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Proportion & Variation
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CAT 2025 Lesson : Proportion & Variation - Continued Proportion

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1.1 Continued proportion

If three numbers, say aa, bb and cc, are said to be in continued proportion, then a:b::b:c\bm{a : b :: b : c}

⇒ab=bc⇒b2=ac⇒\dfrac{a}{b} = \dfrac{b}{c} ⇒ b^{2} = ac

Here, b\bm{b} is called the mean proportional, of a\bm{a} and c\bm{c}.
And, c\bm{c} is called the third proportional of a\bm{a} and b\bm{b}.

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

Therefore, if nn terms are in continued proportion, then they can be written as a,ar,ar2,ar3a, ar, ar^{2}, ar^{3}, ... , arn−1ar^{n - 1}.

Example 33

The mean proportional between 88 and 242242 is _______.

Solution

bb is the mean proportional of aa and cc, if aa, bb and cc are in continued proportion.

∴ b2=ac=8×242=8×2×121b^{2} = ac = 8 \times 242 = 8 \times 2 \times 121

⇒b=24×112=44⇒b = \sqrt{2^{4} \times 11^{2}} = 44

Answer: 4444 cm


Example 44

If four positive terms are in continued proportion, and the ratio of the first and the third terms is 1:161 : 16, then what is the ratio of the first and fourth terms?

Solution

Let the four terms be a,ar,ar2a, ar, ar^{2} and ar3ar^{3}.

aar2=116⇒1r2=116⇒r=4\dfrac{a}{ar^{2}} = \dfrac{1}{16} ⇒\dfrac{1}{r^{2}} = \dfrac{1}{16} ⇒r = 4

Ratio of 1st1^{\text{st}} and 4th4^{\text{th}} terms =aar3=143=143=164= \dfrac{a}{ar^{3}} = \dfrac{1}{4^{3}} = \dfrac{1}{4^{3}} = \dfrac{1}{64}

Answer: 1:641 : 64


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