Mathematics 4024 · Syllabus tracker
Proportionality
Direct and inverse proportion, including squares, square roots, cubes and cube roots, written with ∝ and k.
How to use this
Read each "I can…" statement and rate yourself honestly. Then try its quick check. If you get a check wrong on something you rated Got it, we'll suggest moving it to Shaky. My progress compares your ratings with your past-paper results to find topics you think you know but keep losing marks on.
Loading the tracker…
Quick revision
Proportionality on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Direct and inverse proportion with ∝ and k, including squares, cubes and roots.
1Direct proportion
- y ∝ x means y = kx: when x doubles, y doubles. The graph is a straight line through the origin.
- Other direct forms: y ∝ x2 (y = kx2), y ∝ x3, y ∝ √x, y ∝ ∛x.
- Find k from a given pair of values, then use the formula.
Method for every proportion question
- Write the formula with k.
- Substitute the pair you are given to find k.
- Write the formula again with the number for k.
- Substitute the new value.
Formulas
Worked example
y is directly proportional to the square of x. y = 18 when x = 3. Find y when x = 5.
- y = kx2.
- 18 = k × 9, so k = 2.
- y = 2x2.
- y = 2 × 25.
y = 50
2Inverse proportion
- y ∝ 1/x means y = k/x: when x doubles, y halves. xy is always the same.
- Other inverse forms: y ∝ 1/x2, y ∝ 1/√x.
- The graph of y = k/x is a curve that gets close to both axes.
Formulas
Worked example
The force F is inversely proportional to the square of the distance d. F = 20 when d = 3. Find F when d = 6.
- F = k/d2.
- 20 = k/9, so k = 180.
- F = 180/36.
F = 5
3The effect of a change
- If y ∝ xn and x is multiplied by m, then y is multiplied by mn.
- Inverse: if y ∝ 1/xn and x is multiplied by m, y is divided by mn.
- Percentage changes work the same way: x up 10% means x × 1.1.
| Relationship | x doubled → y | x tripled → y |
|---|---|---|
| y ∝ x | × 2 | × 3 |
| y ∝ x2 | × 4 | × 9 |
| y ∝ x3 | × 8 | × 27 |
| y ∝ 1/x | ÷ 2 | ÷ 3 |
| y ∝ 1/x2 | ÷ 4 | ÷ 9 |
Worked example
y ∝ x3. x is increased by 10%. By what percentage does y increase?
- x is multiplied by 1.1.
- y is multiplied by 1.13 = 1.331.
33.1%
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Using y = kx for 'proportional to the square' | Square it: y = kx2 |
| Inverse proportion written y = x/k | It is y = k/x |
| Not squaring when substituting into y = kx2 | 18 = k × 32, not 18 = k × 3 |
| Forgetting to write the formula with k | Write y = 2x2 — it is often worth a mark |
★In the exam
- Read the wording carefully: 'square', 'square root', 'cube', 'inversely'.
- Always write the formula with your value of k in it before the final step.
- Check: a larger x should make y bigger (direct) or smaller (inverse).