Ratio and proportion (sharing, scaling, best value)

Key learnings

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  Sharing in a ratio: add the parts, divide, then multiply

  To split a quantity in the ratio $a:b:c$, add the parts to get the total number of parts, divide the quantity by that total to find the value of one part, then multiply each share by its number of parts. E.g. $\pounds 240$ shared $3:5$ has $3+5=8$ parts, so one part $=240\div 8=30$, giving $\pounds 90$ and $\pounds 150$.

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  Finding one part when another is known

  If you are told the value of one share, find the value of a single part first ($\text{share}\div\text{its number of parts}$), then scale up to whatever is asked. The total quantity is the value of one part $\times$ the total number of parts.

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  Scaling recipes, maps and quantities by a multiplier

  Direct proportion means multiplying every quantity by the same scale factor. For a recipe for $4$ people made for $10$, multiply by $\tfrac{10}{4}=2.5$. On a $1:25\,000$ map, $1\text{ cm}$ represents $25\,000\text{ cm}=0.25\text{ km}$.