Ratio: simplifying ratios, dividing a quantity in a given ratio, expressing ratios as fractions and unit ratios, combining ratios, and using scale factors, maps and scale drawings.

What this dot point is asking

A ratio compares two or more quantities. Edexcel expects you to simplify ratios, divide a quantity in a given ratio, move between ratios and fractions, combine ratios, and use scale factors in maps and scale drawings. Ratio is one of the most heavily examined areas because it appears in recipes, money sharing, mixtures, maps and similar shapes, so fluency repays itself many times over.

Simplifying and writing ratios

A ratio is simplified by dividing all parts by a common factor until no further factor remains. For $12 : 18$, divide both by $6$ to get $2 : 3$. A ratio can also be written in the form $1 : n$ by dividing both parts by the first: $5 : 8$ becomes $1 : 1.6$, which is useful for comparison.

Dividing in a given ratio

Sharing a quantity in a ratio is a three-step method, shown in the exam question above: add the parts, find one part, then scale up.

A common twist gives one person's share or the difference between shares and asks for the total. If Beth's $3$ parts are $\pounds 270$, one part is $\pounds 90$, so the total of $8$ parts is $\pounds 720$. Working from one part is the key.

Ratios and fractions

Ratios and fractions describe the same split from different angles. In a class with boys to girls in the ratio $2 : 3$, there are $5$ parts in total, so boys are $\dfrac{2}{5}$ of the class and girls $\dfrac{3}{5}$. Reading "$\dfrac{2}{5}$ are boys" back as a ratio gives $2 : 3$ (boys to girls), not $2 : 5$, because the fraction is part-to-whole while the ratio is part-to-part.

Combining ratios

Two ratios that share a common quantity can be combined into a single three-part ratio by scaling so the shared part matches. If $A : B = 2 : 3$ and $B : C = 6 : 7$, scale the first to make $B$ equal $6$: $A : B = 4 : 6$. Now $A : B : C = 4 : 6 : 7$. This technique appears in multi-step problems and rewards careful bookkeeping.

Scale factors and scale drawings

A scale links a drawing or map to real life. A map scale $1 : 25000$ means $1\,\text{cm}$ on the map is $25000\,\text{cm}$ in reality. To find a real distance, multiply the map distance by the scale; to find a map distance, divide the real distance by the scale. Scale drawings of rooms, gardens and routes use the same idea, and the only common difficulty is unit conversion (remember $100\,\text{cm} = 1\,\text{m}$ and $100000\,\text{cm} = 1\,\text{km}$).

Ratio in recipes and mixtures

A very common context for ratio is recipes and mixtures, where ingredients are combined in a fixed ratio. If a recipe for $4$ people uses flour and sugar in the ratio $3 : 1$ and needs $600\,\text{g}$ of flour, then one part is $200\,\text{g}$, so the sugar (one part) is $200\,\text{g}$. Scaling a recipe up or down keeps the ratio fixed: for $6$ people instead of $4$, multiply every ingredient by $\tfrac{6}{4} = 1.5$. Mixtures such as paint colours, concrete and squash-to-water work the same way, and a frequent question asks whether two mixtures will be the same shade or strength, which they are only if their ratios simplify to the same thing.

Try this

Q1. Simplify the ratio $24 : 36$. [1 mark]

• Cue. Divide both by $12$: $2 : 3$.

Q2. Share $£64$ in the ratio $5 : 3$. [3 marks]

• Cue. $8$ parts, so one part is $£8$; the shares are $£40$ and $£24$.