Ratio and proportion, Year 9: direct and inverse proportion

Height is not proportional to age

提示Ask whether doubling one of these two quantities really forces the other one to double.

为什么People grow quickly at some ages, slowly at others, and then stop. For two quantities to be in proportion, multiplying one must always multiply the other by the same number, and growth does not behave like that.

The cost of petrol and the litres bought

提示In a true proportion, none of one thing means none of the other.

为什么Twice the litres costs exactly twice as much, and no litres costs nothing. The taxi fare rises steadily but starts at £3, so doubling the miles does not double the fare.

4

提示Does each egg need the pan to itself?

为什么The eggs cook at the same time, so the time does not depend on how many there are. Before multiplying, always ask whether one quantity really scales with the other.

15

提示Think about what stays fixed between two people as the years go by.

为什么The gap between their ages stays at 5 years, so this is an adding situation, not a multiplying one. The ratio of their ages keeps changing: 2 : 1 today, but 4 : 3 when Tom is 20.

same

提示Double one and the other doubles; treble one and the other trebles.

为什么This is the test to run before choosing a method. If it fails, as it does for a taxi fare with a fixed charge, the unitary method and scale factors will give a wrong answer.

The constant of proportionality

提示Its name says that it never changes and names the kind of relationship it belongs to.

为什么In y = kx, dividing any y by its x always gives k. If 3 kg of apples cost £6 then k = 2, the price in pounds for each kilogram.

4

提示Put the known pair into the formula and see what the multiplier must be.

为什么24 = k × 6, so k = 24 ÷ 6 = 4 and the formula is y = 4x. One known pair is enough to fix k, because the same multiplier works for every pair.

y = 56

提示Find the multiplier that turns the first x into its y, then use it again.

为什么k = 35 ÷ 5 = 7, so y = 7x, and when x = 8, y = 56. Finding k first turns every later question into a single multiplication.

Ribbon costs £1.50 per metre

提示Try n = 1 and see what the formula gives.

为什么When n = 1 the formula gives C = 1.5, so the constant is the cost of one metre. In any direct proportion, k is the amount of y for each single unit of x: a unit rate.

Yes: y ÷ x is 2.5 for every pair

提示Compare each y-value with its partner using division.

为什么5 ÷ 2, 10 ÷ 4 and 25 ÷ 10 all equal 2.5, so one multiplier links every pair and y = 2.5x. If even one pair gave a different result, the relationship would not be a direct proportion.

Inverse proportion

提示Its name contains a word meaning "turned the other way round".

为什么Speed and journey time are a typical pair: drive twice as fast and the same journey takes half as long. The two quantities always multiply to give the same number.

2

提示First find the one thing that does not change: how far the journey is.

为什么The distance is 40 × 3 = 120 miles, and 120 ÷ 60 = 2 hours. Speed and time are inversely proportional, so their product, the distance, stays the same.

The number of people and the amount each one gets

提示Look for the pair where more of one must mean less of the other.

为什么Two people get £30 each and four people get £15 each: doubling the people halves the share, and people × share is always 60. Tickets and total cost rise together, which is direct proportion.

product

提示It is the result of multiplying the two quantities together.

为什么If y = kx then y ÷ x is always k. If two quantities are inversely proportional then x × y is always the same number, such as the 24 worker-hours in a job.

A falling curve that flattens out and never touches the axes

提示As one quantity grows the other shrinks, quickly at first and then ever more slowly.

为什么Doubling x halves y, so the early drops are large and the later ones small. The result is a smooth curve, quite unlike the straight line of direct proportion.

y = 3

提示Check what the two coordinates of each point multiply to.

为什么2 × 18 = 36 and 4 × 9 = 36, so every point on the curve has coordinates that multiply to 36. When x = 12, y = 36 ÷ 12 = 3.

y would have to be 0, and 12 divided by a number is never 0

提示Ask what height a point sitting exactly on that axis has.

为什么On the x-axis y is 0, but 12 ÷ x only gets smaller and smaller as x grows. In the same way x can never be 0, because dividing by zero is impossible, so the curve misses the y-axis too.

reciprocal

提示The word begins with r, and it is also what you get by turning a fraction upside down.

为什么The simplest example is y = 1/x, where each y is the reciprocal of its x. Other inverse proportions, such as y = 12/x, are the same shape stretched.

Doubling x does not halve y, so x × y is not constant

提示Test it: compare the height at x = 1 with the height at x = 2.

为什么On this line y falls by 2 for every 1 across: 8 when x = 1 and 6 when x = 2, where halving would need 4. A falling straight line subtracts the same amount each step; inverse proportion divides, giving a curve that reaches neither axis.