E1.11 Ratio and proportion

Learning objectives

E1.11.1Understand and use ratio and proportion to: • give ratios in their• Understand and use ratio and proportion to: • give ratios in their simplest form • divide a quantity in a given ratio • use proportional reasoning and ratios in context. e.g. 20: 30:

A ratio compares quantities as equal-sized parts. In $a:b:c$ , the three quantities can be written $ka$ , $kb$ and $kc$ for the same multiplier $k$ . Preserve the stated order and convert quantities to the same units before comparing them.

Information given	Use the ratio parts	Example
simplify	divide every term by the same highest common factor	$45:75:120=3:5:8$
total shared in $a:b:c$	divide the total by $a+b+c$ to find one part	share 540 in $2:3:4$ : one part $=540\div9=60$ , so the shares are 120, 180 and 240
one quantity known	divide it by its matching ratio number	blue:red $=5:7$ and blue $=45$ : one part $=9$ , so red $=63$
difference known	divide the difference by the difference in ratio parts	$A:B=4:9$ and $B-A=35$ : five parts $=35$ , so $A=28$ and $B=63$

$\text{one part}=\frac{\text{known quantity}}{\text{matching number of ratio parts}}$

To combine linked ratios, make the shared quantity use the same number of parts. If $P:Q=2:5$ and $Q:R=3:4$ , scale the first ratio by 3 and the second by 5: $P:Q:R=6:15:20$ .

Direct proportion keeps a constant multiplier. A map scale of $1:250000$ uses the same units, so 3.2 cm represents $3.2\times250000=800000$ cm, or 8 km. For recipes or best value, scale to the required quantity or compare each option at one common quantity.

The ratio $2:3$ does not mean the first share is $2/3$ of the total: there are $2+3=5$ parts, so the shares are $2/5$ and $3/5$ . Adding the same amount to both quantities does not preserve a ratio; proportional scaling multiplies both by the same factor.

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