## Percentages

The term “percentage” was adapted from the Latin word “per centum”, which means “by the hundred”. Percentages are fractions with 100 as the denominator. In other words, it is the relation between part and whole where the value of whole is always taken as 100.

## What is Percentage?

Percentage is a fraction or a ratio in which the value of whole is always 100. For example, if Sam scored 30% marks in his math test, it means that he scored 30 marks out of 100. It is written as 30/100 in the fraction form and 30:100 in terms of ratio.

**Percentage Definition:**

Percentage is defined as a given part or amount in every hundred. It is a fraction with 100 as the denominator and is represented by the symbol “%”.

## Calculation of Percentage

Calculating percentage means to find the share of a whole, in terms of 100. There are two ways to find a percentage:

- By using the unitary method.
- By changing the denominator of the fraction to 100.

It should be noted that the second method for calculating percentage is not used in situations where the denominator is not a factor of 100. For such cases we use the unitary method.

### How to get a Percentage?

Percent is another name for indicating hundredths. Thus, 1% is one-hundredth, that means 1%=1/100=0.01.

Let’s calculate percentage using the two methods given above.

When we have two or more values that add up to 100, then the percentage of those individual values to the total value is that number itself. For example, Sally bought tiles of three different colors for her house. The details of the purchase are given in the following table.

Colour | Number of Tiles | Rate per Hundred | Fraction | Written as | Read as |

Yellow | 39 | 39 | 39/100 | 39% | 39 percent |

Green | 26 | 26 | 26/100 | 26% | 26 percent |

Red | 35 | 35 | 35/100 | 35% | 35 percent |

Since the total number of items adds up to 100, the percentages can be easily calculated.

What if the total number of items do not add up to 100? In such cases, we convert the fractions to equivalent fractions with the denominator as 100.

For example, Emma has a bracelet which is made up of 20 beads of two different colours, red and blue. Observe the following table which shows the percentage of red and blue beads out of the 20 beads.

Emma’s sisters, Nora and Jenny, calculated the percentages as well, but in different ways.

Nora used the unitary method. Using the unitary method for calculating percentage, we say that out of 20 beads, the number of red beads are 8. Hence, out of 100, the number of red beads will be 8/20 × 100= 40%.

Jenny converted the fraction 8/20 into an equivalent fraction 40/100 by multiplying the numerator and denominator with 5/5.

So, 8/20= (8×5)/(20×5)

= 40/100

= 40%

### Formula to Calculate Percentage

The percentage formula is used to find the share of a whole in terms of 100. Using this formula, you can represent a number as a fraction of 100. If you observe carefully, all the three ways to get percentage shown above can be easily calculated by using the formula given below:

**Percentage= (Value/Total Value)×100**

**Percentage Difference Between Two Numbers**

Percentage difference is the change in the value of a quantity over a period of time in terms of percentage. Sometimes we need to know the increase or decrease in some quantity as percentages, which is also referred to as Percentage Change. For example, an increase in population, a decrease in poverty, and so on.

We have the formula to show the change in quantity as a percentage. There are two cases that might arise while calculating percentage difference and those are:

- Calculate percentage increase
- Calculate percentage decrease

### How to Calculate Percentage Increase?

Percentage increase refers to the perchange change in the value when it is increased over a period of time. For example, population increase, increase in the number of bacteria on a surface, etc. Percentage increase can be calculated by using the following formula:

**Percentage Increase= (Increased Value-Original value)/Original value × 100**

### How to Calculate Percentage Decrease?

Percentage decrease refers to the perchange change in the value when it is decreased over a period of time. For example, decrease in the level of rainfall, decrease in the number of Covid patients, etc. Percentage decrease can be calculated by using the following formula:

**Percentage Decrease= (Original value-Decreased Value)/Original Value × 100**

**Points to Remember:**

- To find the percentage of a whole, work out the value of 1% and then multiply it by the percent we need to find.
- An increase or decrease in any quantity can be expressed as a percentage.
- Fractions can be converted into percentages and vice-versa.
- Percentages are reversible. For example, 25% of 40 is the same as 40% of 25.

## How to Calculate Percentages

There are many formulas for percentage problems. You can think of the most basic as X/Y = P x 100. The formulas below are all mathematical variations of this formula.

Let’s explore the three basic percentage problems. X and Y are numbers and P is the percentage:

- Find P percent of X
- Find what percent of X is Y
- Find X if P percent of it is Y

Read on to learn more about how to figure percentages.

