Direct and Inverse Proportion With Non-Linear Powers
Two quantities are in direct proportion when one is a constant multiple of the other, and in inverse proportion when one is a constant multiple of the reciprocal of the other. At IGCSE the powers are not restricted to 1: y may be proportional to the square of x, to the cube of x, to the square root of x, or inversely proportional to the square of x. Every one of these is handled the same way. Write the relationship with a constant k, substitute the given pair of values to find k, rewrite the formula with k known, and then use it to answer whatever the question asks. The shape of the graph follows from the power, and an inverse square relationship falls away much faster than an inverse linear one.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Translate the sentence into an equation with k. Directly proportional to the square of x gives y = kx squared; inversely proportional to the cube of x gives y = k divided by x cubed; proportional to the square root of x gives y = k times the square root of x.
- Substitute the given pair of values and solve for k. Keep k exact, as a fraction if necessary, rather than rounding it.
- Rewrite the formula with the value of k in it, and state it clearly, because this is usually a mark in its own right.
- Use the completed formula for the question asked, substituting the new value and solving. If y is given and x is wanted, rearrange before substituting.
- For a proportional change question, work with the multiplier rather than the numbers: if y is proportional to x squared and x is trebled, y is multiplied by 9; if y is inversely proportional to x squared and x is doubled, y is divided by 4.
- Check the direction of the answer: in direct proportion, increasing one increases the other; in inverse proportion, increasing one decreases the other. An answer moving the wrong way means the relationship was written the wrong way round.
Worked example
y is directly proportional to the square of x. When x = 3, y = 18. (a) Find a formula for y in terms of x. (b) Find y when x = 5. (c) Find x when y = 200.
- Write the relationship with a constant: y = kx squared.
- Substitute the known pair: 18 = k times 3 squared = 9k, so k = 2.
- State the formula: y = 2x squared.
- For part (b), substitute x = 5: y = 2 times 25 = 50.
- For part (c), substitute y = 200: 200 = 2x squared, so x squared = 100 and x = 10, taking the positive root since x is a quantity here.
- Sense check: x has doubled from 5 to 10 and y has quadrupled from 50 to 200, which is what a square relationship predicts.
Practice questions
Try each question, then tap to reveal the answer.
Q1Write an equation for: p is inversely proportional to the square of q.Show answer
Answer: p = k divided by q squared.
Q2y is proportional to x cubed and y = 24 when x = 2. Find k.Show answer
Answer: 24 = k times 8, so k = 3.
Q3y is inversely proportional to x. If x is multiplied by 5, what happens to y?Show answer
Answer: y is divided by 5.
Q4y is proportional to the square of x. If x is halved, what happens to y?Show answer
Answer: y is divided by 4, since a half squared is a quarter.
Q5y is inversely proportional to the square of x, and y = 4 when x = 2. Find y when x = 4.Show answer
Answer: k = 4 times 4 = 16, so y = 16/16 = 1.
Q6Why should k be kept exact rather than rounded?Show answer
Answer: Every later answer is calculated from k, so rounding it early carries an error into all of them.
Q7y is proportional to the square root of x, and y = 12 when x = 9. Find y when x = 25.Show answer
Answer: 12 = k times 3, so k = 4, and y = 4 times 5 = 20.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
The force F between two magnets is inversely proportional to the square of the distance d between them. When d = 4 cm, F = 12 newtons. (a) Find a formula for F in terms of d. (b) Find F when d = 6 cm. (c) Find d when F = 48 newtons.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 7 available
y is directly proportional to the cube of x. Without finding k, explain what happens to y when x is multiplied by 4, and justify your answer.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 3 available
Free printable worksheet
Want more practice on paper? Download the direct and inverse proportion with non-linear powers worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Next topics
Not quite what you needed?
Tell us what is missing on direct and inverse proportion with non-linear powers, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.