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Cambridge IGCSE0580

Proportion

Mathematics 0580 Chapter Notes

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Proportion
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Proportion notes

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In this chapter

  1. 1Direct proportion
  2. 2Inverse proportion
  3. 3Square and square root proportion
  4. 4Cube and cube root proportion
  5. 5Finding an equation and an unknown
  6. 6Describing the effect of a change

1. Direct proportion

If y is directly proportional to x, then y is a fixed multiple of x. Double x and y doubles. The symbol ∝ means "is proportional to".

To find an unknown quantity, turn the proportion into an equation with a constant of proportionality k. Do this in the first line of working.

y ∝ x

y = kx

Examiner insight

“Some candidates chose to keep the proportion symbol in all the lines of their working which, if their answer had been incorrect, would not have scored any partial marks.” “It was also common to see attempts without using a constant of proportionality and so failing to score.”

Worked example 13 marks

y is directly proportional to x. y = 12 when x = 4. Find y when x = 7.

  1. 1

    y = kx [1]

  2. 2

    12 = 4k, so k = 3 [1]

  3. 3

    y = 3 × 7 = 21 [1]

2. Inverse proportion

If y is inversely proportional to x, then y is a fixed number divided by x. Double x and y halves.

The equation has the constant k divided by x, so the product xy always equals k.

y ∝ 1/x

y = k/x

Examiner insight

“A small number of candidates took x as directly, rather than inversely, proportional to y, and there was the occasional arithmetic slip, but very few errors were seen.” “Many demonstrated a lack of understanding of inverse proportionality by saying that it would also halve or that it would stay the same.”

Worked example 13 marks

y is inversely proportional to x. y = 6 when x = 4. Find y when x = 8.

  1. 1

    y = k/x [1]

  2. 2

    6 = k/4, so k = 24 [1]

  3. 3

    y = 24/8 = 3 [1]

3. Square and square root proportion

Read the wording closely. "The square of x" means x². "The square root of x" means √x.

Put the power or root on x, not on k. For an inverse relationship, k is divided by that power or root.

y = kx²

y = k√x

y = k/x²

y = k/(√x)

Examiner insight

“Those who did not gain any marks were usually unable to set up a correct proportionality relationship or the one they set up used inverse proportion, omitted the square root or used the square rather than square root.” “Some read ‘square’ as ‘square root’.”

Worked example 13 marks

y is inversely proportional to the square of x. y = 5 when x = 4. Find y when x = 10.

  1. 1

    y = k/x² [1]

  2. 2

    5 = k/16, so k = 80 [1]

  3. 3

    y = 80/100 = 0.8 [1]

4. Cube and cube root proportion

"The cube of x" means x³. "The cube root of x" means √[3]x.

The method is the same as for squares. Choose the power or root the question names, then multiply or divide by k.

y = kx³

y = k√[3]x

y = k/x³

y = k/(√[3]x)

Examiner insight

“Many omitted the cube root, used the cube, square or square root or used inverse proportion.” “Those who did not gain any marks were usually considering the wrong type of relationship, generally direct proportion or not taking the cube into account.”

Worked example 13 marks

y is directly proportional to the cube root of x. y = 6 when x = 8. Find y when x = 27.

  1. 1

    y = k√[3]x [1]

  2. 2

    6 = k × 2, so k = 3 [1]

  3. 3

    y = 3 × 3 = 9 [1]

5. Finding an equation and an unknown

Use the same four steps every time. Write the proportion. Write the equation with k. Substitute the given pair of values to find k. Then put the value of k back into the equation and use it.

If the question asks for y in terms of x, the answer is the equation with k replaced by its value. In "the square of (x + 1)", the whole bracket is squared, not x alone.

Examiner insight

“Candidates should be aware that they may be asked to give an equation as a final answer as in part (a), rather than a value of y, which they may be more familiar with.” “In order to gain full credit candidates had to find the value of k and then write the expression using this value of k which many did not do.”

Worked example 14 marks

y is inversely proportional to the square of (x + 1). y = 2 when x = 3.

(a) Find y in terms of x.

(b) Find y when x = 7.

  1. 1

    y = k/((x+1)²) [1]

  2. 2

    2 = k/4², so k = 32 [1]

  3. 3

    (a) y = 32/((x+1)²) [1]

  4. 4

    (b) y = 32/8² = 32/64 = 1/2 [1]

6. Describing the effect of a change

Say what happens as a factor: the quantity is multiplied by or divided by a number. Do not say only that it increases or decreases.

Put the change into the equation. If y = k√x and x becomes 4x, then y becomes k√4x = 2k√x. So y is multiplied by 2, not by 4.

Examiner insight

“Many said that F increases but did not give a factor.” “A few did not gain the mark because they said ‘it increases by 4’ which suggests an addition.”

Worked example 12 marks

m is inversely proportional to the square of t. t is multiplied by 3. Describe the effect on m.

  1. 1

    m = k/t², so the new m = k/((3t)²) = k/9t² [1]

  2. 2

    m is divided by 9 [1]

Recap

  • Extended: y ∝ x means y = kx. y ∝ 1/x means y = k/x.
  • Extended: put the square, square root, cube or cube root on the quantity named, such as x or (x + 1), not on k.
  • Extended: write the proportion, write the equation with k, find k, then substitute again.
  • Extended: if asked for y in terms of x, give the equation with the value of k in it.
  • Extended: describe a change in y as a factor: multiplied by or divided by a number.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. y is directly proportional to x. Write an equation connecting y, x and a constant k.1 mark
  2. y is directly proportional to x. y = 15 when x = 5. Find y when x = 8.3 marks
  3. Do not use a calculator. y is inversely proportional to x. y = 6 when x = 4. Find x when y = 3.3 marks
  4. Do not use a calculator. y is directly proportional to the square of x. y = 50 when x = 5. Find y when x = 3.3 marks
  5. Do not use a calculator. y is directly proportional to the square root of x. y = 12 when x = 9. Find y when x = 25.3 marks
  6. m is inversely proportional to the square of (t + 2). m = 4 when t = 3. Calculate m when t = 5. Give your answer correct to 3 significant figures.3 marks
  7. y is directly proportional to the cube of x. y = 54 when x = 3. Find y when x = 2.3 marks
  8. Do not use a calculator. y is inversely proportional to the cube root of x. y = 5 when x = 8. (a) Find y in terms of x. (b) Find y when x = 125.4 marks
  9. Do not use a calculator. The time, T seconds, for one swing of a pendulum is directly proportional to the square root of its length, L cm. T = 1.2 when L = 36. (a) Find a formula for T in terms of L. (b) Find L when T = 2. (c) The length is changed so that T is halved. The new length is pL. Find the value of p.5 marks
  10. In a model of a storm, the force on a wall, F newtons, is directly proportional to the square of the wind speed, v metres per second. The wind speed is inversely proportional to the square root of the distance, d km, from the centre of the storm. F = 450 when v = 15. v = 15 when d = 16. (a) Find a formula for F in terms of v. (b) Find a formula for v in terms of d. (c) Find a formula for F in terms of d. Give your answer in its simplest form. (d) Calculate d when F = 800.7 marks

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