## 1. How to calculate percentage of a number. Use the percentage formula: P% * X = Y

Example: What is 10% of 150?

- Convert the problem to an equation using the percentage formula: P% * X = Y
- P is 10%, X is 150, so the equation is 10% * 150 = Y
- Convert 10% to a decimal by removing the percent sign and dividing by 100: 10/100 = 0.10
- Substitute 0.10 for 10% in the equation: 10% * 150 = Y becomes 0.10 * 150 = Y
- Do the math: 0.10 * 150 = 15
- Y = 15
- So 10% of 150 is 15
- Double check your answer with the original question: What is 10% of 150? Multiply 0.10 * 150 =
**15**

## 2. How to find what percent of X is Y. Use the percentage formula: Y/X = P%

Example: What percent of 60 is 12?

- Convert the problem to an equation using the percentage formula: Y/X = P%
- X is 60, Y is 12, so the equation is 12/60 = P%
- Do the math: 12/60 = 0.20
- Important! The result will always be in decimal form, not percentage form. You need to multiply the result by 100 to get the percentage.
- Converting 0.20 to a percent: 0.20 * 100 = 20%
- So 20% of 60 is 12.
- Double check your answer with the original question: What percent of 60 is 12? 12/60 = 0.20, and multiplying by 100 to get percentage, 0.20 * 100 =
**20%**

## 3. How to find X if P percent of it is Y. Use the percentage formula Y/P% = X

Example: 25 is 20% of what number?

- Convert the problem to an equation using the percentage formula: Y/P% = X
- Y is 25, P% is 20, so the equation is 25/20% = X
- Convert the percentage to a decimal by dividing by 100.
- Converting 20% to a decimal: 20/100 = 0.20
- Substitute 0.20 for 20% in the equation: 25/0.20 = X
- Do the math: 25/0.20 = X
- X = 125
- So 25 is 20% of 125
- Double check your answer with the original question: 25 is 20% of what number? 25/0.20 =
**125**

### Remember: How to convert a percentage to a decimal

- Remove the percentage sign and divide by 100
- 15.6% = 15.6/100 = 0.156

### Remember: How to convert a decimal to a percentage

- Multiply by 100 and add a percentage sign
- 0.876 = 0.876 * 100 = 87.6%

## Percentage Chart

The percentage chart is given here for fractions converted into percentage.

Fractions | Percentage |

1/2 | 50% |

1/3 | 33.33% |

1/4 | 25% |

1/5 | 20% |

1/6 | 16.66% |

1/7 | 14.28% |

1/8 | 12.5% |

1/9 | 11.11% |

1/10 | 10% |

1/11 | 9.09% |

1/12 | 8.33% |

1/13 | 7.69% |

1/14 | 7.14% |

1/15 | 6.66% |

## Difference between Percentage and Percent

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

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

*E.g., *More than 75% of the participants responded with their positive response to abjure.

The percentage is represented without a number.

*E.g., *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 |

## Percentage in Maths

Every percentage problem has three possible unknowns or variables :

- Percentage
- Part
- Base

In order to solve any percentage 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 percentage.

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.

## Percentage Tricks

To calculate the percentage, we can use the given below tricks.

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.

## Marks Percentage

Students get marks in exams, usually out of 100. The marks are calculated in terms of per cent. If a student has scored out of total marks, then we have to divide the scored mark from total marks and multiply by 100. Let us see some examples here:

Marks obtained | Out of Total Marks | Percentage |

30 | 100 | 30% |

10 | 20 | 50% |

23 | 50 | 46% |

13 | 40 | 32.5% |

90 | 120 | 75% |

## Percentage Problems

There are nine variations on the three basic problems involving percentages. See if you can match your problem to one of the samples below. The problem formats match the input fields in the calculator above. Formulas and examples are included.

### What is P percent of X?

- Written as an equation: Y = P% * X
- The ‘what’ is Y that we want to solve for
- Remember to first convert percentage to decimal, dividing by 100
- Solution: Solve for Y using the percentage formula
**Y = P% * X**

### Example: What is 10% of 25?

- Written using the percentage formula:
**Y = 10% * 25** - First convert percentage to a decimal 10/100 = 0.1
- Y = 0.1 * 25 = 2.5
- So 10% of 25 is 2.5

### Y is what percent of X?

- Written as an equation: Y = P% ? X
- The ‘what’ is P% that we want to solve for
- Divide both sides by X to get P% on one side of the equation
- Y ÷ X = (P% ? X) ÷ X becomes Y ÷ X = P%, which is the same as P% = Y ÷ X
- Solution: Solve for P% using the percentage formula
**P% = Y ÷ X**

