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Parin Sir > Quantitative Aptitude (Maths) > Ratio Proportion & Variation > Proportion > Proportion

Proportion

June 16, 2020
Category: Proportion

☼ Proportion : Let {a, b, c, . . . } and { x, y, z, . . . } be two equivalent sets of numbers. If $\frac{a}{x}$ = $\frac{b}{y}$ = $\frac{c}{z}$ = . . ., then we say that there is a relation of direct proportionality between the two sets.

e.g. The order of the elements in the proportionality is the same as their order in the respective sets. There is a relation of direct proportionality between the sets { 1, 2, 3, 4 } and { 3, 6, 9, 12 }. $\left (\because \frac{1}{3}=\frac{2}{6}=\frac{3}{9}=\frac{4}{12} \right )$

$\large \mapsto$ In a : b = c : d, a and d are called extreme terms while b and c are called mean terms.

a : b = c : d $\large \Leftrightarrow$ Product of extremes = Product of means

$\large \Leftrightarrow$ ad = bc

■ Proportion can be expressed in the following ways :

(i) a, b, c, d are in proportion.

(ii) d is the fourth proportional of a, b, c.

(iii) a : b = c : d

(iv) a : b : : c : d

(v) $\frac{a}{b}=\frac{c}{d}$

■ Laws of proportion :

(i) Alternendo ( the law of alternate proportionals )

If a : b = c : d, then a : c = b : d or

$\frac{a}{b}=\frac{c}{d} \Rightarrow \frac{a}{c}=\frac{b}{d}$

(ii) Invertendo ( the law of inverse proportionals )

If a : b = c : d, then b : a = d : c or

$\frac{a}{b}=\frac{c}{d} \Rightarrow \frac{b}{a}=\frac{d}{c}$

(iii) Componendo ( the law of corresponding addition )

If a : b = c : d, then ( a + b ) : b = ( c + d ) : d or

$\frac{a}{b}=\frac{c}{d} \Rightarrow \frac{a+b}{b}=\frac{c+d}{d}$

(iv) Dividendo ( the law of corresponding subtraction )

If a : b = c : d, then ( a – b ) : b = ( c – d ) : d or

$\frac{a}{b}=\frac{c}{d} \Rightarrow \frac{a-b}{b}=\frac{c-d}{d}$

(v) Componendo and Dividendo ( the law of corresponding + and – )

If a $\ne$ b and c $\ne$ d and if a : b = c : d, then

( a + b ) : ( a – b ) = ( c + d ) : ( c – d ) or

$\frac{a}{b}=\frac{c}{d}\Rightarrow \frac{a+b}{a-c}=\frac{b+d}{b-d}$

■ Some special type of Proportion :

(i) Fourth proportional : If d is a fourth proportional of a, b, c, then

a : b = c : d i.e., $\frac{a}{b}=\frac{c}{d}$

e.g. Find the fourth proportion of 2, 5, 10.

Soln. Let d be the fourth proportion of 2, 5, 10.

$\therefore \frac{2}{5}=\frac{10}{d} \Rightarrow d=25$

(ii) Third proportional : If d is a third proportional of a and b, then

a : b = b : d i.e., $\frac{a}{b}=\frac{b}{d}$

e.g. Find the third proportion of 2 and 10.

Soln. Let d be the third proportion of 2 and 10.

$\therefore \frac{2}{10}=\frac{10}{d} \Rightarrow d=50$

(iii) Continued proportion : The three quantities a, b and c are said to be in continued proportion, if the ratio between first and the second is equal to the ratio between the second and the third.

Hence, if a, b, c are in continued proportion, then

$\frac{a}{b}=\frac{b}{c} \Leftrightarrow b^{2} =ac \Leftrightarrow b=\sqrt{ac}$