fractions, ratios and percentages
- Calculate percentage increase, decrease and change
- Work with best value and unit pricing
- Use direct and inverse proportion
- Calculate with compound measures — speed, density, pressure, rates of pay
- Solve exchange rate problems
- 🔺 Use direct and inverse proportion formulae, and iterative growth and decay
This topic carries around 25% of the marks on a Foundation paper and appears on every single paper at both tiers. If a resit student can only master one topic, this is the one. Much of it — best value, percentages, speed, exchange rates — is arithmetic you already do in daily life without calling it maths.
Part 1 — Understanding ratio
A ratio compares two or more quantities. If a mix uses 2 parts sand to 3 parts cement, we write 2 : 3.
2 : 3 does not mean 2 and 3. It means for every 2 of the first, there are 3 of the second. That could be 2 and 3, or 20 and 30, or 200 and 300 — all are the ratio 2 : 3.
Add the parts to get the total number of parts. 2 : 3 → 2 + 3 = 5 parts altogether. Nearly every ratio question in the entire GCSE is solved by finding the total number of parts, then finding what one part is worth.
Simplifying ratios
Divide all parts by their highest common factor — the same skill as simplifying a fraction.
| Ratio | HCF | Simplified |
|---|---|---|
| 12 : 18 | 6 | 2 : 3 |
| 25 : 40 | 5 | 5 : 8 |
| 8 : 12 : 20 | 4 | 2 : 3 : 5 |
| 0.4 : 1.2 | ×10 first, then ÷4 | 1 : 3 |
| ½ : ¾ | ×4 first | 2 : 3 |
With decimals or fractions: multiply everything up to whole numbers first, then simplify.
Ratios in the form 1 : n
Divide both parts by the first number. To write 4 : 10 in the form 1 : n: 4 ÷ 4 = 1, and 10 ÷ 4 = 2.5, giving 1 : 2.5.
Students often "round" 2.5 to 3 and lose the mark. In the form 1 : n, n is whatever it is — a decimal is fine and usually expected.
Part 2 — Sharing in a ratio
This is the single most common ratio question on the paper. The method is three steps, every time:
- Add the parts.
- Divide the total by the number of parts to find one part.
- Multiply.
Worked example 1 — basic sharing. £450 is shared between two people in the ratio 4 : 5. How much does each receive?
Step 1: 4 + 5 = 9 parts. Step 2: 450 ÷ 9 = £50 per part. Step 3: first person 4 × 50 = £200, second person 5 × 50 = £250. Check: 200 + 250 = 450 ✓
It takes five seconds and it catches almost every arithmetic slip. If your parts don't add back to the total, something is wrong.
Worked example 2 — given one part, not the total. A recipe uses flour and sugar in the ratio 5 : 2. She uses 350 g of flour. How much sugar?
Do not divide 350 by 7 — you have not been given the total. 350 g is 5 parts, so one part = 350 ÷ 5 = 70 g. Sugar is 2 parts: 2 × 70 = 140 g.
"One part" is the key to everything. Once you know what one part is worth, you can find any part, the total, or the difference. Always ask: what am I given, and how many parts is it?
Worked example 3 — given the difference. Two people share in the ratio 3 : 7. One receives £160 more than the other. How much was shared?
Difference in parts: 7 − 3 = 4 parts. So one part = 160 ÷ 4 = £40. Total parts = 3 + 7 = 10. Total = 10 × 40 = £400.
Worked example 4 — three-part ratio. 240 animals are kept in the ratio mammals : reptiles : invertebrates = 3 : 4 : 5. How many reptiles?
3 + 4 + 5 = 12 parts. 240 ÷ 12 = 20 per part. Reptiles: 4 × 20 = 80.
Practice — sharing in a ratio
- Share £72 in the ratio 3 : 5
- Share 180 kg in the ratio 2 : 7
- Share £315 in the ratio 2 : 3 : 4
- A recipe uses oats and honey in the ratio 8 : 3. She uses 400 g of oats. How much honey?
