Direct and inverse proportion as equations
Use a constant of proportionality to model linear, power and root relationships.
Extended only.
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
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.
After trying it yourself, choose your next review. This is your self-assessment.
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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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources