Ratio and proportion

54

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

The whyThe 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

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

The whyBlue 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

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

The whyThe 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

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.

30p and 150p (£1.50)

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

The whyFinding 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

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

The whyPer 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

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

The whyOne 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

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

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.