# Percentage, Fractions And Decimals

The learning objectives are:

1. Conversion of fraction/decimal to percentage and vice versa
2. Solve simple word problems using percentage

## Definition of Percentage

Any number written as a part out of $100$ can be written as a percentage. It can be converted to a fraction or a decimal. The symbol for percentage is $\%$.

Example:

$17$ out of $100$ expressed as a fraction is $\displaystyle{\frac {17}{100}}$.

$17$ out of $100$ expressed as a decimal is $0.17$.

$17$ out of $100$ expressed as a fraction is $17\,\%$.

Hence, ​\begin{align}17\% &= \frac{17}{100} \\[2ex] &= 0.17 ​\end{align} ​.

### 1. Conversion of fraction/decimal to percentage and vice-versa

Question 1:

Express $\displaystyle{\frac {1}{20}}$ as a percentage.

Solution:

\begin{align*} \frac {1}{20}&= \frac {1}{20} \times 100\% \\[2ex] &= \frac {1}{20} \times \frac {100\%}{1} \\[2ex] &= 5\% \end{align*}

$5\%$

Question 2:

Express $\displaystyle{\frac {3}{5}}$ as a percentage.

Solution:

\begin{align*} \frac {3}{5}&= \frac {3}{5} \times 100\% \\[2ex] &= \frac {3}{5} \times \frac {100\%}{1} \\[2ex] &= 60\% \end{align*}

$60\%$

Question 3:

Express $\displaystyle{1\frac {1}{5}}$ as a percentage.

Solution:

\begin{align*} 1\frac {1}{5} &= \frac {6}{5}\\[2ex] &= \frac {6}{5} \times 100\% \\[2ex] &= \frac {6}{5} \times \frac {100\%}{1} \\[2ex] &= 120\% \end{align*}

$120\%$

Question 4:

Express $\displaystyle{5\frac {3}{4}}$ as a percentage.

Solution:

\begin{align*} 5\frac {3}{4} &= \frac {23}{4}\\[2ex] &= \frac {23}{4} \times 100\% \\[2ex] &= \frac {23}{4} \times \frac {100\%}{1} \\[2ex] &= 575\% \end{align*}

$575\%$

Question 5:

Express $1.34$ as a percentage

Solution:

\begin{align*} 1.34 &= 1.34 \times 100\% \\[2ex] &= 134\% \end{align*}

$134\%$

Question 6:

Express $3.4$ as a percentage.

Solution:

\begin{align*} 3.4 &= 3.4 \times 100\% \\[2ex] &= 340\% \end{align*}

$340\%$

Your help is needed. Tim and Jim are in a heated argument about who scored a better grade for their spelling date. Tim scored $\displaystyle\frac{3}{4}$ and Jim scored $\displaystyle\frac{7}{10}$.

Who do you think scored higher?

In order to compare any two fractions, the denominator of both the fractions must be made the same. Once the denominators are the same, we compare the numerators of the two fractions.

Tim Jim
$\displaystyle{\frac {3}{4}}$ $\displaystyle{\frac {7}{10}}$
$\displaystyle{\frac {(3\;\times\;5)}{(4\;\times\;5)}}$ $\displaystyle{\frac {(7\;\times\;2)}{(10\;\times\;2)}}$
$\displaystyle{\frac {15}{20}}$ $\displaystyle{\frac {14}{20}}$

Since, $15$ is greater than $14$, Tim scored higher than Jim.

We can also convert the fractions to percentages to compare.

\begin{align*} \frac {3} {4}&= \frac {3} {4} \times 100\% \\[2ex] &= \frac {3} {4} \times \frac {100\%} {1} \\[2ex] &= 75\% \end{align*}

\begin{align*} \frac {7} {10}&= \frac {7} {10} \times 100\% \\[2ex] &= \frac {7} {10} \times \frac {100\%} {1} \\[2ex] &= 70\% \end{align*}

After converting from fractions to percentages, Tim’s score is $75\%$ while Jim’s score is $70\%$

Tim scored higher than Jim.

Question 7:

Express the following percentage as a fraction in its simplest form.

$48\,\% = \text{__________}$

Solution:

\begin{align*} 48\,\%&= \frac {48} {100} \\[2ex] &= \frac {12} {25} \end{align*}

$\displaystyle{\frac {12} {25}}$

Question 8:

Express the following percentage as a fraction in its simplest form.

