A percentage is a number or a ratio expressed in terms of fractions of 100. The concept of percentage allows us to determine a proportion of something in terms of its total value. It is used widely in educational institutes and business firms to showcase different statistics. Nearly all the other modules in the syllabus of competitive exams like GMAT and GRE are related to this topic in some way or another, and will utilise your knowledge of it.

Calculations

When you calculate percentages, you find the proportion of a whole in terms of 100. There are two methods of calculating percentages:

• By Unitary Method:

The motive of the unitary method is to find the value of a single unit and then multiply that value by the number of units to get the necessary result. If a certain percentage of an amount is given then the unitary method can help in finding 100% of that amount. For example,

5% of an amount is 700,

So 1% will be 140,——————-(divided by 5)

Therefore, 100% will be 14,000———–(multiplied by 100)

• By changing the denominator to 100:

This method works by multiplying both numerator and denominator with a number which brings 100 in the denominator. It can only be used when the denominator is a factor of 100. For example,  \frac{3}{7} can never have 100 in the denominator because 7 is not a factor of 100.

Converting Percentages into Fractions and Decimals

Percentages are fractions with 100 in their denominator. Let’s understand the conversion of percentages into fractions with easy examples.

1. 74% =  \frac{74}{100} = \frac{37}{50}

2. 35% = \frac{35}{100}= \frac{7}{20}

3. 30% = \frac{30}{100}= \frac{3}{10}

4. 25% = \frac{25}{100}= \frac{1}{4}

Percentages can be converted into decimals by making them fractions with 100 in the denominator. For example:

1. 74% = \frac{74}{100}= 0.74

2. 35% =\frac{35}{100}= 0.35

3. 30% =\frac{30}{100}= 0.3

4. 25% =\frac{25}{100}= 0.25

Expressing One Quantity as a Percentage of Another

The concept of percentage is used to determine the proportion of one quantity in another. Let’s understand this with the help of an example.

There are 80 students in class 6th A. 50 of them are boys and 30 are girls. Let’s calculate the percentage of boys and the percentage of girls in the class.

• Percentage of boys

Number of boys = 50

Total number of students = 80

Therefore, percentage of boys in the class is \frac{50}{80}X 100% = 62.5%

• Percentage of girls

Number of girls = 30

Total number of students = 80

Therefore, percentage of girls in the class is \frac{30}{80}X 100% = 37.5%

Percentage Change Comparison

The concept of percentage allows one to determine the increase or decrease in the value of a particular thing. Let’s understand this with the help of some examples.

• Percentage Increase

Sahil is an active buyer of stamps, on Monday he had 600 stamps with him. Throughout the week he bought stamps from different vendors. He checked his collection on Sunday and found he has 900 stamps.

Let’s find the percentage increase in stamps.

So, the stamps increased by 900-600 = 300

Therefore, the percentage increase will be \frac{300}{600} 100% =\frac{1}{2}X 100% = 50%

• Percentage Decrease

Paras is a farmer, he had 500 mangoes with him. He sold most of the mangoes to a vendor and kept 75 mangoes with him to distribute among his friends and family.

Let’s calculate the percentage decrease in mangoes after the trade.

So, the mangoes decreased by 500-75 = 425

Therefore, the percentage decrease will be \frac{425}{500}X 100% = \frac{17}{20}X 100% = 85%

Sample Questions

1. In a normal season sofa was sold for $200 and during the year end sale the same sofa was sold for$160. Mary purchased the sofa during the normal season, and Daniel purchased the sofa during the year end sale. By what percent the price that Mary paid was more than the price that Daniel paid?
1. 10%
2. 20%
3. 25%
4. 40%
5. 50%

Explanation:

Use the percentage change formula to solve the question,

Percentage change = \frac{Mary Amount - Daniel Amount}{Daniel Amount}*100=\frac{200 - 160}{160}*100=25%

1. In a class of 80 students, 62.5% are girls. If 40% of the boys and 10% of the girls signed up for a summer class then how many students didn’t sign up for the summer classes?
1. 40
2. 50
3. 60
4. 63
5. 64

Explanation:

Total number of students = 80

Girls = 62.5% (80) = \frac{62.5}{100}*80=50

Boys = 30

40% of the boys = \frac{40}{100}*30=12

10% of the girls = \frac{10}{100}*50=5

Number of students signed up for the summer class = 12 + 5 = 17

So, number of students didn’t sign up for the class = 80 – 17 = 63