Ratio and proportion, Year 9: direct and inverse proportion

Height is not proportional to age

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

The whyPeople 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

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

The whyTwice 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

HintDoes each egg need the pan to itself?

The whyThe 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

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

The whyThe 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

HintDouble one and the other doubles; treble one and the other trebles.

The whyThis 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

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

The whyIn 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

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

The why24 = 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

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

The whyk = 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

HintTry n = 1 and see what the formula gives.

The whyWhen 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

HintCompare each y-value with its partner using division.

The why5 ÷ 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

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

The whySpeed 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

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

The whyThe 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

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

The whyTwo 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

HintIt is the result of multiplying the two quantities together.

The whyIf 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

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

The whyDoubling 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

HintCheck what the two coordinates of each point multiply to.

The why2 × 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

HintAsk what height a point sitting exactly on that axis has.

The whyOn 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

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

The whyThe 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

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

The whyOn 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.