Inverse proportion

If two variables are inversely proportional to each other, it means that if one of the variables doubles, for example, the other variable will be halved.

In other words, the product of the two variables remains constant.

Importantly, if x is inversely proportional to y, then y is also inversely proportional to x. y is equal to a constant divided by x, and x is equal to a constant divided by y.

Expressing inverse proportion

• If y is inversely proportional to x, we can write this as:
  • x \propto \frac{1}{y}
• Because y is inversely proportional to x, it must be equal to a constant divided by x:
  • y = \frac{k}{x}
  • where k is the constant of proportionality.
• Similarly, because x is inversely proportional to y, it must be equal to a constant divided by y:
  • x = \frac{k}{y}
  • where k is the constant of proportionality in this case.

Proof

We can show that the inverse proportion can be expressed as y = \frac{k}{x} using the rule stated above that xy is always constant in inverse proportion.

• If xy is constant, we can call this constant k:
  • xy = k
• Rearrange to make y the subject:
  • y = \frac{k}{x}
• This shows we can express inverse proportion as y = \frac{k}{x}.

a is inversely proportional to b. Write b as an equation in terms of a and the constant of proportionality k.

• a\propto \frac{1}{b}
• b = \frac{k}{a}
• Answer: b = \frac{k}{a}

y is inversely proportional to x^2. Write y as an equation in terms of x and the constant of proportionality k.

• y \propto \frac{1}{x^2}
• y = \frac{k}{x^2}
• Answer: y = \frac{k}{x^2}

Finding the constant of proportionality

In a question, you will likely be given a pair of values for the two variables that are inversely proportional to each other. You can use these values to find the constant of proportionality - just by substituting them into the equation y = \frac{k}{x}, if y is inversely proportional to x.

y\propto \frac{1}{x}. When x = 4, y = 10. Find the constant of proportionality.

• y = \frac{k}{x}
• Substitute x = 4 and y = 10:
  • 10 = \frac{k}{4}
• Rearranging to find k:
  • k = 10 \times 4 = 40
• Answer: k = 40

a\propto \frac{1}{b}. When b = 12, a = 3. Find the constant of proportionality.

• a = \frac{k}{b}
• Substitute b = 12 and a = 3:
  • 3 = \frac{k}{12}
• Rearranging to find k:
  • k = 3 \times 12 = 36
• Answer: k = 36

Using inverse proportion to find unknown values

Once you have found the constant of proportionality, you can use it to find unknown values of either variable.

y\propto \frac{1}{x}. The constant of proportionality is 24. Find y when x = 6.

• y = \frac{k}{x}
• Substitute k = 24 and x = 6:
  • y = \frac{24}{6}
• Calculating y:
  • y = 4
• Answer: y = 4

a\propto \frac{1}{b}. When b = 8, a = 5. Find a when b = 20.

• First, find the constant of proportionality:
  • a = \frac{k}{b}
  • Substitute b = 8 and a = 5:
    • 5 = \frac{k}{8}
  • Rearranging to find k:
    • k = 5 \times 8 = 40
• Now, use k to find a when b = 20:
  • a = \frac{k}{b}
  • Substitute k = 40 and b = 20:
    • a = \frac{40}{20}
  • Calculating a:
    • a = 2
• Answer: a = 2