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Arithmetic I

Proportion & Variation

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CAT 2025 Lesson : Proportion & Variation - Continued Proportion

1.1 Continued proportion

If three numbers, say

a

b

and

c

, are said to be in continued proportion, then

\bm{a : b :: b : c}

⇒\dfrac{a}{b} = \dfrac{b}{c} ⇒ b^{2} = ac

Here,

\bm{b}

is called the mean proportional, of

\bm{a}

and

\bm{c}

.
And,

\bm{c}

is called the third proportional of

\bm{a}

and

\bm{b}

.

As the ratio of any two consecutive terms are equal, continued proportions are Geometric Progressions (Refer to Sequences & Progression Lesson).

Therefore, if

n

terms are in continued proportion, then they can be written as

a, ar, ar^{2}, ar^{3}

, ... ,

ar^{n - 1}

Example $3$

The mean proportional between

8

and

242

is _______.

Solution

b

is the mean proportional of

a

and

c

, if

a

b

and

c

are in continued proportion.

∴

b^{2} = ac = 8 \times 242 = 8 \times 2 \times 121

⇒b = \sqrt{2^{4} \times 11^{2}} = 44

Answer:

44

Example $4$

If four positive terms are in continued proportion, and the ratio of the first and the third terms is

1 : 16

, then what is the ratio of the first and fourth terms?

Solution

Let the four terms be

a, ar, ar^{2}

and

ar^{3}

\dfrac{a}{ar^{2}} = \dfrac{1}{16} ⇒\dfrac{1}{r^{2}} = \dfrac{1}{16} ⇒r = 4

Ratio of

1^{\text{st}}

and

4^{\text{th}}

terms

= \dfrac{a}{ar^{3}} = \dfrac{1}{4^{3}} = \dfrac{1}{4^{3}} = \dfrac{1}{64}

Answer:

1 : 64

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CAT 2025 Lesson : Proportion & Variation - Continued Proportion

1.1 Continued proportion

Example 333

Solution

Example 444

Solution

Example $3$

Example $4$