Ratio and proportion

54

提示The gap between the two shares is worth exactly one share, so find what a single share holds first.

为什么The parts differ by 5 − 4 = 1, so one part is 6 pupils. Girls are 4 × 6 = 24 and boys 5 × 6 = 30, giving 54 altogether. Matching a difference to the difference in parts is the key move when a ratio question gives a gap rather than a total.

40

提示Three of the equal shares hold twenty-four between them, so work out what a single share holds.

为什么Blue takes 3 of the 5 parts, so one part is 24 ÷ 3 = 8 counters. All five parts give 5 × 8 = 40, of which 16 are red. Going from one share back to the whole is the move most ratio word problems turn on.

8 and £15

提示How many equal slices does the ratio cut the money into? Start by counting them.

为什么The two shares come to £15 and £25, which adds back to £40 as a check — sharing never creates or loses money.

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.

30p and 150p (£1.50)

提示What would just one of them cost you? Everything follows from that.

为什么Finding the value of one first is the unitary method, and it solves almost any proportion question — once a single pen is priced, any number of pens is a single multiplication.

One eighth

提示Write it over a hundred first, then keep cancelling; doubling your answer would give a quarter.

为什么Per cent means out of a hundred, so 12.5% is 12.5 out of 100, the same as 125 out of 1000, which cancels to 1/8. Knowing 12.5% = 1/8, 20% = 1/5 and 25% = 1/4 turns many percentage questions into quick fraction work.

10000

提示A metre is 100 cm, and a square is measured in two directions at once.

为什么One square metre is a 100 cm by 100 cm square, so it holds 100 × 100 = 10 000 cm². Area units scale by the square of the length factor — the reason 1 m² is not 100 cm² — and volume units by its cube, so 1 m³ = 1 000 000 cm³.

75

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

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.