9.2Ratios and Equivalent Ratios

A ratio compares quantities of the same kind. Write $a$ to $b$ as $a : b$ , in the same unit, then simplify by dividing every part by their HCF.

Equivalent ratios are obtained by multiplying or dividing every part by the same non-zero number
A ratio has no units, convert to the same unit first, then drop the unit

Simplifying a ratio of whole numbers

Write $\displaystyle 45 : 60$ in its simplest form.

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The HCF of

45

and

60

15

Divide both parts by the HCF:

45 \div 15 = 3

and

60 \div 15 = 4

45 : 60 = 3 : 4

A ratio is in its simplest form when the two parts share no factor other than

1

Simplifying a ratio with fractions

Simplify $\displaystyle \dfrac{3}{4} : \dfrac{9}{16}$ .

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Multiply both parts by a number both denominators divide into, here

16

\dfrac{3}{4} \times 16 : \dfrac{9}{16} \times 16

= 12 : 9

= 4 : 3 \quad \scriptsize\textit{HCF is 3}

Ratio with unit conversion

A recipe calls for 750 g of flour and 1.2 kg of rice. Write the ratio of flour to rice in its simplest form.

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Convert to the same unit:

1.2 \text{ kg} = 1200 \text{ g}

\displaystyle \text{Ratio}

\displaystyle = 750 : 1200

\displaystyle = \dfrac{750}{150} : \dfrac{1200}{150} \quad \scriptsize\textit{HCF is 150}

\displaystyle = 5 : 8

Now you try

Simplify the ratio $\frac{3}{2} : \frac{1}{2}$ .

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\frac{3}{2} : \frac{1}{2}

= 3 : 1 \quad \scriptsize\textit{multiply both terms by 2}

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