Concept of Percentages

Comparison of fractions is not an easy task, especially when the two fractions have different denominators.
For example; if 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, denominators of  \frac{3}{{10}} and \frac{9}{{20}}are different.
To compare these fraction, first we make their denominators same
\therefore  {\text{ }}\frac{{{\text{3 x 2}}}}{{{\text{10 x 2}}}} = \frac{6}{{20}}
Now, we have two fraction as \frac{6}{{20}}, \frac{9}{{20}} with same denominator.
Since numerator of \frac{9}{{20}}is greater than numerator of \frac{6}{{20}}.
\therefore  \frac{9}{{20}}is greater than \frac{6}{{20}}
But comparison becomes easier if common denominator is 100.

The fraction with 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” that means “Out of Hundred”.
Example: In a paper of math, out of total marks 50, Waqas got 35 marks, Usman got 43 marks and Shakeel got marks 32.
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 decimal
(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 Percentage:
We can change a fraction into percentage by multiplying the fraction by 100% and simplify 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 \% }}

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