C1.11 Ratio and proportion

Learning objectives

C1.11.1Understand 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: 40 in its simplest form is 2: 3: 4. e.g. adapt recipes; use map scales; determine best value.

A ratio compares quantities as multiplicative parts. In $a:b:c$ , each quantity is the same scale factor times $a$ , $b$ or $c$ . Keep the quantities in the stated order and convert them to the same units before comparing.

Learning job	Ratio-parts method	Example
Simplify a ratio	Divide every term by the same highest common factor.	$20:30:40=2:3:4$
Share a total $T$ in $a:b:c$	Find $p=a+b+c$ , then one part is $T\div p$ . Multiply by $a$ , $b$ and $c$ .	Share 190 in $12:5:2$ : one part $=190\div19=10$ ; shares are 120, 50 and 20.
Use a known difference	Subtract the corresponding ratio parts, then divide the quantity difference by that part difference.	Red:green:blue $=12:5:2$ and red exceeds green by 112. Seven parts $=112$ , so one part $=16$ and blue $=2\times16=32$ .

$\text{one part}=\frac{\text{known total}}{\text{sum of ratio parts}}\quad\text{or}\quad\text{one part}=\frac{\text{known difference}}{\text{difference of ratio parts}}$

For a direct proportion, every quantity changes by the same scale factor. A recipe using 550 g for 8 cakes needs $550\times(360\div8)=24\,750$ g, or 24.75 kg, for 360 cakes. Scale from a known pair; do not add the same amount.

For map scales, first use matching units. A scale of 1 cm to 8 km is $1:800\,000$ because 8 km is 800 000 cm. At a scale of $1:250\,000$ , 3.5 cm represents $3.5\times250\,000=875\,000$ cm, or 8.75 km.

To determine best value, compare like with like: calculate the cost for one common unit, or scale every option to the same quantity. The lowest cost per common unit is the best value, provided the units and quantities are equivalent.

A ratio of $2:3$ does not mean $2/3$ of the whole. There are $2+3=5$ parts, so the shares are $2/5$ and $3/5$ . Ratio order matters, and a context involving whole items may require a final whole-number decision only after the proportional calculation.

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