Direct and Inverse Proportion
Two quantities are in direct proportion when one is a constant multiple of the other, written y is proportional to x, which becomes the equation y = kx where k is the constant of proportionality. They are in inverse proportion when one is a constant multiple of the reciprocal of the other, written y is proportional to 1/x, giving y = k/x. Level 2 Further Maths extends this beyond the linear case to powers and roots: y proportional to x squared gives y = kx^2, y proportional to the square root of x gives y = k times root x, and y inversely proportional to x squared gives y = k/x^2. The graph shows which model applies: direct proportion is a straight line through the origin, y = kx^2 is a curve through the origin that steepens, and inverse proportion is a reciprocal curve with the axes as asymptotes. Every question follows the same three moves: write the proportionality statement as an equation with k, substitute the given pair of values to find k, then use the completed equation to answer whatever is asked.
Before you start
Make sure you're comfortable with these topics first:
Method
- Turn the words into a proportionality statement. Directly proportional to the square means y is proportional to x^2; inversely proportional to the square root means y is proportional to 1 over root x.
- Write it as an equation with an unknown constant: y = kx^2, or y = k / root x. Never work with the proportionality symbol beyond this line.
- Substitute the pair of values you are given and solve for k. Keep k exact as a fraction or a surd rather than rounding it, because rounding here corrupts every later answer.
- Rewrite the full equation with k replaced by its value. This equation is the answer to any part that asks for a formula.
- Substitute the new value to find what is asked. If you are given y and need x, rearrange first, and remember that a square root produces two possible values unless the context rules one out.
- For proportional-change questions you do not need k at all: if y = kx^2 and x is multiplied by 3, y is multiplied by 3^2 = 9. Reason with the multiplier directly.
Worked example
y is inversely proportional to the square of x. When x = 4, y = 5. (a) Find an equation connecting y and x. (b) Find y when x = 10. (c) Find the positive value of x when y = 45. (d) State what happens to y when x is halved.
- Write the proportionality as an equation: y is proportional to 1/x^2, so y = k / x^2.
- Substitute the known pair x = 4, y = 5: 5 = k / 4^2 = k / 16.
- Solve for k: k = 5 x 16 = 80. The equation is y = 80 / x^2.
- For part (b), substitute x = 10: y = 80 / 100 = 0.8.
- For part (c), substitute y = 45: 45 = 80 / x^2, so x^2 = 80 / 45 = 16/9, giving x = 4/3 as the positive value.
- For part (d), reason with multipliers. Halving x replaces x^2 by (x/2)^2 = x^2/4, and dividing by a quarter multiplies y by 4. So y becomes four times larger.
Practice questions
Try each question, then tap to reveal the answer.
Q1y is directly proportional to x. When x = 6, y = 21. Find y when x = 10.Show answer
Answer: y = kx, so 21 = 6k and k = 3.5. When x = 10, y = 35.
Q2P is proportional to the cube of q. When q = 2, P = 40. Find the equation connecting P and q.Show answer
Answer: P = kq^3, so 40 = 8k and k = 5. The equation is P = 5q^3.
Q3T is inversely proportional to n. When n = 8, T = 3. Find n when T = 12.Show answer
Answer: T = k/n, so 3 = k/8 and k = 24. When T = 12, 12 = 24/n, so n = 2.
Q4y is proportional to the square root of x. When x = 9, y = 12. Find y when x = 25.Show answer
Answer: y = k root x, so 12 = 3k and k = 4. When x = 25, y = 4 x 5 = 20.
Q5y is proportional to x squared. If x is multiplied by 4, what happens to y?Show answer
Answer: y is multiplied by 4^2 = 16.
Q6y is inversely proportional to x. If x is increased by 25 per cent, find the percentage change in y.Show answer
Answer: x is multiplied by 1.25, so y is multiplied by 1/1.25 = 0.8. y decreases by 20 per cent.
Q7Explain how the graph of y against x differs between y = kx and y = k/x.Show answer
Answer: y = kx is a straight line passing through the origin with gradient k. y = k/x is a reciprocal curve in two separate parts that never touches either axis, since the axes are asymptotes.
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
The force F newtons between two magnets is inversely proportional to the square of the distance d cm between them. When d = 3, F = 20. (a) Find a formula for F in terms of d. (b) Calculate F when d = 12. (c) Calculate d when F = 5. (d) The distance is increased from 3 cm to 6 cm. State the effect on the force, justifying your answer without further calculation.
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A student claims that because y is proportional to the square root of x, doubling x will double y. (a) Explain why the student is wrong. (b) Determine the exact factor by which y is multiplied when x is doubled. (c) y = 30 when x = 16. Find the value of x for which y = 45.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
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