Proportion Problems and Best Buys

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.

Key facts before you start

Proportion problems and best buys infographic

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.

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.

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.

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.

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.

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.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

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?

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.

Next lesson

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