- Two people share a prize in the ratio 5 : 9. The larger share is £288 more than the smaller. What was the total prize?
Answers are in the mark scheme.
Part 3 — Ratio, fractions and percentages
These are three ways of saying the same thing, and questions move between them. Take a group split in the ratio 3 : 5:
| Form | First group | Second group |
|---|---|---|
| Ratio | 3 parts of 8 | 5 parts of 8 |
| Fraction | ⅜ | ⅝ |
| Percentage | 37.5% | 62.5% |
| Decimal | 0.375 | 0.625 |
A ratio of 3 : 5 does not mean ⅗. The fraction denominator is the total parts (8), not the other part of the ratio. This mistake appears every year and costs whole questions.
Going backwards — fraction to ratio
If ⅖ of the animals are reptiles, then ⅗ are not. So the ratio of reptiles to non-reptiles is 2 : 3.
Part 4 — Percentages
Percentage of an amount — non-calculator
Build from the easy ones: 50% = ÷2, 25% = ÷4, 10% = ÷10, 5% = half of 10%, 1% = ÷100.
Find 35% of £240: 10% = £24, so 30% = 3 × 24 = £72; 5% = half of 24 = £12; 35% = 72 + 12 = £84. (Calculator: 0.35 × 240 = £84.)
Increase and decrease — use multipliers
Faster and far less error-prone than finding the percentage and adding it on.
| Change | Multiplier |
|---|---|
| Increase by 20% | × 1.2 |
| Increase by 5% | × 1.05 |
| Increase by 17.5% | × 1.175 |
| Decrease by 20% | × 0.8 |
| Decrease by 5% | × 0.95 |
| Decrease by 35% | × 0.65 |
A price of £450 increased by 12%: 450 × 1.12 = £504. Decreased by 12%: 450 × 0.88 = £396.
Percentage change
Percentage change = (change ÷ original) × 100
A tortoise weighed 84 kg; a year later, 91 kg. Change = 91 − 84 = 7. (7 ÷ 84) × 100 = 8.33% (2 d.p.).
Not the new value. Not the change. The original. This is the most commonly dropped mark in the entire percentages topic.
Reverse percentages — finding the original
You are given the value after a change and must find what it was before. After a 20% increase, a price is £96 — what was the original?
The wrong method: find 20% of 96 and subtract (gives £76.80 — wrong). The right method: £96 is 120% of the original, so original = 96 ÷ 1.2 = £80. Check: 80 × 1.2 = 96 ✓
A price reduced by 30% in a sale to £77 was: 77 ÷ 0.7 = £110 before.
Look for "before", "originally" or "was" combined with a value after the change. Given the final amount and asked for the start — divide by the multiplier. Never subtract.
Part 5 — Best value and unit pricing
Which is better value? Compare like with like. Two methods, both fine — but you must say which you used and interpret it correctly.
Pack A: 400 g for £3.60. Pack B: 750 g for £6.45.
Method 1 — cost per unit: A = 360p ÷ 400 = 0.9p/g; B = 645p ÷ 750 = 0.86p/g. Pack B is better — each gram costs less.
Method 2 — units per pound: A = 400 ÷ 3.60 = 111.1 g/£; B = 750 ÷ 6.45 = 116.3 g/£. Pack B gives more for your money.
With cost per unit, smaller is better. With units per pound, larger is better. Students who calculate correctly then pick the wrong pack lose the final mark. Write it out: "Pack B is better value because each gram costs less."
Multi-buy offers
Bags cost £20.99 each, or 5 bags for £89. A customer needs 12 bags — cheapest way?
| Option | Working | Cost |
|---|---|---|
| All singles | 12 × 20.99 | £251.88 |
| Two multi-buys + 2 singles | 2 × 89 + 2 × 20.99 | £219.98 |
| Three multi-buys (15 bags) | 3 × 89 | £267.00 |
Cheapest: £219.98. Note the third option — buying more of a cheaper deal is not always cheaper overall. A question will sometimes make it so; you must check.
Part 6 — Compound measures
A compound measure combines two units. Learn the triangle for each — cover what you want, and what's left is the calculation.
