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Arithmetic I

Proportion & Variation

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CAT 2025 Lesson : Proportion & Variation - Combined Variation

2.3 Joint Variation

Joint variation is a form of variation involving two or more variables.

D = s \times t

, an increase of speed or time will result in an increase in the distance covered. So, distance jointly varies with speed and time.

In the case of volume of a cylinder,

V = \pi r^{2} h

. Here,

V \propto r^{2}

and V

\propto h

.

So,

V

jointly varies with

r^{2}

and

h

.

This is represented as

V = K r^{2}h

. In this case the value of

K

\pi

2.4 Combined Variation

Combined variations contain combinations of Direct, Inverse and Joint variations.

For instance, if

a \propto b

and

a \propto \dfrac{1}{c}

and

a \propto d

, then the same can be combined and expressed as

a \propto \dfrac{bd}{c}

a = \dfrac{Kbd}{c}

Example 15

x

varies directly with the cube of

y

x

is also inversely proportional to the square root of

z

. When

x

32

y

4

and

z

64

. What is the value of

x

when

y

3

and

z

16

Solution

x \propto y^{3}

and

x \propto \dfrac{1}{\sqrt{z}}

⇒x = \dfrac{Ky^{3}}{\sqrt{z}}

Substituting

x

32

y

4

and

z

64

32 = \dfrac{K \times 4^{3}}{\sqrt{64}}

⇒K = 4

When

y

3

and

z

16

x = \dfrac{4 \times 3^{3}}{\sqrt{16}} = 27

Answer:

27

Example 16

x

varies as the cube of

y

, and

y

varies as the fifth root of

z

, then

x

varies as the

n^{th}

power of

z

, where

n

is:
[FMS 2011]
(1)

\dfrac{1}{15}

(2)

\dfrac{5}{3}

(3)

\dfrac{3}{5}

(4)

15

Solution

x \propto y^{3}

and

y \propto z^{\frac{1}{5}}

are given in the question

y \propto z^{\frac{1}{5}} ⇒ y^{3} \propto z^{\frac{3}{5}}

∴

x \propto y^{3} \propto z^{\frac{3}{5}}

Answer:

\frac{3}{5}

Example 17

The value of a gem is directly proportional to the square of its weight. When Anu dropped the gem, it broke into three pieces whose weights were in the ratio

1 : 2 : 3

. What is the percentage reduction in the value?
[FMS 2011]

Solution

v = Kw^{2}

where

v

and

w

are the value and weight of a gem respectively.

If the weight of the gem is

6

gm, then the

3

smaller pieces weight

1

gm,

2

gm and

3

gm.

Value of

6

gm gem

= K \times 6^{2} = 36K

Sum of value of

3

smaller gems

= K \times 1^{2} + K \times 2^{2} + K \times 3^{2} = 14K

\%

Reduction in value

= \dfrac{36K - 14K}{36K} \times 100 \% = \dfrac{11}{18} \times 100 \% = 61.11 \%

Answer:

61.11 \%

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