### Example: 12 is what percent of 40?

- Written using the formula:
**P% = 12 ÷ 40** - P% = 12 ÷ 40 = 0.3
- Convert the decimal to percent
- P% = 0.3 × 100 = 30%
- So 12 is 30% of 40

### Y is P percent of what?

- Written as an equation: Y = P% * X
- The ‘what’ is X that we want to solve for
- Divide both sides by P% to get X on one side of the equation
- Y ÷ P% = (P% × X) ÷ P% becomes Y ÷ P% = X, which is the same as X = Y ÷ P%
- Solution: Solve for X using the percentage formula
**X = Y ÷ P%**

### Example: 9 is 60% of what?

- Writen using the formula:
**X = 9 ÷ 60%** - Convert percent to decimal
- 60% ÷ 100 = 0.6
- X = 9 ÷ 0.6
- X = 15
- So 9 is 60% of 15

### What percent of X is Y?

- Written as an equation: P% * X = Y
- The ‘what’ is P% that we want to solve for
- Divide both sides by X to get P% on one side of the equation
- (P% * X) ÷ X = Y ÷ X becomes P% = Y ÷ X
- Solution: Solve for P% using the percentage formula
**P% = Y ÷ X**

### Example: What percent of 27 is 6?

- Written using the formula:
**P% = 6 ÷ 27** - 6 ÷ 27 = 0.2222
- Convert decimal to percent
- P% = 0.2222 × 100
- P% = 22.22%
- So 22.22% of 27 is 6

### P percent of what is Y?

- Written as an equation: P% × X = Y
- The ‘what’ is X that we want to solve for
- Divide both sides by P% to get X on one side of the equation
- (P% × X) ÷ P% = Y ÷ P% becomes X = Y ÷ P%
- Solution: Solve for X using the percentage formula
**X = Y ÷ P%**

### Example: 20% of what is 7?

- Written using the formula:
**X = 7 ÷ 20%** - Convert the percent to a decimal
- 20% ÷ 100 = 0.2
- X = 7 ÷ 0.2
- X = 35
- So 20% of 35 is 7.

### P percent of X is what?

- Written as an equation: P% * X = Y
- The ‘what’ is Y that we want to solve for
- Solution: Solve for Y using the percentage formula
**Y = P% * X**

### Example: 5% of 29 is what?

- Written using the formula:
**5% * 29 = Y** - Convert the percent to a decimal
- 5% ÷ 100 = 0.05
- Y = 0.05 * 29
- Y = 1.45
- So 5% of 29 is 1.45

### Y of what is P percent?

- Written as an equation: Y / X = P%
- The ‘what’ is X that we want to solve for
- Multiply both sides by X to get X out of the denominator
- (Y / X) * X = P% * X becomes Y = P% * X
- Divide both sides by P% so that X is on one side of the equation
- Y ÷ P% = (P% * X) ÷ P% becomes Y ÷ P% = X
- Solution: Solve for X using the percentage formula
**X = Y ÷ P%**

### Example: 4 of what is 12%?

- Written using the formula:
**X = 4 ÷ 12%** - Solve for X: X = Y ÷ P%
- Convert the percent to a decimal
- 12% ÷ 100 = 0.12
- X = 4 ÷ 0.12
- X = 33.3333
- 4 of 33.3333 is 12%

### What of X is P percent?

- Written as an equation: Y / X = P%
- The ‘what’ is Y that we want to solve for
- Multiply both sides by X to get Y on one side of the equation
- (Y ÷ X) * X = P% * X becomes Y = P% * X
- Solution: Solve for Y using the percentage formula
**Y = P% * X**

### Example: What of 25 is 11%?

- Written using the formula:
**Y = 11% * 25** - Convert the percent to a decimal
- 11% ÷ 100 = 0.11
- Y = 0.11 * 25
- Y = 2.75
- So 2.75 of 25 is 11%

### Y of X is what percent?

- Written as an equation: Y / X = P%
- The ‘what’ is P% that we want to solve for
- Solution: Solve for P% using the percentage formula
**P% = Y / X**

### Example: 9 of 13 is what percent?

- Written using the formula:
**P% = Y / X** - 9 ÷ 13 = P%
- 9 ÷ 13 = 0.6923
- Convert decimal to percent by multiplying by 100
- 0.6923 * 100 = 69.23%
- 9 ÷ 13 = 69.23%
- So 9 of 13 is 69.23%

### Problems and Solutions

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 =
(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%. |

#### Word Problems

**Q.1: A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, he had how many apples?**

Solution: Let he had N apples, originally.

