Ratio: simplifying, dividing and equivalent ratios

What is a ratio?

A ratio compares two or more quantities of the same kind. It is written with a colon: 3 : 5. It shows "for every 3 of one thing, there are 5 of another."

Simplifying ratios

Divide all parts of the ratio by their highest common factor (HCF).

Examples:

• 12 : 18 → HCF = 6 → 2 : 3.
• 15 : 25 : 35 → HCF = 5 → 3 : 5 : 7.

Ratios with units: convert to the same unit first.

• 45 cm : 1.2 m = 45 cm : 120 cm → HCF = 15 → 3 : 8.

Ratios with decimals: multiply to clear decimals, then simplify.

• 1.5 : 2.4 → multiply by 10 → 15 : 24 → HCF = 3 → 5 : 8.

Ratios with fractions: find a common denominator or multiply all parts by the LCM of denominators.

Dividing in a given ratio

To split a quantity in the ratio a : b:

1. Find the total number of parts: a + b.
2. Find the value of one part: total quantity ÷ (a + b).
3. Multiply by a and b respectively.

Example: Share £840 in the ratio 3 : 4 : 5. Total parts = 12. One part = 840 ÷ 12 = £70. Shares: 3 × £70 = £210; 4 × £70 = £280; 5 × £70 = £350. Check: 210 + 280 + 350 = £840 ✓.

Finding the original quantity from a part

If you know the value of one part of a ratio, you can find the total.

Example: A and B share money in the ratio 5 : 3. A receives £200. How much does B receive? What was the total? One part = 200 ÷ 5 = £40. B receives 3 × £40 = £120. Total = £200 + £120 = £320.

Equivalent ratios and scaling

Two ratios are equivalent if one is a scalar multiple of the other: 2 : 3 = 6 : 9 = 14 : 21.

This is the basis for solving "unitary method" proportion problems: If 5 workers can build a wall in 12 days, how many days for 3 workers? 5 × 12 = 60 worker-days. 3 workers: 60 ÷ 3 = 20 days.

CCEA context

CCEA Paper 1 tests dividing in a ratio. CCEA Paper 2 tests ratio in real-life contexts — mixing paint, scaling recipes, map scales, currency exchange.