Speed = Distance ÷ Time
A journey of 87 miles takes 1 hour 45 minutes. First convert time to hours: 1 h 45 min = 1.75 hours (45 ÷ 60 = 0.75, not 1.45). Then 87 ÷ 1.75 = 49.7 mph (1 d.p.).
1 hour 45 minutes is 1.75 hours, not 1.45. Minutes to hours: divide by 60. 20 min = 0.333 h · 30 min = 0.5 h · 45 min = 0.75 h · 15 min = 0.25 h. More marks are lost to this than to any other error in the whole topic.
Density = Mass ÷ Volume
Same triangle, with M, D and V. A block of mass 480 g and volume 60 cm³: 480 ÷ 60 = 8 g/cm³.
Rates of pay
£11.44 per hour, 37.5 hours, then 6 hours overtime at time-and-a-half. Basic: 37.5 × 11.44 = £429.00. Overtime rate: 11.44 × 1.5 = £17.16; overtime pay: 6 × 17.16 = £102.96. Total = £531.96.
| Measure | Formula | Units |
|---|---|---|
| Speed | distance ÷ time | mph, m/s, km/h |
| Density | mass ÷ volume | g/cm³, kg/m³ |
| Pressure | force ÷ area | N/m², Pa |
| Fuel consumption | distance ÷ fuel | mpg, km/l |
| Population density | population ÷ area | people/km² |
| Flow rate | volume ÷ time | litres/min |
Part 7 — Exchange rates
Exchange rate questions are direct proportion in disguise. With £1 = €1.17:
(a) £250 to euros: 250 × 1.17 = €292.50. (b) €500 to pounds: 500 ÷ 1.17 = £427.35.
£1 buys €1.17, so euros are "worth less" individually — converting to euros gives a bigger number, so multiply. Converting from euros to pounds gives a smaller number, so divide. Sense-check every answer: if £250 came out as €213, you divided when you should have multiplied.
Part 8 — Direct and inverse proportion
Direct proportion
When one quantity doubles, the other doubles; the graph is a straight line through the origin. 7 identical items cost £15.75 — find the cost of 12. Unitary method: one item = 15.75 ÷ 7 = £2.25; twelve = 12 × 2.25 = £27.00.
Inverse proportion
When one doubles, the other halves; the product stays constant. 4 people take 6 hours to do a job — how long for 3 people at the same rate? Find the total work: 4 × 6 = 24 person-hours. Then 24 ÷ 3 = 8 hours.
Fewer people means more time, not less. If your answer went down when the number of workers went down, you've treated it as direct. Sense-check: more workers → less time; more taps → tank fills faster; higher speed → shorter journey.
🔺 Higher tier — proportion formulae
Direct: y ∝ x, so y = kx. Inverse: y ∝ 1/x, so y = k/x. Method: write the statement → convert to an equation with k → substitute given values to find k → rewrite the formula → use it.
Worked example. y is inversely proportional to x; when x = 4, y = 15. Find y when x = 10. y = k/x → 15 = k/4 → k = 60 → y = 60/x → y = 60/10 = 6. Other forms — y ∝ x², y ∝ √x, y ∝ 1/x² — use the identical method; only the equation changes.
Part 9 🔺 Higher — growth and decay
Repeated percentage change uses a power of the multiplier: Final = Initial × (multiplier)ⁿ, where n is the number of periods.
Compound interest. £2,000 at 3.5% for 6 years: 2000 × 1.035⁶ = £2,458.51.
Depreciation. A van costs £24,000 and depreciates 18% per year: 24000 × 0.82⁴ = £10,857.71 after 4 years.
Finding the number of years. £5,000 at 4% — after how many years does it first exceed £6,000? 5000 × 1.04⁴ = £5,849.29 (not yet); 5000 × 1.04⁵ = £6,083.26 (exceeds). Answer: 5 years.
Do not multiply the single-year interest by the number of years. 3.5% of £2,000 is £70; six years at £70 is £2,420 — but that's simple interest, wrong by £38.51. Compound means the interest earns interest. Use the power.
