R5 - Sharing in a ratio

Sharing a total in a part to part ratio

Asha and Ben share £84 in the ratio $3:4$. Asha suggests dividing £84 by $3$. Ben suggests dividing it by $7$. Which starting step works, and what should each person receive?

The ratio $3:4$ means Asha gets $3$ equal parts and Ben gets $4$ of those same-sized parts. The whole £84 must fill $3+4=7$ parts.

[DIAGRAM: asset_name: Sharing 84 pounds in a three to four ratio; asset_slug: aqa8300_r02_sharing_bar; caption: Seven equal parts divide 84 pounds into shares of 36 pounds and 48 pounds; description: A horizontal bar has seven equal boxes, the first three blue and the next four clay. Every box is labelled 12 pounds. A bracket over all seven says 7 equal parts equals 84 pounds. Labels below the groups show 3 parts equals 36 pounds and 4 parts equals 48 pounds. A final line shows one part equals 84 divided by 7 equals 12 pounds.]

The bar makes the divisor visible. There are seven parts altogether.

Change only the total. The ratio and the number of parts stay the same, but each part becomes more valuable.

There is a neat consequence: sharing £84 in $6:8$ gives the same amounts as $3:4$. There are twice as many parts, but each is worth half as much: $84\div14=6$, so the shares are $6\times6=36$ and $8\times6=48$.

Always check the total and the ratio; either check alone can miss a mistake.

Sharing with a part to whole ratio

A ratio comparing one share with the whole already includes the total part count.

If $\text{Asha's share}:\text{total}=3:7$, the total has $7$ parts. Asha has $3$, leaving $7-3=4$ for Ben. Do not add $3+7$: that counts Asha's part twice.

Write the labels before the numbers. The symbols $3:7$ alone do not tell you which interpretation is intended. This distinction and the one-part sharing method are not on the formulae sheet: learn them.