Concept of Percentages

Comparing fractions is not an easy task, especially when the two fractions have different denominators.

For example, you are asked which of the fractions \frac{3}{{10}} and \frac{9}{{20}} is greater than the other, i.e. we want to compare whether  \frac{3}{{10}} is greater than or less than \frac{9}{{20}}.

Since the denominators of  \frac{3}{{10}} and \frac{9}{{20}}are different, to compare these fraction, first we make their denominators the same.

\therefore {\text{ }}\frac{{{\text{3 x 2}}}}{{{\text{10 x 2}}}} = \frac{6}{{20}}

Now, we have two fractions as \frac{6}{{20}}, \frac{9}{{20}} with the same denominator.

Since the numerator of \frac{9}{{20}}is greater than the numerator of \frac{6}{{20}}.
\therefore \frac{9}{{20}} is greater than \frac{6}{{20}}

But comparison becomes easier if the common denominator is 100.

A fraction with the denominator 100 is called a percentage, denoted by a % or a /100. The sign % is called percent.

For example, \frac{3}{{100}}= 3%, \frac{5}{{100}}= 5%

The term percent is a short form of the Latin word “Per Centum” which means “Out of Hundred”.

Example: On a math paper, out of a total score of 50 Waqas got 35, Usman got 43 and Shakeel got 32.7

Waqas got 35 out of 50 marks
i.e. \frac{{35}}{{50}} = \frac{{35}}{{50}}{\text{x}}\frac{{\text{2}}}{{\text{2}}} = \frac{{70}}{{100}} = 70\%

Usman got 43 out of 50
i.e. \frac{{43}}{{50}} = \frac{{86}}{{100}} = 86 \%

Shakeel got 32 out of 50
i.e. \frac{{32}}{{50}} = \frac{{64}}{{100}} = 64 \%

Example:
15% = \frac{{15}}{{100}} = \frac{3}{{20}}       (replace % by 1/100)

Example:
48% = \frac{{4.8}}{{100}} = \frac{{4.8}}{{100{\text{ x 10}}}} = \frac{6}{{125}}

Example:
8\frac{3}{7} \% = \frac{{59}}{7} \% = \frac{{59}}{7}{\text{ x }}\frac{{\text{1}}}{{{\text{100}}}} = \frac{{59}}{{700}}

Example:
Express in decimals
(1) 8\frac{1}{4}\% = \frac{{33}}{4}\% = \frac{{33}}{{4{\text{ x 100}}}} = \frac{{33}}{{400}} = 0.0825
(2) 0.4\% = \frac{{0.4}}{{100}} = 0.004
(3) 185\% = \frac{{185}}{{100}} = 1.85

Changing a Fraction into a Percentage
We can change a fraction into a percentage by multiplying the fraction by 100% and simplifying it, if possible.

Examples:
Express as percentages:

\frac{1}{{80}} = \frac{1}{{80}}{\text{x 100 \% = }}\frac{{\text{1}}}{{{\text{80}}}}{\text{ x 100 \% = 1}}{\text{.25\% }}

2\frac{1}{8} = \frac{{17}}{8} = \frac{{17}}{8}{\text{ x 100 \% = 212}}{\text{.5 \% }}

0.03 = {\text{ 0}}{\text{.03 x 100 \% = }}\frac{{\text{3}}}{{{\text{100}}}}{\text{ x 100 \% = 3 \% }}

1.12 = {\text{ 1}}{\text{.12 x 100 \% = }}\frac{{{\text{112}}}}{{{\text{100}}}}{\text{x 100 \% = 112 \% }}