Inverse Proportion

SPEC R10 • RATIO, PROPORTION & RATES OF CHANGE

Constructing, interpreting and solving y = k/x — and its extensions to x², x³ and √x.

What Inverse Proportion Means

In the previous lesson we studied direct proportion — where y grows as x grows. Inverse proportion is the opposite: as one quantity goes up, the other goes down, in a very specific mathematical way.

Definition

Inverse proportion. Two quantities are in inverse proportion if one is always a constant multiple of the reciprocal of the other. Written "y ∝ 1/x", which means y = k/x for a fixed constant k.

The Spec Definition — R13

"X is inversely proportional to Y" is exactly equivalent to "X is proportional to 1/Y". Both mean X = k/Y. Examiners may phrase it either way — recognise both.

Everyday examples of inverse proportion:

At fixed distance, the time taken is inversely proportional to speed (twice the speed → half the time).

The time to complete a job is inversely proportional to the number of workers (twice the workers → half the time).

Spotting Inverse Proportion

For the basic case y = k/x, the defining feature is that the product x × y is the same for every pair — that product is the constant k.

y = k/x ⟺ xy = k

So to spot this type of inverse proportion in a table, multiply each (x, y) pair and check you always get the same number:

x2346

y12864

x × y24242424

Reading the Table

Every pair gives xy = 24, so k = 24 and the relationship is y = 24/x. Notice also the doubling–halving pattern: x going from 2 → 4 (doubled) sends y from 12 → 6 (halved).

Higher Powers

The "constant product" rule changes for higher powers. For y = k/x², it is x²y that stays constant across all pairs. For y = k/x³, it is x³y. Always match the power in the rule to the power in the equation.

Direct vs Inverse — The Check

For direct proportion, it's y ÷ x that stays constant. For inverse proportion (basic case), it's x × y that stays constant. Always check which one before writing your equation.

The Graph of y = k/x

The graph of an inverse proportion relationship is a reciprocal curve. In the first quadrant (positive x and y) it looks like this:

Reading the Graph

The curve shown has equation y = 4/x (so k = 4). The marked point (2, 2) satisfies 2 = 4/2 ✓.

As x → 0, the curve shoots up to infinity — it never touches the y-axis.

As x → ∞, the curve flattens towards zero — it never touches the x-axis.

Note: mathematically the graph also has a second branch in the bottom-left quadrant for negative x and y. GCSE proportion problems use positive quantities, so only the first quadrant appears in exam questions.

The MCE Procedure

Every inverse proportion question uses the same three-step pattern as direct proportion — only the equation changes. Find k, then apply the full relationship.

Method

Translate the words into an equation with k. "y inversely proportional to x" → y = k/x.

Calculate

Substitute the given pair to find k (remember: k = x × y for y = k/x). Rewrite the full relationship with k's value.

Evaluate

Substitute the new x-value into the full relationship and state the final answer.

A Useful Shortcut

For the basic inverse case y = k/x, the product xy is always k. So if (x₁, y₁) and (x₂, y₂) are both on the curve, then x₁y₁ = x₂y₂. This gives the same answer as the MCE method.

Worked Example — Basic Inverse Proportion01

y is inversely proportional to x. When x = 5, y = 12. Find the value of y when x = 20. [3 marks]

M

"y inversely proportional to x" means y = k/x. Find k from the given pair, then use the full relationship.

C

Substitute x = 5, y = 12 into y = k/x: 12 = k/5 → k = 60.

Full relationship: y = 60/x.

E

When x = 20: y = 60/20 = 3. Answer: y = 3.

Inverse Proportion to a Power

Just like direct proportion, the relationship can involve a power of x. The wording tells you which power goes in the denominator.

All the Inverse Forms

Phrase in the questionEquation to writey is inversely proportional to xy = k/xy is inversely proportional to x² (the square of x)y = k/x²y is inversely proportional to x³ (the cube of x)y = k/x³y is inversely proportional to √xy = k/√x

Scaling Behaviour

y = k/x: x doubles → y halves (÷ 2).

y = k/x²: x doubles → y quarters (÷ 4).

y = k/x³: x doubles → y ÷ 8.

Read Every Word

"Inversely proportional to the square of x" is y = k/x², not y = k/x. Write the full equation only after reading the complete phrase.

Graph Shapes by Power

All inverse proportion graphs have the same general shape — a curve that approaches both axes but never touches them. The power in the denominator changes how steeply the curve falls.

Common Feature

All three curves approach both axes but never touch them — and all three fall from left to right. Higher powers make the curve drop off more steeply.

Worked Example — Inversely Proportional to a Square02

y is inversely proportional to the square of x. When x = 3, y = 16. Find the value of y when x = 12. [4 marks]

M

"Inversely proportional to the square of x" means y = k/x². Find k from the given pair, then use the full relationship.

C

Substitute x = 3, y = 16 into y = k/x²: 16 = k/9 → k = 144.

Full relationship: y = 144/x².

E

When x = 12: y = 144/144 = 1. Answer: y = 1.

03Worked Example — Worded Context

A team of 6 builders can finish a wall in 20 days. The time t (in days) taken to build the wall is inversely proportional to the number of builders b.

(a) Find an equation connecting t and b. [2 marks]

(b) How long would it take 8 builders working at the same rate? [2 marks]

M

"t is inversely proportional to b" means t = k/b. Use the given pair (b = 6, t = 20) to find k.

C

20 = k/6 → k = 120. (a) t = 120/b.

E

(b) When b = 8: t = 120/8 = 15 days.

Sanity check: more builders → less time. 8 > 6, so t should be smaller than 20. 15 < 20 ✓.

