Ratio, proportion & percentages

Sharing in a ratio, direct and inverse proportion, percentage change and compound growth and decay.

8 min read

These topics turn a worded situation into a multiplier. The skill the ESAT rewards is choosing the right multiplicative relationship quickly, especially for percentage change and compound interest.

To share in a ratio, add the parts to get the total number of shares, find one share, then scale.

Worked example

Share £ $96$ in the ratio $3 : 5$ .

Total shares $= 3 + 5 = 8$ , so one share $= \frac{96}{8} = £12$ .

Amounts: $3 \times 12 = £36$ and $5 \times 12 = £60$ .

A ratio $a : b$ is the same proportion as the fraction $\frac{a}{a + b}$ of the whole, which is useful for converting between ratio and fraction problems.

Direct: $y \propto x$ means $y = k x$ ; double $x$ and $y$ doubles. The graph is a straight line through the origin.

Inverse: $y \propto \frac{1}{x}$ means $y = \frac{k}{x}$ ; double $x$ and $y$ halves. Proportion can involve powers, e.g. $y \propto x^{2}$ or $y \propto \frac{1}{x}$ .

Worked example

$y$ is inversely proportional to $x^{2}$ , and $y = 8$ when $x = 3$ . Find $y$ when $x = 6$ .

$y = \frac{k}{x ^{2}} \Rightarrow 8 = \frac{k}{9} \Rightarrow k = 72.$ Then $y = \frac{72}{36} = 2.$

A percentage is "per hundred". Treat percentage change as a multiplier: a $15%$ increase is $\times 1.15$ ; a $15%$ decrease is $\times 0.85$ .

Reverse percentage: if a price *after* a $20%$ rise is £ $72$ , the original is $72 \div 1.2 = £60$ . Always divide by the multiplier; never subtract $20%$ .

Worked example

A jacket costs £ $80$ and is reduced by $25%$ , then a further $10%$ at the till. Final price?

$80 \times 0.75 \times 0.90 = 80 \times 0.675 = £54.$

Exam tip

Successive percentage changes multiply, they do not add. A $25%$ then $10%$ reduction is not $35%$ off.

Repeated proportional change uses a power of the multiplier: after $n$ steps, $value = P \times (multiplier)^{n}$ .

Worked example

£ $500$ is invested at $4%$ compound interest per year. Value after $3$ years?

$500 \times 1.0 4^{3} = 500 \times 1.124864 = £562.43.$

Exam tip

Without a calculator you usually only need the method plus a rough size, or a value of $n$ small enough to expand by hand. Set up $P \times m^{n}$ and the options will do the rest.

Doing reverse percentages by subtracting instead of dividing by the multiplier.
Adding successive percentage changes.
Forgetting the constant $k$ in proportion (always find it first).

Ratio, proportion & percentages

Overview

Ratio

Direct & inverse proportion

Percentages & change

Compound growth & decay

Common mistakes