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GCSE Maths · EdexcelChapter 16 · Units and compound measures
GCSE Maths · Units and Compound Measures · Higher · Lesson 16.7

GCSE Maths: Direct and inverse proportion (Higher)

Part of the GCSE Maths units and compound measures chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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16.7

Direct and inverse proportion

Edexcel 1MA1: R10, R13
Grade 3-7

Two quantities are in direct proportion when they increase or decrease at the same rate: if one doubles, the other doubles; if one is halved, the other is halved. The cost of fuel and the number of litres bought is a good example: 20 litres cost twice as much as 10 litres. The key fact is that the ratio (y ÷ x) is always the same number.

That fixed number is the constant of proportionality, called k. Here, k is the price of one litre. The relationship is written y = kx, and we say “y is directly proportional to x” or y ∝ x. To test a table of values for direct proportion, work out y ÷ x for every column: they must all be equal.

Litres, x10203550
Cost (£), y14284970
y ÷ x1.41.41.41.4
Every ratio y ÷ x is 1.4, so the cost is directly proportional to the litres: y = 1.4x.

Two ways to solve a direct proportion problem

Unitary method: find the value for one, then multiply. Equation method: use a known pair to find k (k = y ÷ x), write y = kx, then substitute. The equation method is the best for algebra, and it is the one you need for Higher problems.

Inverse proportion

Two quantities are in inverse proportion when one increases as the other decreases in such a way that if one doubles the other is halved. More workers finish a job in fewer days; a faster speed gives a shorter time for the same distance; sharing a prize among more people gives each person less. In inverse proportion the product x × y is always the same number k. We write y = k ÷ x, and say “y is inversely proportional to x”, which means y is proportional to 1/x.

Workers, n12346
Days, d126432
n × d1212121212
Every product is 12, so the days are inversely proportional to the workers: d = 12 ÷ n.

Spot the difference

Direct: y = kx. Both go up together, and the graph is a straight line through the origin. Inverse: y = k/x. As one goes up the other goes down, and the graph is a curve. Be careful: not every pair that both increase is proportional. A taxi that charges a fixed fee plus a price per mile is not directly proportional to distance, because doubling the distance does not double the cost.

Key ideas

  • Direct proportion: y = kx, so y ÷ x is constant; doubling x doubles y.
  • Inverse proportion: y = k ÷ x, so x × y is constant; doubling x halves y.
  • To solve: use a known pair to find k, write the equation, then substitute.
  • A graph that does not go through the origin is not direct proportion.
Worked example 16.7.1
Is y directly proportional to x? x: 2, 3, 5 and y: 6, 9, 15.

Method. Test for directly proportional: check that y ÷ x is the same for every pair.

  1. Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).
    6 ÷ 2 = 3
    9 ÷ 3 = 3
    15 ÷ 5 = 3
  2. Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.
    The answers are all the same.
  3. State the conclusion.
    Yes, directly proportional (k = 3)

Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.

Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.

Worked example 16.7.2
Is y directly proportional to x? x: 1, 2, 3 and y: 3, 5, 7.

Method. Test for directly proportional: check that y ÷ x is the same for every pair.

  1. Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).
    3 ÷ 1 = 3
    5 ÷ 2 = 2.5
    7 ÷ 3 = 73
  2. Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.
    The answers are not all the same.
  3. State the conclusion.
    No, not directly proportional

Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.

Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.

Worked example 16.7.3
A recipe for 4 people uses 300 g of flour. How much flour is needed for 10 people?

Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.

  1. Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.
    4 people goes with 300 g
  2. Find the flour for one person. Dividing by 4 gives the amount for 1.
    300 g ÷ 4 = 75 g
  3. Multiply by 10. We want 10 lots of one person.
    75 g × 10 = 750 g

Check. Compare with the original: 10 people is 2.5 times 4 people, so the flour is 2.5 times 300 g = 750 g. ✓

Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.

Worked example 16.7.4
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.

Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.

Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.

  1. Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).
    y = kx
  2. Use the known pair to find k. Put x = 4 and y = 20 into the equation, then solve for k.
    20 = k × 4
    k = 20 ÷ 4 = 5
  3. Write the equation with the value of k. Now the equation links every pair of values.
    y = 5x
  4. Substitute x = 7. We want y when x is 7.
    y = 5 × 7 = 35
  5. Write the answer.
    y = 35

Check. For the new pair, 35 ÷ 7 = 5, the same constant k. ✓

Watch out. In direct proportion the ratio y ÷ x is always the same number, k.

Worked example 16.7.5
The number of days d a job takes is inversely proportional to the number of workers n. With 6 workers the job takes 10 days. How many days does it take with 4 workers?

Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.

Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.

  1. Write the proportion as an equation with a constant k. "d is inversely proportional to n" means d ∝ 1n. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).
    d = kn
  2. Use the known pair to find k. Put n = 6 and d = 10 into the equation, then solve for k.
    10 = k6
    k = 10 × 6 = 60
  3. Write the equation with the value of k. Now the equation links every pair of values.
    d = 60n
  4. Substitute n = 4. We want d when n is 4.
    d = 60 ÷ 4 = 15
  5. Write the answer.
    d = 15 days

Check. For the new pair, 4 × 15 = 60, the same constant k = 60. ✓

Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.

Worked example 16.7.6
p is inversely proportional to q. When q = 3, p = 8. (a) Find a formula for p in terms of q. (b) Find q when p = 2.

Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.

Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.

  1. Write the proportion as an equation with a constant k. "p is inversely proportional to q" means p ∝ 1q. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).
    p = kq
  2. Use the known pair to find k. Put q = 3 and p = 8 into the equation, then solve for k.
    8 = k3
    k = 8 × 3 = 24
  3. Write the equation with the value of k. Now the equation links every pair of values.
    p = 24q
  4. Substitute p = 2 and solve for q. We know p and want q, so rearrange the equation.
    2 = 24 ÷ q
    q = 24 ÷ 2 = 12
  5. Write the answer.
    q = 12

Check. For the new pair, 12 × 2 = 24, the same constant k = 24. ✓

Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.

