In these lessons, we will learn about direct variation and how to solve applications that involve direct variation.
The following diagrams show Direct Variation and Indirect Variation. Scroll down the page for examples and solutions.
There are many situations in our daily lives that involve direct variation.
For example, a worker 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.
We say that 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 ∝ x 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:
a) the equation connecting x and y
b) the value of y when x = 15
c) 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
The cost of a taxi fare (C) varies directly as the distance (D) travelled. When the distance is 60 km, the cost is $35. Find the cost when the distance is 95 km.
C ∝ D i.e. C = kD, where k is a constant.
Substitute C = 35 and D = 60 into the equation
35 = 60k ⇒ k =
Therefore, C = D
Substitute D = 95 into the equation: C = 55.42
The cost for 95 km is $55.42
Some examples of direct variation problems in real life:
The following statements are equivalent:
A direct variation is a linear equation that can be written in the form y = kx , where k is a nonzero constant. The number k is called the constant of proportionality or constant of variation.
Graphically, we have a line that passes through the origin with the slope of k.
Example: y varies directly with x. Given that y = 6 when x = 30, determine the direct variation equation and then determine y when x = 8.
The total cost of filling up your car with gas varies directly with the number of gallons of gasoline that you are purchasing. If a gallon of gas costs $2.25, how many gallons could you purchase for $18?
The area A of a circle of radius r is given by the equation A = pr2, where p is a constant
In this situation, A is not directly proportional to r but A is directly proportional to r2. We say that ‘A varies directly as the square of r’ or A ∝ r2.
Given that y varies directly as the cube of x and that y = 21 when x = 3, calculate the value of y when x = 8.
y ∝ x3 that is y = kx3 where k is a constant
Substitute x = 3 and y = 21 into the equation:
21 = k(33) ⇒ k =
So, y = x3
When x = 8,
The following video gives some practical examples of direct variation and indirect/inverse variation.
Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.