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Arithmetic I

Proportion & Variation

ALL MODULES

CAT 2025 Lesson : Proportion & Variation - Concepts & Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in the Proportion & Variation lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

7. Cheatsheet

1) Proportion involves equality of ratios that have

2

terms. If

\bm{a, b, c}

and

\bm{d}

are in proportion, then the following are different ways to write the same.

a : b = c : d

a : b :: c : d

\dfrac{a}{b} = \dfrac{c}{d}

\bm{a}

and

\bm{d}

are called extremes, and

\bm{b}

and

\bm{c}

are called means. Upon simplifying, we note that the product of extremes equals the product of means.

3) If three numbers, say

a, b

and

c

, are said to be in continued proportion, then

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

Numbers in continued proportion are in Geometric Progression.

4) Properties of Proportions

4.1) If

\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} =

..., and

p, q, r

and

s

are real numbers, then

\dfrac{pa + qb}{ra + sb} = \dfrac{pc + qd}{rc + sd} = \dfrac{pe + qf}{re + sf} =

...

4.2) If

\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = .... = k

, then

k = \dfrac{a + c + e + ...}{b + d + f + ...}

4.3) If

\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = ... = k

, then

k = \dfrac{pa}{pb} = \dfrac{qc}{qd} = \dfrac{re}{rf} = ... = \dfrac{pa + qc + re + ...}{pb + qd + rf + ...}

5) If

x

directly varies with

y

, then

x \propto y, x = Ky

6) If

x

inversely varies with

y

, then

x \propto \dfrac{1}{y}, x = \dfrac{K}{y}

7) If

a \propto b

and

a \propto \dfrac{1}{c}

and

a \propto d

, then

a \propto \dfrac{bd}{c}

a = \dfrac{Kbd}{c}

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