Direct Variationn


Direct variation may be understood by scenarios from our daily life.

For example: An employee who works for hourly wages may be paid according to the number of hours he worked. The two quantities x (the number of hours worked) and y (the amount paid) are related in such a way that when x changes, y changes proportionately such that the ratio  remains a constant.

I.e., y varies directly with x. Let us represent the constant by k, i.e.

 or y = kx  ( where k ≠ 0)

If y varies directly as x, this relation is written as y ∝ and read as y varies as x. The sign “ ∝ ” is read “varies as” and is called the sign of variation.


If y varies directly as x and given y = 9 when x = 5, find:

  • the equation connecting x and y
  • the value of y when x = 15
  • the value of x when y = 6


a) y x i.e. y = kx where k is a constant

Substitute x = 5 and y = 9 into the equation:

y =  x


b) Substitute x = 15 into the equation

y = = 27


c) Substitute y = 6 into the equation