# Concept of Proportion

A statement of equality of two ratios is called proportion.

Four numbers, a, b, c, d, are said to be in proportion when the ratio of the first two, a and b, is equal to the ratio of the last two, c and d.

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

e.g. $\frac{2}{3} = \frac{6}{9}$     or         2 : 3 = 6 : 9

Some authors use the notation a : b :: c : d for proportion, but this notation is not preferred.

Here $\frac{{\text{a}}}{{\text{b}}} = \frac{{\text{c}}}{{\text{d}}}$ or a : b = c : d

If four numbers are in proportion, then we can also derive some other proportions from them.

Let  $\frac{{\text{a}}}{{\text{b}}} = \frac{{\text{c}}}{{\text{d}}}$ be the given proportion, then

(1) $\frac{{{\text{a + b}}}}{{\text{b}}} = {\text{ }}\frac{{{\text{c + d}}}}{{\text{d}}}$
(2) $\frac{{{\text{a - b}}}}{{\text{b}}} = {\text{ }}\frac{{{\text{c - d}}}}{{\text{d}}}$
(3) $\frac{{{\text{a + b}}}}{{{\text{a- b}}}} = {\text{ }}\frac{{{\text{c +; d}}}}{{{\text{c - d}}}}$
(4) $\frac{{\text{a}}}{{\text{c}}} = \frac{{\text{b}}}{{\text{d}}}$
(5) $\frac{{\text{b}}}{{\text{a}}} = \frac{{\text{d}}}{{\text{c}}}$

are called the Derived Proportions.

Here    $\frac{{\text{a}}}{{\text{b}}} = \frac{{\text{c}}}{{\text{d}}}$                   or         a : b = c : d, then

The numbers a and d are called extremes of proportion, and the numbers b and c are called means of proportion.

Hence Product of Extremes = Product of Means.

To solve the proportion, we use the above principal, A single term in the proportion is called a proportional.

a” is the 1st proportional.
b” is the 2nd proportional.
c” is the 3rd proportional.
d” is the 4th proportional.

Example:
Find the 3rd proportional in 2 : 3 = $x$: 15

Solution:
Let       2 : 3 = $x$: 15
i.e. $\frac{2}{3} = \frac{x}{{15}}$
$\Rightarrow$     3 x $x$ = 2 x 15                  (by the principle of proportion)
$\Rightarrow$     3$x$ = 30
$\Rightarrow$     $\frac{{3x}}{3} = \frac{{30}}{3}$
$\Rightarrow$     $\boxed{x = 10}$

Example:
Find the missing value in $x$: 8 = 9 : 12

Solution:
Let       $x$: 8 = 9 : 12
$\Rightarrow$     $\frac{x}{8} = \frac{9}{{12}}$
$\Rightarrow$     12$x$ = 9 x 8                      (by the principle of proportion)

$\Rightarrow$    12$x$ = 72
$\Rightarrow$    $\frac{{12x}}{{12}} = \frac{{72}}{{12}}$
$\Rightarrow$    $\boxed{x = 6}$

Example:
Find the 2nd proportional in 4, 20, 30.

Solution:
Let $x$ be the 2nd proportional
$\therefore$ 4 :$x$ =20 : 30
$\frac{4}{x} = \frac{{20}}{{30}}$
$x$ x 20 = 4 x 30
$x$ = $\frac{{{\text{4 x 30}}}}{{20}}$ = 6