## Percentages

### 1. Introduction

Percentages is applied in several chapters in math (including Profit & Loss, Interest & Growth, Ratio & Proportion). Quite a few Data Interpretation questions include percentage in some form.

% is the symbol used to denote percentage. For instance, $40$ percent is written as 40%.

The value of this symbol $\% = \dfrac{1}{100}$.

Note that $100\% = 100 \times \dfrac{1}{100} = 1$

Percentage is useful in comparing values, which can be observed in the following example.

### Example 1

Ram scored 33, 39 and 54 marks in Maths, Science and English tests. If the total marks for these tests were 75, 91 and 120 respectively, in which subject did he score the lowest percentage?

### Solution

Percentage of marks in

Maths $= \dfrac{33}{75} \times 100\% = 44\%$

Science $= \dfrac{39}{91} \times 100\% = 42.86\%$

English $= \dfrac{54}{120} \times 100\% = 45\%$

### 2. Conversions

#### 2.1 Conversion of Decimal to Percent

To convert a decimal or fraction, we simply multiply it by 100 and retain the '%' symbol.

### Example 2

Express 0.0023, 0.23 and 230 as a percentage.

### Solution

$0.0023 = 0.0023 \times 1 = 0.0023 \times 100\% = 0.23\%$

$0.23 = 0.23 \times 100\% = 23\%$

$230 = 230 \times 100\% = 23000\%$

In short, the number written with the % symbol is 100 times the number when written without %.

#### 2.2 Conversion of Fraction to Percent

In case of fraction, the output can be written as a proper fraction, mixed fraction or as a near decimal.

### Example 3

Express $\dfrac{2}{11}$ as a percentage.

### Solution

$\dfrac{2}{11} = \dfrac{2}{11} \times 100\%$

$= \dfrac{200}{11} \%$ (Proper Fraction)

$= 18\frac{2}{11} \%$ (Mixed Fraction)

$\sim 18.18\%$ (Near Decimal)

#### 2.3 Conversion of Ratio to Percent

Ratio is a relation between two quantities.

In these questions, we are typically asked to find the percentage of an item in the total.

### Example 4

The ratio of copper to iron in an alloy is 2 : 3. What percent of the alloy is copper?

### Solution

Copper content in the alloy is $2$ parts for every $5$ parts.

Copper $\%$ in the alloy $= \dfrac{2}{5} \times 100\% = 40\%$

Now, it is correct to state that $40\%$ of the alloy is copper.

#### 2.4 Reconversion from Percent

Conversion from percent is easy. Simply replace % with $\dfrac{1}{100}$.

### Example 5

Express 45% as a decimal.

### Solution

$45\% = 45 \times \dfrac{1}{100} = \dfrac{9}{20} = 0.45$