Ratios and Sharing in a Ratio

What you will learn

• Write and simplify a ratio.
• Find a missing amount when a ratio is given.
• Share an amount in a ratio.

Before you start: you should be able to simplify fractions and find common factors. See Equivalent Fractions and Simplifying and HCF and LCM.

The idea

A ratio compares quantities. If a drink is made with $2$ cups of juice for every $3$ cups of water, the ratio of juice to water is $2 : 3$. Order matters: $2 : 3$ is not the same as $3 : 2$.

Simplifying

Like a fraction, a ratio can be divided by a common factor. $12 : 18$ simplifies to $2 : 3$ by dividing both parts by $6$. A ratio is in its simplest form when only $1$ divides every part.

Equivalent ratios

Multiplying both parts by the same number gives an equal ratio: $2 : 3 = 4 : 6 = 6 : 9$. Recipe tables use this.

Sharing in a ratio

To share £$40$ between Amy and Ben in the ratio $3 : 2$:

1. Add the parts: $3 + 2 = 5$.
2. Find one part: $40 \div 5 = 8$.
3. Multiply: Amy gets $3 \times 8 = 24$ and Ben gets $2 \times 8 = 16$.
4. Check: $24 + 16 = 40$.

Worked examples

Example 1: Three-way share

Share £$60$ in the ratio $1 : 2 : 3$.

1. $1 + 2 + 3 = 6$ parts.
2. One part is $60 \div 6 = 10$.
3. Shares: $10$, $20$ and $30$.
4. Check: $10 + 20 + 30 = 60$.

Answer: £$10$, £$20$, £$30$

Example 2: Simplify

Simplify $24 : 36$.

1. The highest common factor of $24$ and $36$ is $12$.
2. $24 \div 12 = 2$ and $36 \div 12 = 3$.

Answer: $2 : 3$

Example 3: Find a missing amount

Flour to sugar is $3 : 2$. A cake uses $450$ g of flour. How much sugar?

1. $3$ parts are $450$ g, so one part is $150$ g.
2. Sugar is $2$ parts: $300$ g.

Answer: $300$ g

Watch out: a ratio of $2 : 3$ means $\frac{2}{5}$ and $\frac{3}{5}$ of the total, not $\frac{2}{3}$.

Why it matters

Ratios turn up in recipes, mixing paint or concrete, maps and scale drawings, and sharing costs fairly. They lead directly to proportion, rates and similar shapes.