Watch out

A table can go up steadily and still not show proportion. Check the ratio (or product) in every column. And do not confuse the two types: in direct proportion you divide (y ÷ x) to find k; in inverse proportion you multiply (x × y).

Exercise 16I  Direct proportion

Calculator allowed
1
FIn each table, decide whether y is directly proportional to x. If it is, write down the value of k.
(a)x: 2, 3, 5 and y: 6, 9, 15
(b)x: 1, 2, 3 and y: 4, 8, 11
(c)x: 4, 6, 10 and y: 10, 15, 25
(d)x: 2, 4, 6 and y: 5, 8, 11
2
Fy is directly proportional to x. When x = 2, y = 6. Find y when:
(a)x = 5
(b)x = 8
(c)x = 10
3
FA recipe for 4 people uses 300 g of flour. How much flour is needed for 10 people?
4
FIn each part y is directly proportional to x. Find the missing value.
(a)When x = 3, y = 12. Find y when x = 10.
(b)When x = 5, y = 35. Find y when x = 8.
(c)When x = 8, y = 20. Find x when y = 35.
(d)When x = 6, y = 9. Find y when x = 14.
5
PSThe extension e cm of a spring is directly proportional to the force F newtons pulling it. A force of 12 N gives an extension of 3 cm. Find (a) the extension for a force of 20 N, (b) the force needed for an extension of 7.5 cm.
(a)Extension for 20 N
(b)Force for 7.5 cm
6
PSThe cost C pounds of n tickets is given by C = 12.5n. (a) What does 12.5 represent? (b) Find the cost of 8 tickets. (c) How many tickets can be bought for £200?
(a)Meaning of 12.5
(b)Cost of 8
(c)Tickets for £200
7
PSPay is directly proportional to the hours worked. For 15 hours Lena is paid £127.50. Work out (a) her pay for 22 hours, (b) the number of hours she works for pay of £204.
(a)Pay for 22 hours
(b)Hours for £204
8
PSA map is drawn so that 2 cm on the map represents 5 km on the ground. Find (a) the real distance for 7.5 cm on the map, (b) the length on the map for a real distance of 32.5 km.
(a)Real distance
(b)Map length
9
RA taxi company charges £3 plus £2 for every mile. Pat says “The cost is directly proportional to the distance.” Show that Pat is wrong.
10
RPat says “y is proportional to x because y goes up when x goes up.” The table shows x: 1, 2, 3 and y: 3, 5, 7. Is Pat correct?
11
PSPrinting 250 leaflets costs £17.50. Work out the cost of printing 600 leaflets at the same rate.
12
Ry is directly proportional to x. Show that if x is increased by 20%, then y is also increased by 20%.
13
Fy is directly proportional to x. When x = 5, y = 3. (a) Write y in terms of x. (b) Find x when y = 2.4.
(a)Equation
(b)Find x
14
PSA machine fills 360 bottles in 45 minutes. At the same rate, how many hours will it take to fill 2400 bottles?

Exercise 16J  Inverse proportion

Calculator allowed
1
FIn each table, decide whether y is inversely proportional to x. If it is, write down k.
(a)x: 1, 2, 3, 4 and y: 24, 12, 8, 6
(b)x: 2, 4, 5 and y: 10, 5, 3
(c)x: 2, 5, 10 and y: 15, 6, 3
(d)x: 1, 2, 3 and y: 10, 8, 6
2
FThe number of days d a job takes is inversely proportional to the number of workers n. With 6 workers the job takes 10 days. How many days does it take with:
(a)4 workers
(b)12 workers
(c)15 workers
(d)5 workers
3
FA journey of 180 km takes 3 hours at 60 km/h. For a fixed distance, time t hours is inversely proportional to speed v km/h. Find the time at:
(a)90 km/h
(b)45 km/h
(c)120 km/h
4
Fy is inversely proportional to x and y = 24 when x = 1. Find the missing values.
(a)Find y when x = 2
(b)Find y when x = 3
(c)Find y when x = 4
(d)Find y when x = 8
5
RSay whether each pair of quantities is in direct proportion, inverse proportion, or neither. Give a reason.
(a)The number of taps filling a bath (all the same) and the time to fill it
(b)A child's age and the child's height
(c)The time for a 100 km journey and the speed
(d)The number of people sharing £60 and the amount each gets
(e)The number of books bought and the total cost at £4.50 each
6
PSA bag of 60 sweets is shared equally between n children, so each child gets s sweets, where s is inversely proportional to n. When n = 10, s = 6. Find (a) s when n = 12, (b) n when s = 4.
(a)Sweets each for 12 children
(b)Children when each gets 4
7
Ry is inversely proportional to x. Explain what happens to y when x is multiplied by 4.
8
PSThe volume V litres of a gas is inversely proportional to the pressure P kPa. When P = 150, V = 12. Find V when P = 200.
9
Fy is inversely proportional to x. When x = 8, y = 5. Find the value of y when x = 20.
10
Fy is proportional to 1/x. When x = 0.5, y = 40. Find x when y = 16.
11
PSThe force F newtons between two magnets is inversely proportional to the distance d cm between them. When d = 4, F = 30. (a) Write a formula for F in terms of d. (b) Find F when d = 6. (c) Find d when F = 12.
(a)Formula
(b)F when d = 6
(c)d when F = 12
12
R10 workers can build a wall in 12 days. After 4 days, 2 of the workers leave. How many more days will it take to finish the wall?

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