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Arithmetic I

Proportion & Variation

ALL MODULES

CAT 2025 Lesson : Proportion & Variation - Componedo & Dividendo

1.2 Basic Properties

a

b

c

and

d

are in proportion such that the Base Identity is

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

, then the following are true.

Properties	Identity	Operation on Base identity
Invertendo	$\dfrac{b}{a} = \dfrac{d}{c}$	Reciprocal
Alternendo	$\dfrac{a}{c} = \dfrac{b}{d}$	Switching numerator for denominator
Componendo	$\dfrac{a + b}{b} = \dfrac{c + d}{d}$	Adding $1$ to both sides
Dividendo	$\dfrac{a - b}{b} = \dfrac{c - d}{d}$	Subtracting $1$ from both sides
Componendo & Dividendo	$\dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}$	Dividing Componendo and Dividendo

Note: You need not memorise the names of these identities.

1.3 Other Properties

1.3.1 Componendo & Dividendo with different coefficients

Property: If

2

or more ratios are equal, such that

\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} =

...

Note the following for the above identity
1) There is no constant term added or subtracted.
2) The coefficients of variables, i.e.

p, q, r

and

s

, are applied in a uniform way in all the ratios.
3) A ratio will remain unchanged if a and b are replaced with c and d respectively or e and f respectively.

For instance if,

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

..., then

\dfrac{4a + 3b}{7a - 5b} = \dfrac{4c + 3d}{7c - 5d} = \dfrac{4e + 3f}{7e - 5f} =

....

Note: The ratios derived using componendo and dividendo need not equal the original ratio of

\dfrac{a}{b}

Example 5

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

, then

\dfrac{5b}{7a - 4b} =

?
(1)

\dfrac{a}{b}

(2)

\dfrac{a}{c}

(3)

\dfrac{5c}{7c - 4d}

(4)

\dfrac{5d}{7c - 4d}

Solution

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

, in the ratio

\dfrac{5b}{7a - 4b}

, the variables a and b can be replaced by c and d respectively.

∴

\dfrac{5b}{7a - 4b} = \dfrac{5d}{7c - 4d}

Answer: (4)

\dfrac{5d}{7c - 4d}

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