Exam-style questions
Answer all questions. Marks are shown in brackets. Calculators may be used unless a question says otherwise. Total: 77 marks.
Question 1
Write each ratio in its simplest form.
(Total 7 marks)
Question 2
(Total 10 marks)
Question 3 — non-calculator
(Total 6 marks)
Question 4
(Total 8 marks)
Question 5
A shop sells bedding in three sizes.
| Size | Amount | Price |
|---|---|---|
| Small | 12 litres | £8.99 |
| Medium | 30 litres | £20.99 |
| Large | 50 litres | £33.50 |
(Total 7 marks)
Question 6
(Total 9 marks)
Question 7
The exchange rate is £1 = $1.26.
(Total 7 marks)
Question 8
(Total 6 marks)
Question 9 🔺 Higher
(Total 9 marks)
Question 10 🔺 Higher
y is directly proportional to the square of x. When x = 3, y = 45.
(Total 8 marks)
TOTAL FOR PAPER: 77 MARKS
Mark scheme
Practice — sharing in a ratio (Part 2)
1. 72 ÷ 8 = 9 → £27 and £45. 2. 180 ÷ 9 = 20 → 40 kg and 140 kg. 3. 315 ÷ 9 = 35 → £70, £105, £140. 4. 400 ÷ 8 = 50 → honey = 3 × 50 = 150 g. 5. Difference = 4 parts = £288, so 1 part = £72; total = 14 × 72 = £1,008.
Question 1
(a) 1 — 3 : 4.
(b) 2 — HCF 9 (1) → 5 : 3 : 7 (1).
(c) 2 — ×10 to give 6 : 15 (1) → 2 : 5 (1).
(d) 2 — divide both by 5 (1) → 1 : 1.6 (1). Do not accept 1 : 2.
Question 2
| (a) 3. 4 + 5 = 9 parts | 1 |
| 540 ÷ 9 = 60 | 1 |
| Amir £240, Beth £300 | 1 |
| (b) 3. Recognises 168 g = 7 parts | 1 |
| 168 ÷ 7 = 24 g per part | 1 |
| 3 × 24 = 72 g | 1 |
A common error is 168 ÷ 10 = 16.8 — treats 168 as the total and scores 0.
| (c) 4. Difference = 9 − 2 = 7 parts | 1 |
| 315 ÷ 7 = £45 per part | 1 |
| Total parts = 11 | 1 |
| 11 × 45 = £495 | 1 |
Question 3
(a) 2 — 10% = 32, 5% = 16 (1) → £48 (1). (b) 2 — 25% = 120 (1) → £600 (1). (c) 2 — 35% = 84 (1) → 156 kg (1).
Question 4
(a) 2 — 745 × 1.065 (1) = £793.43 (1).
| (b) 3. Recognises £119 = 85% of the original | 1 |
| 119 ÷ 0.85 | 1 |
| £140 | 1 |
119 × 1.15 = £136.85 is the classic wrong answer. Award 0.
| (c) 3. Change = 412 − 385 = 27 | 1 |
| (27 ÷ 412) × 100 | 1 |
| 6.6% | 1 |
Dividing by 385 gives 7.0% — the wrong-denominator error. Award 2 for method.
Question 5
| (a) 4. Small: 899 ÷ 12 = 74.9p/litre | 1 |
| Medium: 2099 ÷ 30 = 70.0p/litre | 1 |
| Large: 3350 ÷ 50 = 67p/litre | 1 |
| States Large is best value (lowest cost per litre) | 1 |
The final mark needs the conclusion and the reason. Three correct calculations with no conclusion scores 3.
| (b) 3. Considers combinations totalling ≥ 90 litres | 1 |
| e.g. 3 medium = 90 litres = £62.97 (vs 1 large + 1 medium + 1 small = 92 litres = £63.48) | 1 |
| 3 medium, £62.97 | 1 |
Accept any valid cheapest combination reaching at least 90 litres with correct costing.
