Ratio and Proportion

Ratio and proportion questions are popular in the CEM 11+ exam. They test whether you can compare quantities, share amounts fairly, and scale values up or down. CEM may present these as straightforward ratio problems or hide them inside word problems and real-life scenarios. This lesson will make sure you are fully prepared.

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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.

• The ratio of boys to girls is 3 : 5 means for every 3 boys there are 5 girls.
• Ratios can be written with a colon (3 : 5) or as a fraction (3/5).

Simplifying Ratios

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

Original Ratio	HCF	Simplified
12 : 18	6	2 : 3
20 : 35	5	4 : 7
8 : 12 : 20	4	2 : 3 : 5

Converting Units Before Simplifying

If the quantities are in different units, convert to the same unit first:

Simplify the ratio 1.5 kg : 600 g

• 1.5 kg = 1,500 g
• 1,500 : 600 → divide both by 300 → 5 : 2

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Sharing in a Given Ratio

Two-Part Ratio

Share £40 in the ratio 3 : 5.

1. Add the parts: 3 + 5 = 8 parts
2. Find the value of one part: £40 ÷ 8 = £5
3. Multiply each part: 3 × £5 = £15, 5 × £5 = £25

Check: £15 + £25 = £40

Three-Part Ratio

Share 72 marbles in the ratio 2 : 3 : 4.

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

Check: 16 + 24 + 32 = 72

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Using Ratios to Find Missing Quantities

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

• 7 parts = 28, so 1 part = 28 ÷ 7 = 4
• Cats = 4 parts = 4 × 4 = 16

CEM Tip: CEM questions may not always use the word "ratio" — they might say "for every 4 cats there are 7 dogs" instead. Recognise this as a ratio problem!

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Proportion