CAT 2025 Lesson : Proportion & Variation - Componedo & Dividendo

1.2 Basic Properties

If $a$, $b$, $c$ 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 $\dfrac{a}{b}$.