Direct Proportion
Two quantities are in direct proportion when one is always a constant multiple of the other. As one quantity is multiplied by a factor, the other is multiplied by the same factor.
Foundation & Higher. Ratio calculations and interpreting tables, graphs and given simple equations are shared. Board guidance: constructing an equation such as y = kx from proportional data is Higher for AQA and Edexcel, but is also on the OCR Foundation route. Forming power and root models is Higher. Useful links: Rearranging Formulae and Linear Graphs.
Recognising direct proportion
If y is directly proportional to x, the ratio y/x is constant for x ≠ 0. For example:
| x | 2 | 4 | 6 | 10 |
|---|---|---|---|---|
| y | 6 | 12 | 18 | 30 |
| y/x | 3 | 3 | 3 | 3 |
The constant of proportionality is 3.
Writing y = kx
Direct proportion between y and x is written y ∝ x. This means y = kx for some constant k.
If y = 20 when x = 5, then 20 = 5k, so k = 4. Therefore y = 4x.
Worked example — find the equation, then use it
y is directly proportional to x. When x = 8, y = 28.
Start with y = kx. Substitute the known pair: 28 = 8k, so k = 3.5.
The model is y = 3.5x. If x = 14, then y = 3.5 × 14 = 49.
If instead y = 70 and x is unknown, solve 70 = 3.5x to get x = 20.
Direct proportion graphs
A graph of y = kx is a straight line through the origin. Its gradient is k.
A straight line that does not pass through the origin, such as y = 3x + 5, is not direct proportion because the fixed charge breaks the constant ratio.
In the diagram, the blue solid line is y = 3x; the red dashed line is y = 3x + 5. On the blue line, moving from (2,6) to (4,12) gives gradient (12−6)/(4−2) = 3. Both lines have gradient 3, but only the blue line has zero intercept.
Exam reasoning — proportional or just linear?
| x | y = 3x | y = 3x + 5 |
|---|---|---|
| 0 | 0 | 5 |
| 2 | 6 | 11 |
| 4 | 12 | 17 |
For y = 3x, the ratio y/x is 3 whenever x ≠ 0, and the graph passes through (0,0). At the origin, y/x is undefined; use the equation y = 3x there. For y = 3x + 5, the ratios change and the graph starts at y = 5.
This distinction matters in contexts such as taxi fares: a fixed starting charge creates a linear relationship, but not direct proportion.
Using the model
Once the equation is known, substitute either variable to find the other. If y = 4x and y = 52, then x = 52 ÷ 4 = 13.
Higher — powers and roots
Higher proportional models may involve powers or roots. If y ∝ x², then y = kx². If y ∝ √x, then y = k√x.
The scaling effect follows the power. If y ∝ x² and x doubles, y is multiplied by 2² = 4.
Higher worked example
y ∝ x² and y = 45 when x = 3. Then 45 = 9k, so k = 5. The model is y = 5x². When x = 4, y = 5 × 16 = 80.
Higher worked example — a root model
Suppose y ∝ √x and y = 10 when x = 4. Write y = k√x, so 10 = 2k and k = 5. Thus y = 5√x. If y = 30, then √x = 6 and x = 36. If x is multiplied by 9, y is multiplied by √9 = 3.
For a power model, reverse the power only after isolating it. If y = 5x² and y = 80, then x² = 16. Algebraically x = ±4; if x represents a positive length, x = 4. For y ∝ x³, doubling x multiplies y by 8. The same constant-ratio idea applies to y/x², y/x³ or y/√x as appropriate.
Think Like a Mathematician
A table may look roughly linear without being proportional. The quickest checks are whether y/x stays constant and whether the graph passes through the origin.
Common Mistakes
Confusing linear with proportional: y = 2x + 7 is linear but not direct proportion.
Using x/y instead of y/x inconsistently: choose one ratio and keep the order fixed.
Higher: if y ∝ x², doubling x does not merely double y.
Practice — direct proportion
1. y is directly proportional to x. If y=18 when x=6, find k.
k = 18 ÷ 6 = 3.
2. Using the same relationship, find y when x=11.
y = 3x, so y = 33.
3. y ∝ x and y=35 when x=7. Find x when y=60.
k=5, so 60=5x and x=12.
4. Which is direct proportion: y=4x, y=4x+2, or y=x²?
For direct proportion to x, y=4x.
5. A direct-proportion graph has gradient 2.5. Write its equation.
y = 2.5x.
6. The pairs (2,8), (5,20), (7,28) are given. Show they are directly proportional.
y/x = 4 for every pair, so y = 4x.
7. Higher: y ∝ x² and y=32 when x=4. Find y when x=7.
32=16k, so k=2. y=2×49=98.
8. Higher: y ∝ √x and y=12 when x=9. Find y when x=25.
12=3k so k=4. y=4√25=20.
9. Higher: z ∝ t³. If t is multiplied by 3, by what factor is z multiplied?
3³ = 27.
10. Explain why a taxi fare y=2x+4 is not directly proportional to distance x.
The fixed £4 means y/x is not constant and the graph does not pass through the origin.
Frequently Asked Questions
Can I test y/x at the origin?
No: 0/0 is undefined. Use non-zero x-values for the ratio test and check that the model y = kx gives y = 0 when x = 0.
Are all proportional equations Higher-only?
No. Ratio calculations and interpreting given simple equations are shared. Constructing the equation from proportional data is Higher for AQA and Edexcel; OCR also includes simple y = kx and y = k/x construction at Foundation. Forming models involving powers or roots is Higher.