Ratio and proportion

What a ratio tells you

A ratio compares the sizes of two or more quantities. If a fruit bowl holds 3 apples for every 5 oranges, the ratio of apples to oranges is $3:5$. It does not say how many pieces of fruit there are, only how they compare: 6 apples and 10 oranges, or 30 apples and 50 oranges, are both in the ratio $3:5$.

Three ideas sit underneath every ratio question:

• Order matters. Apples to oranges is $3:5$; oranges to apples is $5:3$. Write the parts in the order the question names them.
• A ratio gives you fractions of the whole. In $3:5$ there are $3+5=8$ parts altogether, so the apples are $\frac{3}{8}$ of the fruit and the oranges are $\frac{5}{8}$. Writing $\frac{3}{5}$ for the fraction of apples is a very common slip: $\frac{3}{5}$ compares apples with oranges, not apples with all the fruit.
• A ratio has no units, so the quantities must be in the same units. You can compare 45 minutes with 2 hours, but only after writing both in minutes.

Writing a ratio in its simplest form

A ratio is in its simplest form when every part is a whole number and the parts have no common factor other than 1. Dividing every part by the same number does not change the ratio, for the same reason that $\frac{6}{8}=\frac{3}{4}$: each part keeps exactly the same share of the whole.

The quickest route is to divide every part by the highest common factor (HCF). The syllabus gives the example $20:30:40$. The HCF of 20, 30 and 40 is 10, so $20:30:40=2:3:4$. If you cannot spot the HCF straight away, divide by any common factor you can see and keep going until no common factor is left.

Two awkward cases come up often:

• Decimals or fractions in the ratio. Multiply every part by the same number to clear them, then simplify. $1.5:2.5$ becomes $15:25$ (multiply by 10), which is $3:5$. For $\frac{1}{2}:\frac{2}{3}$, multiply by 6, the lowest common multiple of the denominators, to get $3:4$.
• Different units. Convert to one unit first. 45 minutes : 2 hours is $45:120$, which simplifies to $3:8$. Writing $45:2$ compares minutes with hours and is simply wrong.

Some questions ask for the form $1:n$, which is how map scales are written. Divide every part by the first part: $8:20=1:2.5$. In this form $n$ is allowed to be a decimal; that is the point of it.

Dividing a quantity in a given ratio

To share an amount in a given ratio, think in parts:

1. Add the numbers in the ratio to find the total number of parts.
2. Divide the amount by the total number of parts to find the value of one part.
3. Multiply the value of one part by each number in the ratio.
4. Check that the shares add up to the amount you started with.

Why it works: in the ratio $2:5$, the first person receives 2 parts for every 5 parts the second receives. Out of every 7 parts handed out, 2 go to the first person, so their share is $\frac{2}{7}$ of the total. Steps 2 and 3 are just a tidy way of finding $\frac{2}{7}$ of the amount.

Questions do not always give you the total. You may be told one person's share, or the difference between two shares. Then work out how many parts that number represents, find one part from it, and carry on as before. The question to ask every time is: how many parts is the number I have been given worth?

Proportional reasoning: recipes and the unitary method

Two quantities are in direct proportion when they change at the same rate: double one and the other doubles, halve one and the other halves. The ingredients in a recipe behave like this, and so does the cost of fuel against the number of litres bought.

There are two reliable methods:

• Scale factor. A recipe for 4 people uses 300 g of rice. For 10 people the scale factor is $10÷4=2.5$, so you need $300\times 2.5=750$ g.
• Unitary method. Find the amount for one, then multiply. 300 g for 4 people is $300÷4=75$ g per person, so 10 people need $75\times 10=750$ g.

The unitary method always works; the scale factor is quicker when the numbers are friendly. What does not work is adding: going from 4 people to 10 people is not adding 6 lots of anything to 300 g.

Some quantities go the other way. If 4 painters take 6 days to paint a building, then 8 painters working at the same rate take 3 days: doubling the workers halves the time. This is inverse proportion. What stays fixed is the product: the job needs $4\times 6=24$ painter-days however you split it. So find that total first, then divide by the new number of painters or days.

Before you calculate, ask: should the answer be bigger or smaller? More people eating needs more rice; more people working needs fewer days. That one question catches most errors. And think about whether the answer must be a whole number, and which way to round: you cannot bake part of a biscuit, so if you have enough butter for 26.7 biscuits, you can make 26.

Map scales

A map scale is a ratio. A scale of $1:50000$ means that 1 cm on the map represents $50000$ cm on the ground. A ratio has no units, so both parts are in the same unit: you work in centimetres and convert at the end.

The conversions you need: 100 cm = 1 m and 1000 m = 1 km, so $100000$ cm = 1 km.

• Map to real: multiply by the scale number, then convert. 6 cm on a $1:50000$ map is $6\times 50000=300000$ cm, which is 3000 m, or 3 km.
• Real to map: convert the real distance into centimetres, then divide by the scale number. 4.5 km is $450000$ cm, and $450000÷50000=9$ cm.

A good habit is to say the scale in words before you start. $1:50000$ means 1 cm stands for 500 m, so 2 cm stands for 1 km. Saying it this way also tells you whether your final answer is sensible: a 9 cm line on this map cannot be 45 km long.

Best value

To decide which of several packs is the best value, compare like with like. There are two standard methods:

• Price per unit: divide the price by the amount, for example the cost of 1 kg or of 100 ml. The smallest price per unit is the best value.
• Amount per unit of money: divide the amount by the price, for example grams per £1. The largest amount per pound is the best value.

Either method earns full marks as long as you do the same calculation for every option, show each result, and then state which is best. An answer of just 'the large one' earns nothing, even when it is right, because the marks are for the comparison.

On a non-calculator paper, look for a convenient common amount instead. For 250 g at £1.50 and 400 g at £2.20, compare the cost of 50 g: $1.50÷5=0.30$ and $2.20÷8=0.275$. The 400 g pack is cheaper per 50 g, so it is the better value. Bigger packs are often, but not always, cheaper per unit, which is exactly why examiners set these questions.