Ratio and proportion

2 : 3

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

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

3/8

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

为什么There 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

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

为什么There 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

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

为什么£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

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

为什么One 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

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

为什么Per 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

提示Each metre is worth one hundred of the smaller unit.

为什么There 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

提示Share the journey out evenly across the time it took.

为什么Speed 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

提示Bring both packs down to the same size before judging them.

为什么Divide 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%

提示Measure the rise against the price it started at, not the price it reached.

为什么The 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

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

为什么The 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

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

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