5.01b - Sharing in a ratio

5.01b - Sharing in a ratio

If two people share £84 in the ratio two to five, why should you divide by seven? Learn to share a total, reconstruct missing amounts from one share or a difference, and divide a quantity between three or more recipients.

Sharing between two quantities

A ratio gives numbers of equal-sized parts, not the actual amounts each person receives.

For a ratio 2:52:5, the first person receives two parts and the second five parts of the same size. There are seven parts altogether. To share a known total, divide by the total number of parts to find one part, then multiply by each person's number of parts.

One ratio part

A ratio part is one common unit of a proportional sharing. In the ratio a:ba:b, the amounts can be written a×one parta\times\text{one part} and b×one partb\times\text{one part}. For ordinary sharing, the total and the ratio entries are positive.

The two required checks are different: the shares must add to the original total, and their ratio must match the stated comparison. A correct sum alone does not guarantee a correct split.

Worked example 1: two unequal shares

Question. Asha and Ben share £84 in the ratio 2:52:5. Find each share.

Plan. Divide £84 into seven equal parts, then allocate two and five parts.

total parts=2+5=7one part=84÷7=12Asha’s share=2×12=24Ben’s share=5×12=60\begin{aligned} \text{total parts}&=2+5=7 \\ \text{one part}&=84\div7=12 \\ \text{Asha's share}&=2\times12=24 \\ \text{Ben's share}&=5\times12=60 \end{aligned}

Answer. Asha receives £24 and Ben receives £60.

Check. 24+60=8424+60=84, and 24:60=2:524:60=2:5.

Keep the same total but change the ratio. The sum of the entries changes, so one part changes too.

Worked example 2: a different division of the same total

Question. Asha and Ben share £84 in the ratio 2:12:1. Find both shares and express the resulting division as a ratio in simplest form.

Plan. There are now three parts, with Asha receiving two of them.

one part=84÷(2+1)=28Asha=2×28=56Ben=1×28=28Asha : Ben=56:28=2:1\begin{aligned} \text{one part}&=84\div(2+1)=28 \\ \text{Asha}&=2\times28=56 \\ \text{Ben}&=1\times28=28 \\ \text{Asha : Ben}&=56:28=2:1 \end{aligned}

Answer. £56 and £28, giving 2:12:1.

Check. 56+28=8456+28=84 and 56÷28=256\div28=2, so Asha receives twice Ben's amount.

You can also form a ratio from any given two-part division: £18 and £30 gives 18:30=3:518:30=3:5. If a ratio is not simplified, the sharing still works: 4:104:10 gives twice as many parts as 2:52:5, each half as large, producing the same final shares.

Recovering a total from a share

A known share is not a known total: divide it by the number of parts that it actually represents.

If the ratio is 2:52:5 and the smaller share is £24, that £24 represents two parts, not seven. If the difference between the shares is £36, that difference represents 5−2=35-2=3 parts. The central decision is to match the given amount to the correct part count.

Worked example 3: one share is known

Question. Asha and Ben receive money in the ratio 2:52:5. Asha receives £24. Find Ben's share and the total.

Plan. Asha's two parts establish the value of one part.

one part=24÷2=12Ben’s share=5×12=60total=(2+5)×12=84\begin{aligned} \text{one part}&=24\div2=12 \\ \text{Ben's share}&=5\times12=60 \\ \text{total}&=(2+5)\times12=84 \end{aligned}

Answer. Ben receives £60; the total is £84.

Check. Ben's share is 52×24=60\frac52\times24=60, and 24+60=8424+60=84. Dividing £24 by seven would wrongly treat Asha's share as the total.

Keep the ratio but replace the known share with a known difference. The given amount now measures the gap between five parts and two parts.

Continue this lesson.

You’re previewing NovaLearn without an account. Create a free account to read the rest.

Create free account