Ratio and Proportion

Ratio and proportion questions are a staple of the CSSE 11+ maths paper. These questions test whether you can compare quantities, share amounts fairly, and solve real-world problems involving scaling. The CSSE is known for embedding ratio concepts within multi-step word problems, so you need to be confident with all the methods in this lesson.

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What Is a Ratio?

A ratio compares two or more quantities. It tells you how much of one thing there is compared to another.

If there are 4 red sweets and 6 blue sweets, the ratio of red to blue is 4 : 6, which simplifies to 2 : 3.

Key Points

• The order matters: 2 : 3 is different from 3 : 2.
• A ratio does not tell you the actual amounts — only the relationship.
• Ratios can have more than two parts: e.g. red : blue : green = 1 : 3 : 2.

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Simplifying Ratios

Simplify a ratio by dividing all parts by their Highest Common Factor (HCF).

Simplify 15 : 25

• HCF of 15 and 25 = 5
• 15 ÷ 5 : 25 ÷ 5 = 3 : 5

Simplify 24 : 36 : 12

• HCF of 24, 36, and 12 = 12
• 24 ÷ 12 : 36 ÷ 12 : 12 ÷ 12 = 2 : 3 : 1

Ratios with Decimals or Fractions

Multiply all parts by the same number to make them whole numbers first.

Simplify 0.4 : 1.2

• Multiply by 10: 4 : 12
• Simplify: 4 ÷ 4 : 12 ÷ 4 = 1 : 3

Simplify 1/3 : 1/2

• Multiply by 6 (LCM of 3 and 2): 2 : 3

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Sharing in a Given Ratio

This is one of the most common CSSE ratio questions.

Share £60 in the ratio 2 : 3

Step 1: Find the total number of parts: 2 + 3 = 5

Step 2: Find the value of one part: £60 ÷ 5 = £12

Step 3: Multiply each part:

• First share: 2 × £12 = £24
• Second share: 3 × £12 = £36

Check: £24 + £36 = £60

Three-Way Sharing

Share 90 sweets in the ratio 1 : 2 : 3

• Total parts: 1 + 2 + 3 = 6
• One part: 90 ÷ 6 = 15
• Shares: 1 × 15 = 15, 2 × 15 = 30, 3 × 15 = 45
• Check: 15 + 30 + 45 = 90

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Using Ratios to Find Missing Values

The ratio of boys to girls in a class is 3 : 5. There are 15 boys. How many girls are there?

• 3 parts = 15 boys
• 1 part = 15 ÷ 3 = 5
• Girls = 5 parts = 5 × 5 = 25 girls

The ratio of cats to dogs at a shelter is 4 : 9. There are 36 dogs. How many animals are there in total?

• 9 parts = 36 dogs
• 1 part = 36 ÷ 9 = 4
• Cats = 4 × 4 = 16
• Total = 16 + 36 = 52 animals

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Proportion

Proportion means that two ratios are equal. It is used when quantities increase or decrease at the same rate.

Direct Proportion (Scaling Up or Down)

A recipe for 4 people uses 300 g of flour. How much flour is needed for 10 people?

Step 1: Find the amount for 1 person: 300 ÷ 4 = 75 g