Direct and inverse proportion as equations
Use a constant of proportionality to model linear, power and root relationships.
Section Paper 4 · Extended · Calculator
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Substitute into powers and roots.
- Rearrange formulas and solve a simple equation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
If x triples, what happens to x²?
What you will learn
- Translate proportional statements into equations with k.
- Find k from a known pair and calculate new values.
- Distinguish direct, inverse, square, cube and root proportion.
Find the constant before predicting
If y is directly proportional to x, write y ∝ x and then y = kx. A known pair y = 18 at x = 6 gives k = 3, so y = 3x. The ratio y/x is constant when x ≠ 0. A graph passes through the origin for unrestricted direct linear proportion; a fixed added fee would break this relationship.
For inverse proportion y ∝ 1/x, write y = k/x with x ≠ 0. The product xy is constant. If y = 12 when x = 5, k = 60; at x = 10, y = 6. Doubling x halves y. Inverse proportion is more specific than simply saying one quantity decreases as another increases.
The power belongs to the stated variable
If y ∝ x², then y = kx². Doubling x multiplies y by four; for y ∝ x³ it multiplies y by eight. If y ∝ √x, then y = k√x for the non-negative real x used here; quadrupling x doubles y. Cube-root proportion y = k∛x responds to an eightfold input with a twofold output.
Inverse power relationships use y = k/x² or y = k/x³, with non-zero x. The scaling factor acts through the full power: tripling x in y = k/x² divides y by nine. To find x from a square relationship, solve for x² first and use the context to decide which sign is allowed.
Check the model and its units
A formula can mix a stated square or root relation with a numerical coefficient; derive that coefficient from the known data. For y = k√x with y = 15 at x = 9, k = 5. At x = 25, y = 25. Do not set k to the known y value unless the input factor actually equals one.
For fixed work and identical workers, completion time may be modelled as inversely proportional to worker count. The assumptions matter: unchanged total work, equal worker rates and no extra coordination delay. Check whether a proportional model is justified by the wording instead of applying a ratio because two quantities are present.
Pause and explain
Which quantity stays constant when y is inversely proportional to x?
Worked example
y is inversely proportional to x². When x = 2, y = 18. Find y when x = 6.
Show the worked solution
- Write y = k/x² and substitute the known pair: 18 = k/4, so k = 72.
- At x = 6, y = 72/36 = 2.
- The input tripled, so the output divides by nine; 18/9 = 2 checks the result.
Answer y = 2
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
y ∝ x³ and y = 16 when x = 2. Find y when x = 3.
Give me a hint
Write the equation with an unknown constant first.
Compare my reasoning
- 16 = k × 2³ gives k = 2.
- The model is y = 2x³.
- At x = 3, y = 2 × 27 = 54.
y = 54
Look for these in your work
- Used the stated cubic relationship.
- Found the constant from the known pair.
For y ∝ √x, y = 12 at x = 16. Find x when y = 21. Also find y at x = 27 if instead y ∝ ∛x and y = 8 at x = 8.
Give me a hint
Use a separate constant for each of the two independent models.
Compare my reasoning
- Square-root model: 12 = 4k gives k = 3; 21 = 3√x implies √x = 7 and x = 49.
- Cube-root model: 8 = 2k gives k = 4.
- At x = 27, its cube root is three, so y = 12.
First model: x = 49; second model: y = 12
Look for these in your work
- Kept the root type and models distinct.
- Reversed a square root by squaring.
Six identical workers complete fixed work in 15 days. Under an inverse-proportion model, how many days would ten identical workers need?
Give me a hint
Worker count multiplied by days stays constant in this stated model.
Compare my reasoning
- The constant is 6 × 15 = 90 worker-days.
- At ten workers, t = 90/10 = 9 days.
- The answer assumes equal rates, fixed work and no coordination losses; more workers reduce time.
9 days under the stated model
Look for these in your work
- Used a constant product rather than a constant ratio.
- Stated the model's assumptions.
Common mistakes
- Using a direct ratio for inverse proportion.
- Ignoring the square or root in the scaling factor.
- Assuming a model's physical conditions without checking them.
What can you explain now?
y ∝ x² and y = 20 at x = 2. Find y at x = 5.
Compare with the explanation
125
k = 20/4 = 5, then y = 5 × 25 = 125.
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Make it stick
For direct square, inverse square, square-root and cube-root proportion, explain how the output changes when the input is multiplied by four or eight.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.