Direct vs Inverse — Side-by-Side

Keep these two relationships cleanly separate. The main exam error is mixing the two — writing y = kx when the question says "inversely" (or vice versa).

FeatureDirect: y ∝ xInverse: y ∝ 1/x

Equationy = kxy = k/x

What doubles x does to yy doublesy halves

Constant quantityy ÷ x = kx × y = k

Graph shapeStraight lineReciprocal curve

Graph and axesPasses through (0, 0)Never touches either axis

Reading a Worded Question

If the question says "more of one → more of the other" (more people buying petrol → more total cost), it's direct. If it says "more of one → less of the other" (more workers → less time), it's inverse.

Common Errors to Avoid

These are the errors that repeatedly lose marks on inverse proportion questions. Learn them so you don't repeat them.

Students often writeCorrect versionWhy it matters

y = kx for inverse proportiony = k/x (or k/x², etc.)Direct and inverse are opposite structures

"As x doubles, y doubles" for inverseAs x doubles, y halves (for y = k/x)Inverse relationships move in opposite directions

"y = k/x²" means y proportional to x²y = k/x² means y inversely proportional to x²The 1/ prefix flips the relationship

Find k and stop — write k as the answerSubstitute the new x into the full relationshipk is an intermediate value, not the final answer

"y proportional to x²" and "inversely proportional to x²" mean the sameDirect → y = kx²; Inverse → y = k/x²These graphs behave in opposite ways

Examiner Alerts

Examiner Alert — Direct vs Inverse Mix-up

The biggest source of lost marks on this topic is writing the wrong form of the equation. Students who read "proportional" too quickly and default to y = kx lose every mark when the question actually says "inversely proportional". Read the full phrase; if you see the word "inversely", write y = k / (something), never y = k × (something).

Examiner Alert — Rearranging for x

Questions frequently ask you to find x given y. For y = k/x², that means rearranging: x² = k/y → x = √(k/y). Students often stop at x² = some number, forgetting to take the root, and lose the final accuracy mark.

Showing All Three Steps

For full marks always write: (1) the equation with k, (2) the calculation that gives k, (3) the full relationship with k substituted — then (4) the final evaluation. Even if you can do it mentally, the method marks are only awarded for working on the page.

Practice Question 1

y is inversely proportional to x. When x = 4, y = 10. Find the value of y when x = 8. [3 marks]

M

y inversely proportional to x: y = k/x.

C

10 = k/4 → k = 40. So y = 40/x.

E

When x = 8: y = 40/8 = 5.

Practice Question 2

A is inversely proportional to the square of d. When d = 5, A = 8. Find the value of A when d = 4. [4 marks]

M

A inversely proportional to d²: A = k/d².

C

8 = k/25 → k = 200. So A = 200/d².

E

When d = 4: A = 200/16 = 12.5.

Practice Question 3 — Worded

A pool can be filled by 5 pumps working together in 4 hours. The time t (in hours) to fill the pool is inversely proportional to the number of pumps p used.

(a) Find an equation connecting t and p. [2 marks]

(b) How long would it take 10 pumps to fill the pool? [1 mark]

(c) How many pumps would be needed to fill the pool in 1 hour? [2 marks]

(a)

t = k/p. Substitute p = 5, t = 4: 4 = k/5 → k = 20. t = 20/p.

(b)

When p = 10: t = 20/10 = 2 hours.

(c)

When t = 1: 1 = 20/p → p = 20 pumps.

Practice Question 4

y is inversely proportional to the cube of x. When x = 2, y = 10.

(a) Find an equation for y in terms of x. [2 marks]

(b) Work out the value of y when x = 5. [2 marks]

(c) Work out the value of x when y = 640. [3 marks]

(a)

y = k/x³. Substitute x = 2, y = 10: 10 = k/8 → k = 80. y = 80/x³.

(b)

y = 80/5³ = 80/125 = 16/25 (= 0.64).

(c)

640 = 80/x³ → x³ = 80/640 = 1/8 → x = 1/2 (= 0.5).

Practice Question 5 — Recognition

A curve passes through the two points (2, 9) and (6, 1). It is known that y is inversely proportional to some power of x — either x, x², or x³.

(a) Which power of x is correct? Show working to justify your answer. [3 marks]

(b) Write down the equation of the curve. [1 mark]

(a)

Test each candidate using k = xⁿ × y for both points. The correct power gives the same k for both:

y = k/x: k₁ = 2×9 = 18; k₂ = 6×1 = 6. Not equal — reject.

y = k/x²: k₁ = 4×9 = 36; k₂ = 36×1 = 36. Match! Accept.

y = k/x³: k₁ = 8×9 = 72; k₂ = 216×1 = 216. Not equal — reject.

(b)

y = 36/x².

Summary

The Core Idea

Inverse proportion: y ∝ 1/x means y = k/x. Equivalently, xy = k (constant product).

All forms: y = k/x, y = k/x², y = k/x³, y = k/√x. Read the exact wording to choose the power.

Graph test: reciprocal curve — falls from left to right, never touches either axis.

The Three-Step Procedure (MCE)

1

M — Write the relationship using k (e.g. y = k/x²).

2

C — Find k by substituting the given pair. Rewrite the full relationship with k's value.

3

E — Apply it : substitute the new value and state the final answer, with units if relevant.

Direct vs Inverse — The Test

DirectInverse

Equationy = kxy = k/x

Constanty ÷ x = kx × y = k

x doublesy doublesy halves

GraphLine through originReciprocal curve

The Three Traps

Direct/Inverse mix-up: if you see "inversely", the k must be in the numerator , with x in the denominator.

Missing the power: "inversely proportional to the square" is y = k/x², not y = k/x.

k is not the answer: always take the third step — substitute the new value and evaluate.