Ratio and Proportion

Ratio and proportion questions are among the most common in the FSCE 11+ exam. They test your ability to compare quantities, share amounts fairly, scale recipes, and work out the best value when shopping. These are practical, real-world skills — and the exam rewards students who can explain their reasoning clearly. This lesson teaches you systematic methods for every type of ratio and proportion problem.

What Is a Ratio?

A ratio compares two or more quantities. If a class has 12 boys and 18 girls, the ratio of boys to girls is 12:18. We can simplify this by dividing both parts by their highest common factor (HCF = 6): 12:18 = 2:3.

This means for every 2 boys, there are 3 girls.

Key Points About Ratios

• The order matters: 2:3 (boys:girls) is different from 3:2 (girls:boys).
• Ratios have no units — they are pure comparisons.
• You can have ratios with more than two parts: red:blue:green = 1:3:2.

Simplifying Ratios

Divide all parts by their HCF:

• 15:25 → HCF = 5 → 3:5
• 8:12:20 → HCF = 4 → 2:3:5
• 0.5:1.5 → Multiply both by 2 → 1:3

To simplify ratios with fractions: 1/2 : 3/4 → multiply both by 4 → 2:3.

Sharing in a Ratio

This is one of the most important skills. Here is the standard method:

Method:

1. Add up the parts of the ratio.
2. Divide the total amount by the number of parts to find the value of one part.
3. Multiply each part of the ratio by the value of one part.

Proportion

Proportion means that two ratios are equal. If 3 apples cost 90p, then 1 apple costs 30p, and 5 apples cost 150p. The relationship between the number of apples and the cost stays the same — they are in direct proportion.

The Unitary Method

The unitary method means finding the value of one unit first, then scaling up.

Example: If 4 books cost £22, how much do 7 books cost?

1. Cost of 1 book: £22 ÷ 4 = £5.50
2. Cost of 7 books: £5.50 x 7 = £38.50

Worked Examples

Worked Example 1: Sharing in a Ratio

Question: Sarah and James share £420 in the ratio 3:4. How much does each person get?

Step-by-step solution:

1. Total parts: 3 + 4 = 7 parts.
2. Value of one part: £420 ÷ 7 = £60.
3. Sarah gets: 3 x £60 = £180.
4. James gets: 4 x £60 = £240.
5. Check: £180 + £240 = £420. Correct!

Answer: Sarah gets £180, James gets £240.

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Worked Example 2: Scaling a Recipe

Question: A recipe for 4 people uses 300g of flour, 200ml of milk, and 2 eggs. How much of each ingredient is needed for 10 people?

Step-by-step solution:

1. Scale factor: 10 ÷ 4 = 2.5
2. Flour: 300g x 2.5 = 750g
3. Milk: 200ml x 2.5 = 500ml
4. Eggs: 2 x 2.5 = 5 eggs
5. Check one: 750g ÷ 10 people = 75g per person. Original: 300g ÷ 4 = 75g per person. Consistent!

Answer: 750g flour, 500ml milk, 5 eggs.

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Worked Example 3: Proportion (Unitary Method)

Question: A car travels 195 miles on 15 litres of petrol. How far can it travel on 23 litres?

Step-by-step solution:

1. Distance per litre: 195 ÷ 15 = 13 miles per litre.
2. Distance on 23 litres: 13 x 23 = 299 miles.

Answer: 299 miles.

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Worked Example 4: Best Buy Problem

Question: A shop sells orange juice in two sizes. Small: 330ml for 85p. Large: 1 litre for £2.40. Which is better value?

Step-by-step solution:

1. Method: Find the price per millilitre for each.
2. Small: 85p ÷ 330ml = 0.2576p per ml (approximately 0.258p per ml).
3. Large: 240p ÷ 1000ml = 0.24p per ml.
4. 0.24p per ml < 0.258p per ml, so the large bottle is better value.

Alternative method — find how much you get per penny:

• Small: 330 ÷ 85 = 3.88ml per penny.
• Large: 1000 ÷ 240 = 4.17ml per penny.
• You get more juice per penny with the large bottle, so it is better value.

Answer: The large bottle (1 litre for £2.40) is better value.

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Worked Example 5: Three-Part Ratio