Question 6
(a) 3 — 2 h 30 min = 2.5 h (1) · 138 ÷ 2.5 (1) · 55.2 mph (1).
(b) 3 — 1 h 45 min = 1.75 h (1) · 46 × 1.75 (1) · 80.5 miles (1).
(c) 3 — 1.44 kg = 1440 g (1) · 1440 ÷ 180 (1) · 8 g/cm³ (1). Failing to convert kg → g gives 0.008 and scores 1 for method only.
Question 7
(a) 2 — 480 × 1.26 (1) = $604.80 (1). (b) 2 — 735 ÷ 1.26 (1) = £583.33 (1).
| (c) 3. 105 ÷ 1.26 = £83.33 | 1 |
| Compares with £89 | 1 |
| USA is cheaper by £5.67 | 1 |
Question 8
| (a) 3. Recognises inverse proportion / 5 × 12 = 60 worker-days | 1 |
| 60 ÷ 8 | 1 |
| 7.5 days | 1 |
An answer greater than 12 shows the candidate treated it as direct proportion. Award 0.
(b) 3 — 22.50 ÷ 6 = £3.75 (1) · 15 × 3.75 (1) · £56.25 (1).
Question 9 🔺
(a) 3 — multiplier 1.042 (1) · 3500 × 1.042⁵ (1) · £4,297.99 (1).
(b) 3 — multiplier 0.78 (1) · 18000 × 0.78³ (1) · £8,543.38 (1).
| (c) 3. Uses 8000 × 1.03ⁿ | 1 |
| Shows 8000 × 1.03⁷ = £9,838.34 and 8000 × 1.03⁸ = £10,133.49 | 1 |
| 8 years | 1 |
Question 10 🔺
(a) 3 — y = kx² (1) · 45 = k × 9, so k = 5 (1) · y = 5x² (1).
(b) 2 — y = 5 × 49 (1) = 245 (1).
(c) 3 — 320 = 5x² (1) · x² = 64 (1) · x = 8 (1). Accept x = ±8 if both given; do not penalise the omitted negative in a context-free question.
The eight mistakes that cost the most marks
1. Dividing by the total when given one part. 350 g of flour in a 5 : 2 ratio is 5 parts, not 7. Ask what you've been given.
2. Ratio 3 : 5 read as the fraction ⅗. It's ⅜ — the denominator is the total.
3. Reverse percentages done by adding back. £96 after a 20% rise is not £96 minus 20%. Divide by 1.2.
4. Percentage change divided by the wrong number. Always the original.
5. 1 hour 45 minutes written as 1.45. It's 1.75. Divide minutes by 60.
6. Units not converted. kg and cm³ in the same density calculation gives an answer out by 1000.
7. Inverse proportion scaled directly. Fewer workers means more time. Sense-check the direction.
8. Best value calculated but not concluded. Three correct calculations and no answer to the actual question loses the final mark every time.
Teaching notes
Teach "one part" as the central idea and return to it constantly. Almost every ratio question is: find what one part is worth, then multiply. Students who hold that idea handle unfamiliar phrasings; those who memorise a separate method per question type cannot.
Multipliers should replace find-and-add entirely. × 1.12 for a 12% increase is faster, is required for compound interest and reverse percentages later, and removes a whole class of arithmetic error. Teach it from the start, not as an upgrade.
Reverse percentages need their own lesson. Students consistently apply the wrong operation because the intuitive method feels right. Give them the check — multiply your answer back and see if you get the given figure — and make it non-negotiable.
The time-conversion error deserves a dedicated starter, repeated. 1 h 45 min = 1.75 h. Ten quick conversions at the start of five consecutive lessons fixes it permanently and recovers marks across the whole paper.
Best value questions must end with a sentence. Train "X is better value because each unit costs less" as an automatic final line. One mark, on most papers, the easiest on this topic to leave behind.
For post-16 resit learners this is the single highest-value topic. It's roughly a quarter of a Foundation paper, appears on all three papers, and much of it — best value, percentages, speed, exchange rates — is arithmetic students already do daily without recognising it as maths. Building from what they can already do beats starting from formal rules.