Study S1 Mathematics Maths - Percentage - Geniebook

# Percentage

In this chapter, we will be discussing the below-mentioned topics in detail:

• Fractions, Ratio, Decimals and Percentage
• Convert a non-integer percentage to fraction/ decimal and vice versa
• Percentages greater than $$100\%$$ or less than $$1\%$$
• Percentage of quantities
• Percentage of a quantity
• Express one quantity as a percentage of another quantity
• Compare two quantities by percentage
• Percentage change

## Fractions, Ratio, Decimals and Percentage

Let’s understand this with the help of some examples:

#### Decimals

\begin{align} \frac{2}{5} \end{align} $$2:5$$ \begin{align} \frac{2}{5} \times 100 = 40\% \end{align} \begin{align*} 40\% &= \frac{40}{100}\\ \\ &= 0.4 \end{align*}
\begin{align} 25\% &= \frac{25}{100}\\\\ &= \frac{1}{4} \end{align} $$1:4$$ $$25\%$$ $$0.25$$
\begin{align} \frac{90}{100} = \frac{9}{10} \end{align} $$9:10$$ \begin{align} 0.9 \times 100\% = 90\% \end{align} $$0.9$$

## Percentages Greater Than $$100\%$$ or Less Than $$1\%$$

Let’s understand this with the help of some examples:

Question 1:

Without using a calculator, express each of the following percentages as a fraction.

1. $$105\%$$
2. $$0.04\%$$

Solution:

1.

\begin{align*} 105\% &= \frac{105}{100}\\ &= \frac{21}{20}\\ &= \frac{11}{20} \end{align*}

1.

\begin{align*} 0.04\% &= \frac{0.04 × 100}{100 × 100}\\ &= \frac{4}{10000}\\ &= \frac{1}{2500} \end{align*}

Question 2:

Without using a calculator, express each of the following percentages as a decimal.

1. $$274\%$$
2. $$0.38\%$$

Solution:

1.

\begin{align*} 274\% &= \frac{274}{100}\\ \\ &= 2.74 \\ \\ \end{align*}

1.

\begin{align*} 0.38\% &= \frac{0.38}{100}\\ \\ &= 0.0038\\ \end{align*}

## Percentage Of A Quantity

Let’s understand this with the help of some examples:

Question 3:

A box contains $$60$$ balls, of which $$16$$ are red.

1. If $$40\%$$ of the balls are green, find the number of green balls.
2. The rest of the balls in the box are yellow.

If $$70\%$$ of the yellow balls are removed, find the number of yellow balls left.

Solution:

1. Number of green balls\begin{align} = \frac{40}{100} \times 60\\ \end{align}

$$\qquad\qquad\qquad\qquad\qquad= 24 \;balls$$

OR

Number of green balls $$= 0.4 × 60$$

$$\qquad\qquad\qquad\qquad\quad\;\;= 24 \;balls$$

1. Number of yellow balls at first\begin{align*} &= 60 \;– 16 \;– 24 \end{align*}

$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad= 20$$

Number of yellow balls remaining $$= \frac{30}{100} × 20$$

$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad= 6 \;balls$$

OR

Number of yellow balls remaining $$= 0.3 × 20$$

$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad= 6 \;balls$$

## Comparing two quantities by percentage

Let’s understand this with the help of some examples:

Question 4:

Shop $$A$$ sold $$225$$ out of $$1000$$ vinyl records in January. Shop $$B$$ sold $$400$$ out of $$1500$$ vinyl records in the same month. Which shop had a higher proportion of sales in January?

Solution:

Percentage of sales by Shop\begin{align} A=\frac{225}{1000} \times 100 \end{align}

\begin{align} \qquad\qquad\qquad\qquad\qquad\qquad\qquad=22.5\% \end{align}

Percentage of sales by Shop\begin{align} B= \frac{400}{1500} \times 100 \end{align}

\begin{align} \qquad\qquad\qquad\qquad\qquad\qquad\qquad= 26 \frac{2}{3}\% \end{align}

Shop $$B$$ had a higher proportion of sales in January.

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