Ratio and proportion

2 : 3

HintDivide both sides by the largest number that goes into each of them exactly.

The whyBoth 12 and 18 divide by 6. A simplified ratio keeps the same relationship but states it in the smallest whole numbers possible.

3/8

HintAdd the parts together to find how many pieces the whole is cut into.

The whyThere are 3 + 5 = 8 parts altogether, and the first share owns 3 of them. A ratio compares part with part, while a fraction compares part with whole.

£15

HintWork out what a single part is worth first, then take three of them.

The whyThere are 8 parts in total, so one part is £40 ÷ 8 = £5. The two shares come to £15 and £25, which adds back to £40 as a check.

Convert both amounts to the same unit

HintPence and pounds cannot be compared side by side until they are written in one currency.

The why£2 is 200p, so the ratio becomes 50 : 200, which simplifies to 1 : 4. Comparing 50 with 2 directly would wrongly suggest 25 : 1.

£1.50

HintFind the price of a single pen before scaling up to the larger order.

The whyOne pen costs 90 ÷ 3 = 30p, so five cost 5 × 30 = 150p. Finding the value of one first is the unitary method, and it solves almost any proportion question.

Out of a hundred

HintEvery percentage compares things on the same fixed base, whatever the real total happens to be.

The whyPer cent comes from the Latin per centum. Putting scores on a common base is what lets 17 out of 20 and 42 out of 50 be compared fairly.

250 cm

HintEach metre is worth one hundred of the smaller unit.

The whyThere are 100 cm in a metre, so 2.5 × 100 gives the answer. Metric units step in powers of ten, which is why converting only shifts the digits along.

75 km/h

HintShare the journey out evenly across the time it took.

The whySpeed is distance ÷ time, so 150 ÷ 2 gives the answer in kilometres per hour. It is an average because the car will have been quicker and slower at different moments.

Compare the cost of one unit

HintBring both packs down to the same size before judging them.

The whyDivide the price by the quantity in each pack to get a price per gram or per millilitre, and the smaller figure wins. Larger packs are usually, but not always, the better deal.

25%

HintMeasure the rise against the price it started at, not the price it reached.

The whyThe rise is £5, and £5 out of £20 is a quarter of the original. Percentage change is always worked out as a share of the starting amount.

1.2

HintKeep the whole of the original, then add the extra fifth on top of it.

The whyThe original amount counts as 1, or 100%, and the increase adds another 0.2. A 15% decrease works the same way, giving 1 − 0.15 = 0.85.

3 hours

HintTwice as many people means each of them has half as much to do.

The whyThe job is 4 × 6 = 24 worker-hours, so eight workers need 24 ÷ 8 each. In inverse proportion, doubling one quantity halves the other.