Direct proportion

If two variables are directly proportional to each other, it means that if one of the variables doubles, for example, the other variable will also double.

In other words, the ratio between the two variables remains constant.

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

Expressing direct proportion

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

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

• a\propto b
• b = ka
• Answer: b = ka

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

• y \propto x^2
• y = kx^2
• Answer: y = kx^2

Finding the constant of proportionality

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

y\propto x. When x = 4, y = 10. Find the constant of proportionality.

• y = kx
• Substitute x = 4 and y = 10:
  • 10 = k \times 4
• Rearranging to find k:
  • k = \frac{10}{4} = 2.5
• Answer: k = 2.5

a\propto b. When b = 12, a = 3. Find the constant of proportionality.

• a = kb
• Substitute b = 12 and a = 3:
  • 3 = k \times 12
• Rearranging to find k:
  • k = \frac{3}{12} = 0.25
• Answer: k = 0.25

Using direct proportion to find unknown values

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

You will probably have a question that tells you:

• that two variables are directly proportional to each other
• gives you a pair of values for the two variables
• asks you to find an unknown value of one of the variables, given a value of the other variable

y\propto x. When x = 2, y = 8. Find y when x = 5.

• First, find the constant of proportionality:
  • y = kx
  • Substitute x = 2 and y = 8:
    • 8 = k \times 2
  • Rearranging to find k:
    • k = \frac{8}{2} = 4
• Now, use k to find y when x = 5:
  • y = kx
  • Substitute k = 4 and x = 5:
    • y = 4 \times 5 = 20
• Answer: y = 20

a\propto \sqrt b. When b = 9, a = 6. Find a when b = 16.

• First, find the constant of proportionality:
  • a = k\sqrt{b}
  • Substitute b = 9 and a = 6:
    • 6 = k \times \sqrt{9}
    • 6 = k \times 3
  • Rearranging to find k:
    • k = \frac{6}{3} = 2
• Now, use k to find a when b = 16:
  • a = k\sqrt{b}
  • Substitute k = 2 and b = 16:
    • a = 2 \times \sqrt{16}
    • a = 2 \times 4 = 8
• Answer: a = 8