Inverse Proportion
In inverse proportion, increasing one quantity causes the other to change so that a fixed product is maintained. A decrease on its own is not enough to prove inverse proportion.
Foundation & Higher. Constant-product calculations and interpreting given simple equations and reciprocal graphs are shared. Board guidance: constructing y = k/x from proportional data is Higher for AQA and Edexcel, but is also on the OCR Foundation route. Forming inverse-power and root models is Higher. Useful links: Functions and Rearranging Formulae.
Recognising inverse proportion
For y inversely proportional to x, the product xy is constant. Example:
| x | 2 | 3 | 4 | 6 |
|---|---|---|---|---|
| y | 12 | 8 | 6 | 4 |
| xy | 24 | 24 | 24 | 24 |
So xy = 24 and y = 24/x.
Worked example — use the constant product
y is inversely proportional to x. When x = 5, y = 18.
The constant product is k = xy = 5 × 18 = 90, so y = 90/x.
When x = 12, y = 90 ÷ 12 = 7.5. If y = 15, then 15 = 90/x, so x = 6.
Writing y = k/x
y ∝ 1/x means y = k/x. If y=10 when x=6, then k=xy=60, so y=60/x.
Fixed-work and fixed-distance contexts
If a fixed job takes 12 hours with 4 equally productive workers, worker-hours = 48. With 6 equally productive workers it would take 48 ÷ 6 = 8 hours.
Always state assumptions such as equal productivity or constant total distance.
For a fixed 120 km journey, time = 120/speed. At 40 km/h the time is 3 h; at 60 km/h it is 2 h. The product speed × time remains 120 km. This assumes the speed is constant during each journey and ignores stops. The workers model also assumes the work can be shared without workers obstructing one another.
The reciprocal graph
For positive x and y, y=k/x gives a curved reciprocal graph. It approaches the axes but does not behave like a straight line through the origin.
Interpret the reciprocal graph
| x | 1 | 2 | 4 | 8 |
|---|---|---|---|---|
| y when k=16 | 16 | 8 | 4 | 2 |
As x grows through positive values, y approaches the horizontal axis without reaching it. As x approaches 0 from above, y grows without bound and the curve approaches the vertical axis. The equation y = 16/x is undefined at x = 0. For k > 0, allowing negative x gives a second branch with negative y; positive-quantity contexts use only the first branch.
Useful check: if doubling x does not divide y by 2 in a basic y = k/x model, the relationship is not inverse proportion.
Plot the table points and join them with a smooth curve, not straight line segments. Also, y = k/x can be written y = k × (1/x): y is directly proportional to the reciprocal 1/x. A graph of y against 1/x would be a straight line through the origin.
Direct, inverse or neither?
Direct proportion: y/x constant. Inverse proportion: xy constant. If neither test works, the relationship is neither.
Higher — inverse powers and roots
Higher relationships may take forms such as y ∝ 1/x² or y ∝ 1/√x.
If y ∝ 1/x² and x doubles, y is divided by 2² = 4.
Higher worked example
y ∝ 1/x² and y=12 when x=2. Then 12=k/4, so k=48. Therefore y=48/x². When x=4, y=48/16=3.
Higher worked example — inverse square root
If y ∝ 1/√x and y = 6 when x = 9, write y = k/√x. Then 6 = k/3, so k = 18 and y = 18/√x. When y = 3, √x = 6, giving x = 36. Here x must be positive.
If x is multiplied by 9 in this model, y is divided by √9 = 3. For inverse cubes, doubling x divides y by 2³ = 8. In a positive-quantity inverse-square model y = 48/x², y = 12 gives x² = 4 and x = 2; without a positive-domain restriction, x = −2 also satisfies the equation.
Think Like a Mathematician
Why is ‘one goes up while the other goes down’ not enough?
Many decreasing relationships are not inverse proportion. The defining feature is a constant product, or its corresponding inverse-power rule.
Common Mistakes
Assuming any decrease means inverse proportion. Check xy.
Using y=kx by habit: inverse proportion uses division.
Ignoring context: worker problems only fit the model if productivity assumptions are reasonable.
Practice — inverse proportion
1. y is inversely proportional to x. y=15 when x=4. Find k.
k=xy=60. k=60.
2. Using y=60/x, find y when x=10.
6.
3. Find x when y=7.5 in the same relationship.
7.5=60/x, so x=60÷7.5=8.
4. A job takes 9 hours with 8 equal workers. How long with 12 workers?
Total worker-hours = 72. 72÷12=6 hours.
5. Are x=2,y=18; x=3,y=12; x=6,y=6 inversely proportional?
Products: 36,36,36. Yes.
6. A relationship has y=20−x. Is it inverse proportion?
No. xy is not constant; it is simply a decreasing linear relationship.
7. Higher: y ∝ 1/x² and y=20 when x=2. Find y when x=5.
k=20×4=80. y=80/25=3.2.
8. Higher: y ∝ 1/√x and y=9 when x=4. Find y when x=9.
9=k/2 so k=18. y=18/3=6.
9. Higher: if y ∝ 1/x³ and x is multiplied by 2, what happens to y?
y is divided by 2³, so it becomes one eighth as large.
10. State one assumption when modelling job time as inversely proportional to number of workers.
For example: all workers are equally productive and can work independently without slowing each other down.
Frequently Asked Questions
Is every decreasing graph inverse proportion?
No. For y = k/x, check that xy is constant. A decreasing straight line such as y = 20 − x fails this test.
Can x be zero in y = k/x?
No: division by zero is undefined. In positive-quantity contexts, use x > 0; the permitted values depend on the model.