Ratio & Proportion

Ratio and proportion are ways of comparing quantities. These topics appear frequently in the GL 11+ exam, often in word problems that test your ability to share amounts fairly, scale recipes, or work with maps and models.

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What Is a Ratio?

A ratio compares two or more quantities using the symbol :

If there are 3 red sweets and 5 blue sweets, the ratio of red to blue is 3 : 5.

Key Points

• The order matters: 3 : 5 is different from 5 : 3.
• A ratio does not tell you the actual amounts — only the relationship between them.
• Ratios can have more than two parts: red : blue : green = 2 : 5 : 3.

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Simplifying Ratios

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

Simplify 12 : 18

• HCF of 12 and 18 = 6
• 12 ÷ 6 : 18 ÷ 6 = 2 : 3

Simplify 15 : 25 : 10

• HCF of 15, 25, and 10 = 5
• 15 ÷ 5 : 25 ÷ 5 : 10 ÷ 5 = 3 : 5 : 2

Simplifying Ratios with Fractions or Decimals

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

Simplify 0.5 : 1.5

• Multiply by 2: 1 : 3

Simplify ¹⁄₃ : ²⁄₅

• Multiply by 15 (LCM of 3 and 5): 5 : 6

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Sharing in a Given Ratio

This is one of the most common 11+ ratio questions.

Share £40 in the ratio 3 : 5

Step 1: Find the total number of parts: 3 + 5 = 8

Step 2: Find the value of one part: £40 ÷ 8 = £5

Step 3: Multiply each part of the ratio:

• First share: 3 × £5 = £15
• Second share: 5 × £5 = £25

Check: £15 + £25 = £40 ✓

Worked Example with Three Parts

Share 72 marbles in the ratio 2 : 3 : 4

• Total parts: 2 + 3 + 4 = 9
• One part: 72 ÷ 9 = 8
• Shares: 2 × 8 = 16, 3 × 8 = 24, 4 × 8 = 32
• Check: 16 + 24 + 32 = 72 ✓

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Using Ratios to Find Missing Values

The ratio of cats to dogs at a shelter is 4 : 7. There are 28 dogs. How many cats are there?

• 7 parts = 28 dogs
• 1 part = 28 ÷ 7 = 4
• Cats = 4 parts = 4 × 4 = 16 cats

The ratio of boys to girls in a class is 3 : 5. There are 12 boys. How many children are there in total?

• 3 parts = 12 boys
• 1 part = 12 ÷ 3 = 4
• Total parts = 3 + 5 = 8
• Total children = 8 × 4 = 32

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What Is Proportion?

Proportion describes how one quantity changes in relation to another. If two quantities are in direct proportion, as one increases the other increases at the same rate.

Scaling Recipes

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