# Percentage

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.

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.
Solution: The quantity of apples and grapes in a bag can be compared in terms of their ratio, i.e. 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?
Solution- Suman’s monthly salary = $1200 Savings of Suman = $(1200 – 280) = $ 920 Fraction of salary she saves = \(\large \frac{920}{1200} \times 100\) Percentage of salary she saves = \(\large \frac{920}{1200} \times 100 = \frac{920}{12} = 76.667\) % Example- Below given are three grids of chocolate. What percent of each White chocolate bar has Dark chocolate bar? Solution- Each grid above has 100 white chocolate blocks. For each white chocolate bar, the ratio of the number of dark chocolate boxes to the total number of white chocolate bars can be represented as a fraction. (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%. |