Direct Variation

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.

Example:

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

Solution:

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

Example:

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.

Solution:

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

Other Forms Of Direct Variation

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 .

Example:

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.

Solution:

that is y = kx 3 where k is a constant

Substitute x = 3 and y = 21 into the equation:

21 = k(3 3) ⇒ k =

So, y = x³

When x = 8,

The following video gives some practical examples of direct variation and indirect/inverse variation.

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