Ratio, Proportion & Scale

Section 1 of 3

Ratio in Simplest Form

A ratio compares two amounts. Simplify it like a fraction — and always get both sides into the same units first.

Simplify a ratio

Divide both sides by their common factor. Change to the same unit before you start. Handy conversions: $10\,\text{mm} = 1\,\text{cm}$, $100\,\text{cm} = 1\,\text{m}$, $1000\,\text{m} = 1\,\text{km}$.

Worked example — basic and units

Write in simplest form: (i) $5 : 10$; (ii) $2$ days $:$ $4$ weeks; (iii) $2\,\text{cm} : 1\,\text{m}$.

(i) $5 : 10 = 1 : 2$ (divide by $5$)

(ii) $2 : 28$ days $= 1 : 14$

(iii) $2 : 100$ cm $= 1 : 50$

$1:2,\ \ 1:14,\ \ 1:50$

Worked example — with fractions

Write in simplest form: $\dfrac{3}{4} : \dfrac{1}{6}$ and $1\tfrac{1}{2} : \dfrac{2}{3}$.

$\tfrac{3}{4} : \tfrac{1}{6} = \tfrac{9}{12} : \tfrac{2}{12} = 9 : 2$

$\tfrac{3}{2} : \tfrac{2}{3} = \tfrac{9}{6} : \tfrac{4}{6} = 9 : 4$

$9:2$ and $9:4$

Section 2 of 3

Dividing in a Ratio

Share it out

Add the numbers to get the total parts. One part $=$ total $\div$ parts. Then multiply for each share.

Worked example — two shares

Divide €650 in the ratio $5 : 8$.

$5 + 8 = 13$ parts; $1$ part $= \tfrac{650}{13} = $ €$50$

$5$ parts $= $ €$250$; $8$ parts $= $ €$400$

€250 and €400

Worked example — three shares

Divide €720 in the ratio $2 : 3 : 4$.

$2 + 3 + 4 = 9$ parts; $1$ part $= \tfrac{720}{9} = $ €$80$

€$160$, €$240$, €$320$

€160, €240, €320

Worked example — worded

Divide €240 between A and B so that B gets twice A’s share.

$A : B = 1 : 2$, so $3$ parts $= $ €$240$, $1$ part $= $ €$80$

$A = $ €$80$, $B = $ €$160$

A = €80, B = €160

You try

Divide €185 in the ratio $1 : \tfrac{2}{3}$.

First clear the fraction: $1 : \tfrac{2}{3} = 3 : 2$. Then $5$ parts $= $ €$185$.

$1 : \tfrac{2}{3} = 3 : 2$; $5$ parts $= $ €$185$, $1$ part $= \tfrac{185}{5} = $ €$37$

$3$ parts $= $ €$111$; $2$ parts $= $ €$74$

€111 and €74

Section 3 of 3

Scale

Scale = a ratio

A scale like $1 : 50$ means $1$ on the model is $50$ in real life. To go from model to real, multiply by the scale; from real to model, divide. Mind the units.

Worked example — model to real

A scale is $1 : 50$. If the model is $6\,\text{cm}$ long, how long is the real thing?

$6 \times 50 = 300\,\text{cm}$

$300 \div 100 = 3\,\text{m}$

$3\,\text{m}$

You try

A map has scale $1 : 25000$. A road is $3.2\,\text{mm}$ on the map. How long is it in real life?

Multiply by the scale, then convert mm to a sensible unit.

$3.2 \times 25000 = 80000\,\text{mm}$

$80000 \div 1000 = 80\,\text{m}$

$80\,\text{m}$

That’s Part 1.

Simplifying ratios, dividing an amount in a ratio, and scale. Next parts: percentages & money, compound interest, income tax, and foreign exchange.