Percentage

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Percentage

A percentage is a fraction of 100. It is a way of expressing a number as a fraction of a whole, where the whole is considered as 100. The term ‘percentage’ comes from the Latin per centum, meaning “by the hundred”.

The term percent is sometimes abbreviated as p.c. The symbol % is often used for the term percent.

Basic formulas

Percentage Change = (\frac{New Value - Old Value}{Old Value}) \times 100 %

Shortcuts

1. To find out by how much B is less than A when A is x% more than B, you can use the following shortcut:

Difference = (\frac{x}{100 + x}) \times 100 %

Example:

Let’s say A is 20% more than B, then B is less than A by:

Difference = (\frac{20}{100 + 20}) \times 100 % = (\frac{20}{120}) \times 100 % = (\frac{1}{6}) \times 100 % = 16.67 %

So, B is 16.67% less than A when A is 20% more than B.

2. To find out by how much B is more than A when A is x% less than B, you can use the following shortcut:

Difference = (\frac{x}{100 - x}) \times 100 %

Example:

Let’s say A is 20% less than B, then B is more than A by:

Difference = (\frac{20}{100 - 20}) \times 100 % = (\frac{20}{80}) \times 100 % = (\frac{1}{4}) \times 100 % = 25 %

So, B is 25% more than A when A is 20% less than B.

Let’s use the following formulas to find the reduction in consumption when the price increases and the increase in consumption when the price decreases:

3. Reduction in Consumption when Price Increases by

$P %$ :

Reduction in Consumption = (\frac{P}{100 + P}) \times 100 %

4. Increase in Consumption when Price Decreases by

$P %$ :

Increase in Consumption = (\frac{P}{100 - P}) \times 100 %

Example: Calculation for Increase in Price:

Let’s say the price of a commodity decreases by 20%.

Increase in Consumption = (\frac{20}{100 - 20}) \times 100 %

Increase in Consumption = (\frac{20}{80}) \times 100 %

Increase in Consumption = 0.25 \times 100 % = 25 %

So, to keep the expenditure constant, the consumption should be increased by 25%.

5. Net Percentage Change Formula:

Net Percentage Change = x + y + \frac{x y}{100} %

Example:

Let’s say a number is successively increased by 20% and then decreased by 10%.

Net Percentage Change = 20 % - 10 % + \frac{20 % \times (- 10 %)}{100} %

Net Percentage Change = 10 % + \frac{- 2 %}{10} %

Net Percentage Change = 10 % - 0.2 %

Net Percentage Change = 9.8 %

So, the net percentage change is a decrease of 9.8%.

To find the population or value of an item after years when the population or value changes at per annum, you can use the following formula:

6. Future Value Formula:

Future Value = P \times {(1 + \frac{r}{100})}^{n}

Example:

Let’s say the present population of a town is $P = 1000$ and it changes at per annum.

Calculation for Population after 3 years:

Future Value = 1000 \times {(1 + \frac{5}{100})}^{3}

Future Value = 1000 \times (1.05)^{3}

Future Value = 1000 \times 1.157625

Future Value = 1157.625

So, the population of the town after 3 years would be approximately 1157.625.

Calculation for Value of Item after 2 years:

Let’s say the present value of an item is and it changes at per annum.

Future Value = 500 \times {(1 + \frac{8}{100})}^{2}

Future Value = 500 \times (1.08)^{2}

Future Value = 500 \times 1.1664

Future Value = 583.2

So, the value of the item after 2 years would be approximately $583.20.

Exercise of percentages

1. If a number is increased by 20%, by what percent should the increased number be decreased to get back to the original number?

Solution:
Using the formula:

Net Percentage Change = x + y + \frac{x y}{100}

Here, x $= 20 %$ and $y = - 20 %$

Net Percentage Change = 20 % - 20 % + \frac{20 % \times (- 20 %)}{100} = 20 % - 20 % - 4 % = - 4 %

So, the increased number should be decreased by 4% to get back to the original number.

2. The population of a town increases by 5% annually. If the present population is 4000, what will be the population after 2 years?

A) 4410
B) 4200
C) 4412
D) 4150

Solution:

Future Value = 4000 \times {(1 + \frac{5}{100})}^{2} = 4000 \times 1.1025 = 4410

Answer: A) 4410

3. A commodity is sold at a profit of 25%. If the cost price increases by 20%, by how much percent should the selling price increase to maintain the same profit percentage?

Solution:
Using the formula:

Net Percentage Change = x + y + \frac{x y}{100}

Here, and

Net Percentage Change = 25 % + 20 % + \frac{25 % \times 20 %}{100} = 45 % + 5 % = 50 %

So, the selling price should increase by 50% to maintain the same profit percentage.

4. If the price of a book is decreased by 10% and then increased by 10%, what is the net percentage change in the price of the book?

Solution:

Net Percentage Change = x + y + \frac{x y}{100}

Here, and

Net Percentage Change = - 10 % + 10 % + \frac{- 10 % \times 10 %}{100} = 0 % + 1 % = 1 %

5. A number is 25% less than another number. By what percent is the second number more than the first number?

Solution:
Using the formula:

Difference = (\frac{x}{100 + x}) \times 100 %

Here,

Difference = (\frac{25}{100 + 25}) \times 100 % = (\frac{25}{125}) \times 100 % = 20 %

6. The price of a toy increases by 15%. If the new price is $115.50, what was its original price?

Solution:
Let the original price be .

P \times (1 + \frac{15}{100}) = 115.50

P \times 1.15 = 115.50

P = \frac{115.50}{1.15} = 100

7. A student scored 480 marks out of 600. What is the percentage of marks obtained by the student?

Solution:

Percentage = (\frac{Marks obtained}{Total marks}) \times 100 %

Percentage = (\frac{480}{600}) \times 100 % = 0.8 \times 100 % = 80 %

8. If the price of an article is decreased by 20% and then increased by 25%, what is the net percentage change in the price?

Solution:

Net Percentage Change = x + y + \frac{x y}{100}

Here, and

Net Percentage Change = - 20 % + 25 % + \frac{- 20 % \times 25 %}{100} = 5 % - 5 % = 0 %

9. A discount of 15% on a shirt results in a selling price of $850. What was the original price of the shirt?

Solution:
Let the original price be .

P \times (1 - \frac{15}{100}) = 850

P \times 0.85 = 850

P = \frac{850}{0.85} = 1000

10. The price of a car is increased by 12%. If the increased price is $56,000, what was its original price?

Solution:
Let the original price be .

P \times (1 + \frac{12}{100}) = 56, 000

P \times 1.12 = 56, 000

P = \frac{56, 000}{1.12} = 50, 000

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Percentage

Table of Contents

Percentage

Basic formulas

1. Percentage Change:

2. Percentage of a Whole:

3. Finding the Part:

4. Finding the Whole:

5. Percentage Increase and Decrease:

6. Simple Interest:

7. Percent Difference:

8. Converting Percentages to Fractions and Decimals:

Shortcuts

Exercise of percentages

Related Links

MCQ’s

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