GCSE MATHS • HIGHER TIER

Inverse Proportion Made Simple

Recognise inverse relationships, find the constant of proportionality and solve real-world problems.

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Inverse Proportion in Four Moves

1

Recognise

Check whether one quantity increases while the other decreases.

2

Find k

Multiply the paired values to find the constant of proportionality.

3

Calculate

Use y = k/x or the constant product to find the missing value.

4

Check

Confirm that the product stays constant and the inverse relationship makes sense.

What You’ll Learn

Understand inverse proportion.

Recognise inverse relationships.

Use y = k/x

Find the constant of proportionality.

Solve worker, machine, speed and time problems.

Interpret inverse proportion graphs.

Distinguish inverse proportion from direct proportion.

Key Rules

In inverse proportion, one quantity increases while the other decreases.

The product of the two quantities remains constant.

Write inverse proportion as:

y∝1​/x​

Find a missing value by dividing the constant by the known value.

Worked Proof: Inverse Proportion

Step-by-Step Method

  • 1. Find the radius or diameter.
  • 2. Calculate the circumference..
  • 3. Find the fraction represented by the angle.
  • 4. Multiply the circumference by this fraction. the angle.
  • 5. Give the answer in exact or decimal form.

More Worked Examples

1

Question: y is inversely proportional to x. When x = 4, y = 12. Find y when x = 6

Step 1: Find k

12 = k/4
k = 48
Step 2: Use the formula.
y = 48/6 = 8
Answer: 8

2

Question: y is inversely proportional to x. When x = 5, y = 20. Find y when x = 10

20 = k/5

k = 100
y = 100/10 = 10
Answer: 10

3

Question: 6 workers complete a job in 8 hours. How long would 12 workers take?

Workers × Hours = Constant

6 × 8 = 48
48 ÷ 12 = 4
Answer: 4 hours

4

Question: A journey takes 10 hours at 60 km/h. How long would it take at 75 km/h?

Speed × Time = Constant

60 × 10 = 600
600 ÷ 75 = 8
Answer: 8 hours
  • Common GCSE Mistakes
  • Confusing direct proportion with inverse proportion.
  • Not checking whether one quantity rises while the other falls.
  • Forgetting that inverse proportion uses multiplication to find the constant.
  • Making arithmetic or calculator errors.
  • Exam Tips
  • Check whether increasing one quantity makes the other decrease.
  • Remember that the product of inversely proportional quantities stays constant.
  • Write y = k/x before substituting in algebraic questions.
  • Find k accurately before calculating the missing value.
  • For workers, machines, speed and time, look for a constant product.
  • Check that your answer follows the inverse relationship.

Skills Used in Inverse Proportion

Inverse Proportion

Ratio

Algebra

Constant of Proportionality

Graphs

Rates of Work

Speed and Time

Problem Solving

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Frequently Asked Questions

Two quantities are inversely proportional when one increases while the other decreases so that their product remains constant.

The source uses y ∝ 1/x and y = k/x.

k is the constant of proportionality.

Use a known pair of values. Since y = k/x, the paired values give the constant product.

It forms a curved line called a hyperbola rather than a straight line through the origin.

In inverse proportion one quantity increases as the other decreases; the source contrasts y = k/x with direct proportion y = kx.