# 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:

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

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

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

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