Proportion

Proportion problems are a staple of the Edexcel GCSE Mathematics (1MA1) specification. They test your ability to scale quantities up or down, compare value for money, and work with the unitary method. Proportion questions appear on all three papers and range from straightforward scaling to multi-step best-buy problems.

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What Is Proportion?

Two quantities are in proportion if they increase (or decrease) at the same rate. When one doubles, the other doubles; when one is halved, the other is halved.

Term	Meaning	Example
Direct proportion	As one increases, the other increases at the same rate	More hours worked = more pay
Unitary method	Finding the value of one unit first, then scaling	Cost of 1 item, then cost of n items
Scaling	Multiplying all quantities by the same factor	Recipe for 4 scaled to recipe for 6

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The Unitary Method

The unitary method is one of the most reliable approaches for proportion questions on Paper 1 (non-calculator).

Method

1. Find the value of one unit by dividing.
2. Multiply by the number of units you need.

Worked Example 1

5 notebooks cost 3.75 pounds. How much do 8 notebooks cost?

1. Cost of 1 notebook = 3.75 / 5 = 0.75 pounds
2. Cost of 8 notebooks = 0.75 x 8 = 6.00 pounds

Worked Example 2

A recipe for 6 pancakes uses 240 g of flour. How much flour is needed for 15 pancakes?

1. Flour for 1 pancake = 240 / 6 = 40 g
2. Flour for 15 pancakes = 40 x 15 = 600 g

Edexcel Exam Tip: On Paper 1 (non-calculator), choose friendly numbers when finding the unit value. For example, if a recipe serves 4 and you need 10, find the value for 2 first (halve), then multiply by 5.

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Scaling Method

Sometimes it is easier to find a scale factor rather than the unit value.

Worked Example 3

A recipe for 4 people requires 300 g of rice, 200 ml of coconut milk and 150 g of chicken. How much of each ingredient is needed for 10 people?

• Scale factor = 10 / 4 = 2.5
• Rice: 300 x 2.5 = 750 g
• Coconut milk: 200 x 2.5 = 500 ml
• Chicken: 150 x 2.5 = 375 g

Worked Example 4

A map scale is 1 : 25,000. Two towns are 8 cm apart on the map. Find the real distance in km.

• Real distance = 8 x 25,000 = 200,000 cm
• Convert to km: 200,000 / 100,000 = 2 km

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Best Buys

Best-buy questions ask you to determine which of two (or more) options offers better value for money. There are two main approaches.

Method 1: Find the Cost per Unit

Divide the price by the quantity for each option. The lower cost per unit is the better buy.

Method 2: Find the Quantity per Pound

Divide the quantity by the price for each option. The higher quantity per pound is the better buy.

Worked Example 5

Which is better value?

• Pack A: 500 g of cereal for 2.40 pounds
• Pack B: 750 g of cereal for 3.30 pounds

Method 1 (cost per gram):

• Pack A: 2.40 / 500 = 0.0048 pounds per gram = 0.48p per gram
• Pack B: 3.30 / 750 = 0.0044 pounds per gram = 0.44p per gram

Pack B is cheaper per gram, so Pack B is better value.

Method 2 (grams per pound):

• Pack A: 500 / 2.40 = 208.3 g per pound
• Pack B: 750 / 3.30 = 227.3 g per pound

Pack B gives more grams per pound, so Pack B is better value.

Worked Example 6

A shop sells orange juice in two sizes.

• Small: 330 ml for 0.89 pounds
• Large: 1 litre for 2.49 pounds

Which is better value?

• Small: 0.89 / 330 = 0.002697 pounds per ml
• Large: 2.49 / 1000 = 0.002490 pounds per ml

The large bottle is cheaper per ml, so the large bottle is better value.

Edexcel Exam Tip: Best-buy questions are usually worth 3-4 marks. You must show your working for both options and write a clear conclusion. Simply stating the answer without working will not earn full marks.

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Recipes and Proportion

Recipe questions require you to scale ingredients up or down. They are a favourite on Edexcel papers.

Worked Example 7

A recipe for 12 biscuits uses:

• 150 g butter
• 200 g sugar
• 350 g flour

Tom wants to make 30 biscuits. How much of each ingredient does he need?

Scale factor = 30 / 12 = 2.5

• Butter: 150 x 2.5 = 375 g
• Sugar: 200 x 2.5 = 500 g
• Flour: 350 x 2.5 = 875 g

Worked Example 8

Priya has 500 g of flour but wants to use the recipe above. What is the maximum number of biscuits she can make (assuming she has enough of the other ingredients)?

• 350 g flour makes 12 biscuits
• 1 g flour makes 12 / 350 biscuits
• 500 g flour makes (12 / 350) x 500 = 17.14...
• She can make a maximum of 17 biscuits (round down since you cannot make a fraction of a biscuit)

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Exchange Rates

Currency conversion is a form of proportion.

Worked Example 9

The exchange rate is 1 pound = 1.15 euros. Convert 60 pounds to euros.

60 x 1.15 = 69 euros

Worked Example 10

Marcus receives 230 euros. How much is this in pounds?

230 / 1.15 = 200 pounds

Edexcel Exam Tip: Exchange rate questions can include commission charges. Read carefully whether the rate has already been adjusted or whether you need to apply a percentage charge separately.

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Practice Problems