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