Model direct and inverse proportion
Build proportional models from known values, choose among linear and power relationships, and check constants and units.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
If y is directly proportional to x, what happens to y when x doubles?
What you will learn
- Recognise direct and inverse proportion
- Find the constant of proportionality
- Model square, square-root and cube variation for Extended
- Check units and a proposed relationship
Use a constant to describe variation
Direct proportion y ∝ x means y = kx; doubling x doubles y. Inverse proportion y ∝ 1/x means y = k/x; doubling x halves y.
Other Extended models include y ∝ x², y ∝ √x and y ∝ x³. Write the equation with k, substitute a known pair to find k, then apply the model.
Check what stays constant
For direct proportion y/x = k; for inverse proportion xy = k. For a power model, compare y with the stated power or root of x.
The syllabus lists C2.8 as Extended content only; E2.8 covers proportion. A single matching pair does not prove a model unless the question or data supports it.
Pause and explain
What stays constant when y is inversely proportional to x?
Worked example
y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.
Show the worked solution
- Write y = kx².
- 45 = 9k, so k = 5.
- At x = 5, y = 5 × 25 = 125.
Answer y = 125.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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 is directly proportional to x. When x = 4, y = 18. Find the formula and y when x = 10.
Give me a hint
Use y = kx and solve for k from the known pair.
Compare my reasoning
- 18 = 4k, so k = 4.5.
- The equation is y = 4.5x.
- At x = 10, y = 45.
y = 4.5x; y = 45.
Look for these in your work
- Direct model chosen
- Constant found
- New value calculated
z is inversely proportional to √x. When x = 25, z = 6. Find z when x = 100.
Give me a hint
Use z = k/√x, then substitute the second x value.
Compare my reasoning
- 6 = k/5, so k = 30.
- z = 30/√x.
- For x = 100, z = 30/10 = 3.
z = 3.
Look for these in your work
- Root model written correctly
- Constant calculated
- Inverse scaling applied
A spring extends 12 mm under 8 N while within its elastic range. If extension is directly proportional to force, predict the extension at 14 N and state a limitation.
Give me a hint
Find k with units, then apply it while respecting the stated range.
Compare my reasoning
- 12 = 8k, so k = 1.5 mm/N.
- At 14 N, extension = 1.5 × 14 = 21 mm.
- Beyond the elastic limit, the spring may no longer follow this proportional model.
21 mm; model may fail beyond the elastic limit.
Look for these in your work
- Proportional model used
- Units on k retained
- Physical limitation stated
Common mistakes
- Assuming direct proportion means y = x
- Using a direct model for inverse variation
- Forgetting the power or root
- Ignoring units in k
What can you explain now?
If y ∝ 1/x and xy = 24, find y when x = 6.
Compare with the explanation
4
The model is y = 24/x.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How can you identify direct or inverse variation, find k, and check whether a model fits its context?
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.