Free GCSE Maths lesson: Ratio, Proportion and Rates
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Lesson 19 · GCSE / Key Stage 4 · Maths · Ratio, Proportion and Rates

Proportion Problems and Best Buys

Learn how to scale quantities, compare unit prices and decide which option is better value without being distracted by different pack sizes.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Ratio, Proportion and Rates

GCSE specification fit

A practical proportion skill for prices, recipes, rates and value for money.

GCSE proportion questions often ask you to compare different-sized quantities fairly. This lesson focuses on the unitary method: find one unit first, then scale to the quantity you need.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandRatio, Proportion and Rates
Tier guidanceFoundation and Higher

What you will learn

  • How to recognise direct proportion in everyday contexts.
  • How to use the unitary method to find one unit first.
  • How to scale quantities up or down by the same factor.
  • How to compare unit prices and best buys.
  • How to compare rates fairly using the same units.
  • How to check whether your answer needs £, p, g, kg, ml, litres or time units.

Why this matters

Proportion is the maths behind supermarket offers, recipes, fuel costs, speed, pay rates, exchange rates and shared work.

A bigger pack is not always better value. A faster-looking rate is not always fair unless both rates are compared over the same time, distance or amount.

Prior knowledge

You should already be comfortable with:

  • multiplying and dividing whole numbers and decimals,
  • using pounds and pence accurately,
  • scaling equivalent ratios,
  • reading worded questions carefully,
  • checking whether an answer is sensible in context.

Clear explanation

Direct proportion

Two quantities are in direct proportion when they increase or decrease by the same scale factor. If 2 notebooks cost £3.00, then 4 notebooks cost £6.00 because the number of notebooks has doubled.

2 notebooks → £3.00 4 notebooks → £6.00 8 notebooks → £12.00

The unitary method

The unitary method means finding the value of one unit first. Once you know one unit, you can multiply by the amount you need.

6 apples cost £2.40 1 apple costs £2.40 ÷ 6 = £0.40 10 apples cost 10 × £0.40 = £4.00

Scaling up and down

You can also scale directly if the factor is clear. A recipe for 4 people uses 300 g pasta. For 6 people, multiply by 6 ÷ 4 = 1.5.

300 g × 1.5 = 450 g

The same scale factor must be used for every ingredient.

Unit prices and best buys

To compare different pack sizes, change each option to the same unit. For food, common units are price per item, price per 100 g, price per kg or price per litre.

Option A: 400 g for £1.60 → £1.60 ÷ 4 = £0.40 per 100 g Option B: 750 g for £2.85 → £2.85 ÷ 7.5 = £0.38 per 100 g Option B is better value because 38p per 100 g is less than 40p per 100 g.

Comparing rates fairly

Rates compare different types of unit, such as miles per hour, pounds per hour or litres per minute. Make the units match before deciding.

Machine A: 150 labels in 5 minutes → 150 ÷ 5 = 30 labels per minute Machine B: 210 labels in 7 minutes → 210 ÷ 7 = 30 labels per minute

The rates are equal, even though Machine B makes more labels overall.

A simple visual check

This comparison changes both prices to pence per 100 g, so the smaller value is the better buy.

Best-buy comparison using pence per 100 g Option A is shown as 40 pence per 100 grams and Option B as 38 pence per 100 grams, so Option B has the smaller unit price. Option A: 40p per 100 g Option B: 38p per 100 g 40p 38p

Worked examples

Example 1: Direct proportion in a price question

5 pens cost £3.50. How much do 8 pens cost?

1 pen costs £3.50 ÷ 5 = £0.70 8 pens cost 8 × £0.70 = £5.60
Answer: £5.60.

Example 2: Scaling a recipe

A recipe for 3 people uses 240 g rice. How much rice is needed for 5 people?

1 person needs 240 g ÷ 3 = 80 g 5 people need 5 × 80 g = 400 g
Answer: 400 g rice.

Example 3: Best buy

Which is better value: 12 bottles for £7.20 or 8 bottles for £5.20?

12 bottles: £7.20 ÷ 12 = £0.60 per bottle 8 bottles: £5.20 ÷ 8 = £0.65 per bottle
Answer: 12 bottles for £7.20 is better value because each bottle costs 60p.

Example 4: Comparing rates

Driver A travels 180 miles in 3 hours. Driver B travels 260 miles in 5 hours. Who has the higher average speed?

Driver A: 180 ÷ 3 = 60 miles per hour Driver B: 260 ÷ 5 = 52 miles per hour
Answer: Driver A has the higher average speed.

Quick checks

Choose an answer, then check your thinking.

1. 4 tickets cost £18. How much does 1 ticket cost?

2. Which is cheaper per item?

3. A printer makes 240 pages in 6 minutes. What is the rate per minute?

Practice questions

Question 1

3 notebooks cost £4.50. Find the cost of 7 notebooks.

Reveal answer and marking guidance

Answer: £10.50.

Marking: 1 notebook costs £4.50 ÷ 3 = £1.50; 7 notebooks cost 7 × £1.50 = £10.50.

Question 2

A recipe for 4 people uses 500 ml stock. How much stock is needed for 10 people?

Reveal answer and marking guidance

Answer: 1250 ml, or 1.25 litres.

Marking: 1 person needs 500 ml ÷ 4 = 125 ml; 10 people need 10 × 125 ml = 1250 ml.

Question 3

Which is better value: 500 g cereal for £1.80 or 750 g cereal for £2.55?

Reveal answer and marking guidance

Answer: 750 g for £2.55 is better value.

Marking: 500 g: £1.80 ÷ 5 = £0.36 per 100 g. 750 g: £2.55 ÷ 7.5 = £0.34 per 100 g.

Question 4

Shop A sells 9 batteries for £6.75. Shop B sells 12 batteries for £9.60. Which shop is cheaper per battery?

Reveal answer and marking guidance

Answer: Shop A.

Marking: Shop A: £6.75 ÷ 9 = £0.75 per battery. Shop B: £9.60 ÷ 12 = £0.80 per battery.

Question 5

A tap fills 18 litres in 3 minutes. Another tap fills 28 litres in 4 minutes. Which tap has the faster flow rate?

Reveal answer and marking guidance

Answer: The second tap.

Marking: First tap: 18 ÷ 3 = 6 litres per minute. Second tap: 28 ÷ 4 = 7 litres per minute.

Question 6

Paint covers 24 m² using 3 litres. How many litres are needed for 40 m² at the same coverage rate?

Reveal answer and marking guidance

Answer: 5 litres.

Marking: 1 litre covers 24 ÷ 3 = 8 m²; 40 m² needs 40 ÷ 8 = 5 litres.

Question 7

Orange juice is sold as 1.5 litres for £2.40 or 900 ml for £1.53. Which is better value?

Reveal answer and marking guidance

Answer: 1.5 litres for £2.40 is better value.

Marking: 1.5 litres = 1500 ml, so £2.40 ÷ 15 = £0.16 per 100 ml. 900 ml: £1.53 ÷ 9 = £0.17 per 100 ml. The 1.5 litre bottle is cheaper per 100 ml.

Question 8

A printer uses 18 sheets in 4.5 minutes. At the same rate, how many sheets will it use in 12 minutes?

Reveal answer and marking guidance

Answer: 48 sheets.

Marking: 18 ÷ 4.5 = 4 sheets per minute. In 12 minutes it uses 12 × 4 = 48 sheets.

Question 9

Potatoes are sold in a 3.6 kg bag for £4.68 or a 2.5 kg bag for £3.35. Which bag is better value, and by how much per kg?

Reveal answer and marking guidance

Answer: The 3.6 kg bag is better value by 4p per kg.

Marking: 3.6 kg bag: £4.68 ÷ 3.6 = £1.30 per kg. 2.5 kg bag: £3.35 ÷ 2.5 = £1.34 per kg. Difference = £0.04 per kg.

Question 10

A 500 ml bottle of sauce costs £1.15. A 2 litre bottle costs £4.00 but has 10% off. After the discount, which bottle is better value, and by how much per litre?

Reveal answer and marking guidance

Answer: The 2 litre bottle is better value by 50p per litre.

Marking: 500 ml = 0.5 litres, so £1.15 ÷ 0.5 = £2.30 per litre. The discounted 2 litre bottle costs £4.00 × 0.90 = £3.60, so £3.60 ÷ 2 = £1.80 per litre. Difference = £0.50 per litre.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For proportion and best-buy problems, marks usually come from finding a fair unit rate, scaling both quantities consistently, converting units before comparing, keeping extra decimal places until the final decision, and writing a final sentence that answers the real-life question.

Common mistakes

  • Comparing total prices only: a lower total price might be worse value if the pack is much smaller.
  • Mixing units: do not compare p per 100 g with £ per kg unless you convert one of them.
  • Scaling only one quantity: in a recipe, every ingredient must be multiplied or divided by the same factor.
  • Rounding too early: keep enough decimal places until the final comparison.
  • Forgetting the decision sentence: best-buy questions usually need a clear final choice with a reason.

Extension challenge

A 1.2 kg bag of rice costs £3.06. A 750 g bag of the same rice costs £2.10. Which bag is better value, and by how many pence per kg?

Reveal answer

Answer: The 1.2 kg bag is better value by 25p per kg.

1.2 kg bag: £3.06 ÷ 1.2 = £2.55 per kg. 750 g = 0.75 kg, so £2.10 ÷ 0.75 = £2.80 per kg. Difference = £2.80 − £2.55 = £0.25 per kg.

Exam-board guidance

Proportion and best-buy methods are common across GCSE Maths specifications. Expect questions involving money, rates, measures, recipes, speed, flow, unit costs and value-for-money decisions.

AQA GCSE Maths

Show the unit value or scale factor clearly, convert to matching units, include any offer or discount, then compare like with like before making the decision.

OCR GCSE Maths

Write the comparison basis, such as price per item, price per 100 g or amount per hour, before choosing and justifying the answer. Keep extra decimal places until the final comparison.

Pearson Edexcel GCSE Maths

Best-buy questions often reward a clear unit-price or equivalent-quantity calculation plus a final sentence explaining the choice.

Eduqas GCSE Maths

Keep the context visible, especially whether the comparison is per item, per kg, per litre or per hour, and round money only at the end.

WJEC Wales

Expect practical numeracy contexts, so check units and give a recommendation that answers the real-life value-for-money question.

CCEA GCSE Maths

Use a neat unitary-method line, then compare equivalent quantities rather than original package sizes so calculator working and the final decision are easy to follow.

Next lesson

Next, build on this with a focused lesson on Direct Proportion.

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