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 first two a and b is equal to ratio of 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 used the notation for proportion as a : b :: c : d, but this notation is not preferred now.
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 derived some other proportion from it.
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 proportion, we use above principal, A single term in the proportion is called 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      3x = 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      12x = 9 x 8                      (by the principle of proportion)

  \Rightarrow     12x = 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

Comments

comments