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Maths Angel

Inverse Proportion Formula and Examples

Year 10–11 (UK)Grade 9–10 (US)
Key concept

Inverse proportion means as one value goes up, the other goes down, so their product xy = k stays constant. Multiply a known pair like 6 × 8 = 48 to find the constant k. Then write y = k ÷ x to work out any missing value.

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Inverse proportion with a definition, notation showing y is proportional to 1 divided by x, and the formula y = k divided by x.Inverse proportion example y ∝ 1/x: x=6 gives y=8, so k=48 and y=12 when x=4.Worked example of inverse proportion y ∝ 1/x²: x=3 gives y=8, so k=72 and y=2 when x=6.Inverse proportion example n ∝ 1/√r: n=12 when r=9, so k=36 and n=9 when r=16.

What Is Inverse Proportion?

  • Inverse proportion means the product stays constant. For example, if x is doubled, y is halved, so xy stays the same.
  • In exams, multiply the variables together to get the constant k. For example, if and , then .

Example: Inverse Proportion to x

  • If y is inversely proportional to x, write xy = k.
  • Multiply the given pair to find k first, then substitute the x value to find the wanted y value.

Example: Inverse Proportion to x²

  • If y is inversely proportional to x², write x²y = k.
  • Multiply y by x² to find k, then substitute the x value to find the wanted y value.

Example: Inverse Proportion to √x

  • If y is inversely proportional to √x, write √.
  • Multiply √x by y to find k, then substitute the x value to find the wanted y value.

Practice Questions

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is inversely proportional to , and when , . What is the formula connecting and ?

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Frequently asked questions

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Two quantities are inversely proportional when one is multiplied by a number and the other is divided by the same number. If one doubles, the other halves; if one triples, the other becomes a third. Their product always stays the same, so with x = 6 and y = 8 the product 48 is fixed, and when x becomes 4, y becomes 12.

y = k ÷ x, where k is a constant. Multiplying both sides by x gives the other common form, x × y = k, which says the product of x and y never changes. The statement “y is inversely proportional to x” is written y ∝ 1/x. You find k from one known pair of values, then use the formula for any other pair.

Find the constant first, then use it. If y is inversely proportional to x and y = 8 when x = 6, multiply the pair: k = 6 × 8 = 48, so the formula is y = 48 ÷ x. To find y when x = 4, substitute: y = 48 ÷ 4 = 12. The two steps, find k and then substitute, are the whole method.

In direct proportion the two quantities rise and fall together: y = kx, so doubling x doubles y. In inverse proportion one rises as the other falls: x × y = k, so doubling x halves y. In both, the GCSE method is the same: use the pair you are given to find k, write the formula, then substitute the new value.

Use the same method with the power or root in the formula. For y inversely proportional to x², write y × x² = k: with x = 3 and y = 8, k = 72, so when x = 6, y = 72 ÷ 36 = 2. For n inversely proportional to √r, write n × √r = k: with r = 9 and n = 12, k = 36, so when r = 16, n = 36 ÷ 4 = 9.

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