In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100, which means, a part per hundred. We often represent percentage by** “%”**. It is to be noted that, percentages have no dimension, hence are dimensionless numbers.

Two quantities are generally expressed on the basis of their **ratios****. **Let us understand the concepts of percentage through example in a much better way.

Examples: Let a bag contain 2 kg Apples and 3 grapes in a bag. Find the ratio of quantities present, and percentage occupied by each.
The actual interpretation of percentages can be understood by the following way: The same quantity can be represented in terms of percentage occupied, which is given as: Total quantity present = 5 kg Ratio of apples (in terms of total quantity) = \(\large \frac{2}{5}\) = \(\large \frac{2}{5} \times \frac{100}{100}\) From the definition of percentage, it is the ratio that is expressed per hundred, \(\large \frac{1}{100} = 1\)% Thus, Percentage of Apples = \(\large \frac{2}{5} \times 100 = 40\) Percentage of Grapes = \(\large \frac{3}{5} \times 100 = 60\) |

## Converting Fractions to Percentage:

A fraction can be represented by \(\large \frac{a}{b}\).

Multiplying and dividing the fraction by 100, we have

\(\large \frac{a}{b} \times \frac{100}{100}\)

\(\large =\left ( \frac{a}{b} \times 100 \right ) \frac{1}{100}\) ………………(i)

From the definition of percentage, we have \(\large = \frac{1}{100}\) = 1%

Thus equation (i) can be written as:

\(\large = \frac{a}{b} \times 100\)%

Thus fraction can be converted to percentage simply by multiplying the given fraction by 100.

## Difference between Percentage and Percent:

The word percentage and percent are related closely to each other.

Percent ( or symbol %) is accompanied with a specific number.

*Eg: *More than 75% of the participants responded with their positive response to abjure.

Percentage is represented without a number, is used generally for general case of percent.

*Eg: *The percentage of the population affected by malaria is between 60% and 65%.

Fractions, Ratios, Percents and Decimals are interrelated with each other. Let us look on to the conversion of one form to other:

S.no | Ratio | Fraction | Percent(%) | Decimal |

1 | 1:1 | 1/1 | 100 | 1 |

2 | 1:2 | 1/2 | 50 | 0.5 |

3 | 1:3 | 1/3 | 33.333 | 0.3333 |

4 | 1:4 | 1/4 | 25 | 0.25 |

5 | 1:5 | 1/5 | 20 | 0.20 |

6 | 1:6 | 1/6 | 16.667 | 0.16667 |

7 | 1:7 | 1/7 | 14.285 | 0.14285 |

8 | 1:8 | 1/8 | 12.5 | 0.125 |

9 | 1:9 | 1/9 | 11.111 | 0.11111 |

10 | 1:10 | 1/10 | 10 | 0.10 |

11 | 1:11 | 1/11 | 9.0909 | 0.0909 |

12 | 1:12 | 1/12 | 8.333 | 0.08333 |

13 | 1:13 | 1/13 | 7.692 | 0.07692 |

14 | 1:14 | 1/14 | 7.142 | 0.07142 |

15 | 1:15 | 1/15 | 6.66 | 0.0666 |

## Every percent problem has three possible unknowns, or variables :

- The percent

- The part

- The base

In order to solve any percent problem, you must be able to identify these variables.

Look at the following examples. All three variables are known:

Example : 70% of 30 is 21

70 is the percent.

30 is the base.

21 is the part.

Example : 25% of 200 is 50

25 is the percent.

200 is the base.

50 is the part.

Example : 6 is 50% of 12

6 is the part.

50 is the percent.

12 is the base.

## Calculation Tricks for percent:

x % of y = y % of x |

Example- Prove that 10% of 30 is equal to 30% of 10.
Solution- 10% of 30 = 3 30% of 10 = 3 Therefore they are equal i.e. x % of y = y % of x holds true. |

## Problems on Percent:

Example- Suman has a monthly salary of $1200. She spends $280 per month on food. What percent of her monthly salary does she save?
Savings of Suman = Fraction of salary she saves = \(\large \frac{920}{1200} \times 100\) Percentage of salary she saves = \(\large \frac{920}{1200} \times 100 = \frac{920}{12} = 76.667\) %
(i) 0 dark and 100 white. i.e. 0 per 100 or 0%. (ii) 50 dark and 50 white. I.e. 50 per 100 or 50%. (iii) 100 dark and 0 white i.e. 100 per 100 or 100%. |

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