Sharing in a Ratio

Statement

To divide a total into the ratio \(a:b\):

\[ \text{Share for } a = \frac{a}{a+b} \times \text{total}, \quad \text{Share for } b = \frac{b}{a+b} \times \text{total} \]

Why it’s true

• A ratio splits a quantity into proportional parts.
• Total parts = \(a+b\).
• Each person’s share is their number of parts divided by total parts, multiplied by the total amount.
• Works the same for ratios with more than two parts (sum all parts in denominator).

Recipe (how to use it)

1. Add up all the parts of the ratio.
2. Divide total by this sum to find value of one part.
3. Multiply by each part’s number to find each share.

Spotting it

Look for wording like “divide £240 in the ratio 2:3” or “share 60 sweets between 3 children in ratio 1:2:3”.

Common pairings

• Money sharing problems.
• Recipe scaling.
• Maps, scale drawings, and proportions.

Mini examples

1. Divide £240 in ratio 2:3 → total parts=5, one part=£48 → shares £96 and £144.
2. Divide 60 sweets in ratio 1:2:3 → total parts=6, each=10 → shares 10, 20, 30.
3. Divide 100g in ratio 4:1 → total parts=5, each=20g → shares 80g and 20g.

Pitfalls

• Forgetting to divide by the sum of the ratio.
• Mixing up which part corresponds to which share.
• Not simplifying ratio before use (if needed).

Exam strategy

• Write total parts clearly before calculation.
• Check answers add up to the total.
• For multi-part ratios, extend formula to \(\frac{\text{part}}{\text{sum of all parts}} \times \text{total}\).

Summary

To share a total in a ratio, divide total into parts by the sum of the ratio, then multiply by each ratio part. Always check the shares sum to the total.