Inverse proportion describes a relationship where one quantity increases as another decreases by the same factor. If you double x, y is halved; if you triple x, y is divided by three. GCSE questions ask you to form an equation using a constant k, find k from given values, and then use it to solve problems.
What does it mean for two quantities to be inversely proportional?
Two quantities x and y are inversely proportional if their product is always the same constant: x × y = k. As one goes up, the other must come down to keep the product fixed.
The proportionality symbol ∝ lets us write this concisely:
$$y \propto \frac{1}{x}$$
This is read as "y is inversely proportional to x." Converting to an equation replaces ∝ with "= k ÷":
$$y = \frac{k}{x}$$
where k is the constant of proportionality.
What types of inverse proportion appear at GCSE?
GCSE Higher extends beyond the simple 1/x relationship. The proportionality symbol can involve any power or root of x.
| Proportionality statement | Equation form |
|---|---|
| y ∝ 1/x | y = k/x |
| y ∝ 1/x² | y = k/x² |
| y ∝ 1/√x | y = k/√x |
| y ∝ 1/x³ | y = k/x³ |
The method for all of these is identical: write the equation form, substitute a known pair of values to find k, then use k to answer the question.
How do you find the constant of proportionality?
Worked example: y is inversely proportional to x². When x = 3, y = 8. Find y when x = 6.
Step 1: Write the equation. $$y = \frac{k}{x^2}$$
Step 2: Substitute the known pair (x = 3, y = 8) to find k. $$8 = \frac{k}{3^2} = \frac{k}{9}$$ $$k = 8 \times 9 = 72$$
Step 3: Write the completed equation. $$y = \frac{72}{x^2}$$
Step 4: Find y when x = 6. $$y = \frac{72}{6^2} = \frac{72}{36} = 2$$
Answer: y = 2 when x = 6. Notice that x doubled (from 3 to 6) and y went from 8 to 2 — it decreased by a factor of 4 = 2². That makes sense for y ∝ 1/x², where doubling x gives y/4.
How do you work backwards to find x?
Worked example (continued): Using y = 72/x², find x when y = 0.5.
$$0.5 = \frac{72}{x^2}$$ $$x^2 = \frac{72}{0.5} = 144$$ $$x = \sqrt{144} = 12$$
Answer: x = 12. Always take the positive square root unless negative values are meaningful in context.
What does an inverse proportion graph look like?
A graph of y = k/x (k > 0) is a hyperbola — a curve with two branches, one in the first quadrant (positive x, positive y) and one in the third quadrant (negative x, negative y).
Key features:
- The curve never crosses the x-axis (y can never reach 0 for finite x).
- The curve never crosses the y-axis (x = 0 is undefined).
- As x → ∞, y → 0: the curve has the x-axis as an asymptote.
- As x → 0⁺, y → ∞: the curve has the y-axis as an asymptote.
A graph of y = k/x² is steeper on the left and falls more quickly — it only exists in the first and second quadrants (since x² is always positive, y is always positive for k > 0).
How do you recognise inverse proportion in a table of values?
Check whether the product x × y remains constant throughout the table. If it does, y ∝ 1/x. If x² × y is constant, y ∝ 1/x².
| x | y | x × y | x² × y |
|---|---|---|---|
| 2 | 18 | 36 | 72 |
| 3 | 8 | 24 | 72 |
| 6 | 2 | 12 | 72 |
| 9 | 0.89 | 8 | 72 |
Here x × y is not constant, but x² × y = 72 throughout, confirming y ∝ 1/x² with k = 72.
Frequently asked questions
How is inverse proportion different from direct proportion?
Direct proportion (y ∝ x) means y increases as x increases. The graph is a straight line through the origin. Inverse proportion (y ∝ 1/x) means y decreases as x increases. The graph is a hyperbola, not a straight line, and it never passes through the origin. A quick test: in direct proportion, doubling x doubles y. In inverse proportion, doubling x halves y.
Can k be negative in inverse proportion?
Yes — k can be any non-zero constant. If k is negative, the hyperbola lies in the second and fourth quadrants (for y = k/x). In most GCSE exam contexts, k is positive because the quantities are physical measurements like speed, time, or current. Always check the context before assuming k is positive.
What if I'm given a ratio rather than individual values?
If a question tells you that y decreased by a factor of 4 when x doubled, you can find the power without being given k. Since x doubled (factor 2) and y was divided by 4 (factor 2²), the relationship is y ∝ 1/x². In general, if multiplying x by a factor f divides y by fⁿ, then y ∝ 1/xⁿ.
Does inverse proportion appear on Foundation tier?
Simple inverse proportion (y ∝ 1/x) can appear on Foundation Higher, but y ∝ 1/x² and other powers are Higher tier only. At Foundation you might be told "y is inversely proportional to x" and asked to complete a table using the product method, without forming an algebraic equation. Forming and using the equation y = k/x² is a Higher skill.
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