Ratio, proportion & rates of change · Ratio

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Sharing a quantity in a ratio

Why a ratio splits a quantity into equal "parts", how to find the value of one part, and how that unlocks difference, one-share-given, and fraction questions.

Ratio, proportion & rates of change · Ratio

Sharing a quantity in a ratio

Why a ratio splits a quantity into equal "parts", how to find the value of one part, and how that unlocks difference, one-share-given, and fraction questions.

Why it works

A ratio like

5 : 3

is not two amounts — it's a recipe for cutting something into equal parts. "Share £480 in the ratio

5 : 3

" means: chop the £480 into

5 + 3 = 8

equal parts, give one person

5

of those parts and the other

3

. So the single most useful move in every ratio question is:

\text{value of one part} = \frac{\text{total}}{\text{number of parts}}.

For £480 in

5 : 3

: one part

= \frac{480}{8} = £60

. Then each share is just its number of parts

\times £60

: the shares are

5 \times 60 = £300

and

3 \times 60 = £180

. (Check:

300 + 180 = 480

. ✓) Find the part, and the whole question falls open.

Why you must add the ratio numbers first. The commonest mistake is to divide by one of the ratio numbers — say

480 \div 5

— or by the number of people. Neither is the number of parts. There are

5 + 3 = 8

parts, so you divide by

8

. Miss the "add them up" step and every share is wrong.

A ratio is not the same as a fraction of the whole. In

5 : 3

, the first person's share is

\frac{5}{8}

of the total — not

\frac{5}{3}

. The denominator of the fraction is the total number of parts (

5 + 3 = 8

), not the other person's number. So

5 : 3

gives fractions

\frac{5}{8}

and

\frac{3}{8}

, which add to

1

— as they must, since together the shares are the whole thing.

Once you have the part, everything else is one step:

Difference between shares — the difference is $(5 - 3) = 2$ parts, so

2 \times 60 = £120

. You don't have to find both shares and subtract; the difference in parts times the part value gets there directly.

One share is given — reverse the move. If the smaller share ( $4$ parts) is

£96, then one part

= 96 \div 4 = £24

, and now you can find the total or the other share.

The habit to build: turn the ratio into a number of parts, find the value of one part, then read off whatever the question asks.