# 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:

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

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

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

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