$80\,\% = \text{ __________}$

Solution:

\begin{align*} 80\, \% &= \frac {80} {100} \\[2ex] &= \frac {4} {5} \end{align*}

$\displaystyle{\frac {4} {5}}$

Question 9:

Express $70\,\%$ as a decimal.

Solution:

\begin{align*} 70\, \% &= \frac {70} {100} \\[2ex] &= 0.7 \end{align*}

$0.7$

Question 10:

Express $550\,\%$ as a decimal.

Solution:

\begin{align*} 550 \% &= \frac {550} {100} \\[2ex] &= 5.5 \end{align*}

$5.5$

Question 11:

Express $0.2\%$ as a fraction in its simplest form.

Solution:

\begin{align*} 0.2 \% &= \frac {0.2} {100} \\[2ex] &= 0.002 \\[2ex] &= \frac {2} {1000} \\[2ex] &= \frac {1} {500} \\ \end{align*}

$\displaystyle{\frac {1} {500}}$

Question 12:

Express $0.5\%$ as a fraction in its simplest form.

Solution:

\begin{align*} 0.5 \% &= \frac {0.5} {100} \\[2ex] &= 0.005 \\[2ex] &= \frac {5} {1000} \\[2ex] &= \frac {1} {200} \\ \end{align*}

$\displaystyle{\frac {1} {200}}$

Question 13:

Express $\displaystyle{4\frac {1} {2}\%}$ as a decimal.

Solution:

\begin{align*} 4 \frac {1} {2} \% &= 4\frac {5} {10}\% \\[2ex] &= 4.5\% \\[2ex] &= \frac {4.5} {100} \\[2ex] &= 0.045 \\ \end{align*}

$0.045$

Question 14:

Express $\displaystyle1\frac{2}{5}\%$ as a decimal.

Solution:

\begin{align*} 1 \frac {2} {5} \% &= 1\frac {4} {10}\% \\[2ex] &= 1.4\% \\[2ex] &= \frac {1.4} {100} \\[2ex] &= 0.014 \end{align*}

$0.014$

### 2. Solve simple word problems using percentage

Question 1:

In a school, the ratio of the number of boys to the number of girls is $1:4$. What percentage of the pupils are boys?

Solution:

\begin{align} \text{Number of Boys} &: \text{Number of Girls}\\[2ex] 1 &: 4 \end{align}

Number of boys $= 1$ unit

Number of girls $= 4$ units

Total number of pupils $= 5$ units

Percentage of the pupils that are boys

$\displaystyle{= \frac {\text{Number of boys}}{\text{Total number of pupils}} \times 100\%}$

$\displaystyle{= \frac {1}{5} \times 100\%}$

$= 20\%$

$20\%$

Question 2:

There are $240$ pupils in Primary $5$. $180$ of them go to school by bus. What percentage of pupils go to school by bus?

Solution:

Total number of pupils $= 240$

Number of pupils who go to school by bus $= 180$

Percentage of pupils who go to school by bus

$\displaystyle{=\frac {\text{Number of pupils who go to school by bus}}{\text{Total number of pupils}} \times 100\%}$

$\displaystyle{= \frac {180}{240} \times 100\%}$

$= 75\%$

$75\%$

Question 3:

In a chicken farm, there are $320$ hens and $180$ roosters. $20\%$ of hens and $15\%$ of roosters have white feathers. What percentage of the chickens have white feathers?

Solution:

Number of hens $= 320$

Number of roosters $= 180$
Total number of chickens\begin{align}\\[2ex] &= 320 + 180\\[2ex] &= 500 \end{align}

Number of hens with white feathers

$= 20\%$ of hens

$\displaystyle{=\frac {20} {100} \times 320}$

$=64$

Number of roosters with white feathers
$= 15 \%$ of roosters
$\displaystyle{= \frac{15}{100} \times 180}$
$= 27$

Total number of chickens with white feathers

$= 64 + 27$

$= 91$

Percentage of chickens with white feathers

$\displaystyle{=\frac {\text{Number of chickens with white feathers}}{\text{Total number of chickens}} \times 100\%}$

$\displaystyle{ = \frac{91}{500} \times 100\%}$

$\displaystyle{= 18.2\%}$

$18.2\%$

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