Sheet № 175 · Higher only · AQA · Edexcel · OCR
Inverse Proportion –
Inverse proportion is a Higher tier topic where one quantity increases as the other decreases. The product of the two quantities remains constant — this is the key relationship that distinguishes inverse proportion from direct proportion.
§Key definitions
Question:
y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10.
Q1 (Foundation):
y is inversely proportional to x. When x = 5, y = 12. Find y when x = 20.
Q2 (Foundation):
8 machines can produce an order in 3 hours. How long would 12 machines take?
Q3 (Higher):
y is inversely proportional to x². When x = 2, y = 50. Find x when y = 2.
§Formulas to memorise
y ∝ 1/x means y = k/x, where k = xy
y ∝ 1/x² means y = k/x², where k = yx²
Convert to an equation with a constant: y = k/x.
y = k/x
k = 15 × 4 = 60
When x = 10: y = 60/10 = 6
y = k/x²
k = 8 × 9 = 72
When x = 6: y = 72/6² = 72/36 = 2
k = workers × time = 6 × 10 = 60
Worked example
y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10.
Working: y = k/x 15 = k/4 k = 15 × 4 = 60 When x = 10: y = 60/10 = 6
⚠ Common mistakes
- ✗Confusing direct and inverse proportion. In direct proportion, both quantities increase together. In inverse proportion, one goes up while the other goes down.
- ✗Forgetting to square x when y ∝ 1/x². If the question says "inversely proportional to the square of x," you must square x before dividing.
- ✗Using subtraction instead of division. Inverse proportion means multiplying gives a constant, not that the difference is constant.
- ✗Not recognising inverse proportion in context. Clues include: more workers = less time, faster speed = shorter journey, wider pipe = lower height of water.
✦ Exam tips
- →The product xy = k is constant for simple inverse proportion — use this as a quick check.
- →Always write the proportionality statement first, then convert to an equation before substituting.
- →For graphs: a reciprocal curve (hyperbola) indicates inverse proportion; a straight line through the origin indicates direct proportion.
- →If you are given a table of values, multiply x by y for each pair — if all products are equal, it is inverse proportion.