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Arithmetic I

Proportion & Variation

ALL MODULES

CAT 2025 Lesson : Proportion & Variation - Basics of Variation

2. Variation

When two or more variables are related, the impact that a change in one variable has on the other is called variation. The following are the three types of variation.

2.1 Direct Variation / Directly Proportional

x

varies directly with

y

' or '

x

is directly proportional to

y

' means an increase in

\bm{x}

results in an increase in

\bm{y}

in the same proportion or a decrease in

\bm{x}

results in a decrease in

\bm{y}

in the same proportion

This is written as

x \propto y

x = Ky

, where

K

is a constant.

For instance, let the perimeter and the length of the side of a square be

p

and

s

respectively. We know that

p = 4s

An increase in the length of the side of the square results in a proportional increase in the perimeter and vice versa. Therefore,

p

and

s

are directly proportional to each other. The value of

K

in this case is

4

Example 11

a

varies directly with the square of

b

, and

a = 12

when

b = 8

, then what is the value of

a

when

b = 3

?
(1)

\dfrac{3}{4}

(2)

\dfrac{9}{2}

(3)

\dfrac{27}{16}

(4)

\dfrac{81}{8}

Solution

a

varies directly with the square of

b ⇒a = kb^{2}

When

a = 12

and

b = 8

12 = K \times 8^{2}

⇒K = \dfrac{12}{64} = \dfrac{3}{16}

When

b = 3

a = Kb^{2}

⇒a = \dfrac{3}{16} \times 3^{2} = \dfrac{27}{16}

Answer: (3)

\dfrac{27}{16}

2.1.1 Unitary Method for Direct Variation

In many chapters (especially Time & Speed and Time & Work) we will come across problems where we would be required to compute a value based on a given variation. This is the recommended method when the relationships are linear, for instance

x

is directly proportional to

y

(not

y^{2}, y^{3}

\sqrt{y}

).

In these types of problems, we must first identify if the given variation is direct or indirect. The application of this is shown in the example below.

Example 12

In a certain amount of time, a person driving at

30

km/h covers

85

km. What is the distance covered by her if she drives at

48

km/h ?

Solution

Observation: Direct variation can be applied as increase in speed results in increase in distance covered.

Downward arrows have been used to symbolise the numerators and denominators in the equation.

⇒\dfrac{85}{x} = \dfrac{30}{48}

⇒x = 136

Answer:

136

2.2 Inverse Variation / Inversely Proportional

x

varies inversely with

y

' or '

x

is inversely proportional to

y

' means an increase in

\bm{x}

results in a decrease in

\bm{y}

in the same proportion or a decrease in

\bm{x}

results in an increase in

\bm{y}

in the same proportion.

This is written as

x \propto \dfrac{1}{y}

x = \dfrac{K}{y}

, where

K

is a constant.

Example 13

Where

x

and

y

are positive real numbers,

x

varies inversely with the square root of

y

. If

x = 8

when

y = 4

, then find

y

when

x = 2

Solution

x

varies inversely with

y ⇒x = \dfrac{k}{\sqrt{y}}

When

x = 8

and

y = 4

8 = \dfrac{k}{\sqrt{4}} ⇒k = 16

When

x = 2

x = \dfrac{k}{\sqrt{y}} ⇒2 = \dfrac{16}{\sqrt{y}}

⇒y = 64

Answer:

64

2.2.1 Unitary Method for Inverse Variation

This is similar to that for direct variation shown in section 2.1.1. However, In inverse variation, the direction of the two arrows are different. This is shown in the example below.