Now as per the given question,

(100 – 40)% of N = 420

⇒ (60/100)x N = 420

⇒ N = (420 x 100/60) = 700

**Q.2: Out of two numbers, 40% of the greater number is equal to 60% of the smaller. If the sum of the numbers is 150, then the greater number is?**

Solution: Let us assume, greater number be X.

∴ Smaller number = 150 – X

According to the question,

(40 x X)/100 = 60(150 – X)/100

⇒ 2p = 3 × 150 – 3X

⇒ 5X = 3 × 150

⇒ X = 90

## Solved Examples on Percentage

**Example 1:**

Robert got a 5% hike in his salary. His current salary is $70,000. Calculate his revised salary after promotion.

**Solution:**

Robert’s current salary = $70,000

5% hike in the salary means: 5% of 70,000= 5/100 × 70,000

5 × 700 = $3500

Robert’s pay hike is $3500.

Thus, his new salary will be $70,000 + $3500 = $73,500.

∴ Robert’s salary after his promotion will be $73,500.

**Example 2:**

Neil bought a new cell phone for $90. The value of the phone decreases by 3% on its original price every year. Find the value of his mobile after 3 years.

**Solution:**

3% of 90 is: $2.7

The phone depreciates by $2.7 every year.

Thus, the value of the mobile after 3 years will be: 90 – (3×2.7) = $81.9

∴ The value of the mobile after 3 years will be $81.9

### How to use the formula for percentage on the left.

**Example #1:**

25 % of 200 is ____

In this problem, **of** = 200, **is** = ?, and **%** = 25

We get:

**is**/200 = 25/100

Since **is** in an unknown, you can replace it by y to make the problem more familiar.

y/200 = 25/100

Cross multiply to get y × 100 = 200 × 25

y × 100 = 5000

Divide 5000 by 100 to get y

Since 5000/100 = 50, y = 50

So, 25 % of 200 is 50

**Example #2:**

What number is 2% of 50 ?

This is just another way of saying 2% of 50 is ___

So, set up the proportion as example #1:

**is**/50 = 2/100

Replace is by y and cross multiply to get:

y × 100 = 50 × 2

y × 100 = 100

Since 1 × 100 = 100, y = 1

Therefore, 1 is 2 % of 50

**Example #3:**

24% of ___ is 36

This time, notice that **is** = 36, but **of** is missing

After you set up the formula, you get:

36/**of** = 24/100

Replace of by y and cross multiply to get:

36/y = 24/100

y × 24 = 36 × 100

y × 24 = 3600

Divide 3600 by 24 to get y

3600/24 = 150, y = 150

Therefore, 24 % of 150 is 36

### How to use the other formula for percentage on the right.

Now, we will take examples to illustrate how to use the formula for percentage on the right

**Example #4:**

To use the other formula that says part and whole, just remember the following:

- The number after
**of**is always the whole. - The number after
**is**is always the part.

If a problem says 25 % of ___ is 60, then, we know that the whole is missing and the part = 60

Your proportion will like this:

60/**whole** = 25/100

After cross multiplying, we get:

**whole** × 25 = 60 × 100

**whole** × 25 = 6000

Divide 6000 by 25 to get whole

6000/25 = 240, so whole = 240

Therefore, 25 % of 240 is 60

**Example #5:**

___% of 45 is 9

Here whole = 45 and part = 9, but **%** is missing

We get:

9/45 = **%**/100

Replacing % by x and cross multiplying gives:

9 × 100 = 45 × x

900 = 45 × x

Divide 900 by 45 to get x

900/45 = 20, so x = 20

**FAQs on Percentage**

### How Do you Minus a Percentage?

To subtract some percentage from a number, just multiply that number by the percentage you want to retain. For example, to subtract 10% of 500, just multiply 90% by 500

### How to Calculate the Average Percentage?

Follow the steps to calculate the average percentage: The average percentage can be calculated by dividing the total items represented in percentages by the overall total of items. In other words,

Calculate the average percentage by dividing the total items represented by percentages by the overall total of items

- Convert the percentage into decimal numbers. For example, to calculate the average of 30% of 50 and 20% of 80, we convert them into their decimal forms that are 0.3 and 0.2 respectively.
- Write the number represented by each decimal number. In this case, it will be 0.3×50=15 and 0.2×80=16 respectively.
- Add the numbers thus obtained. (15+16=31).
- Find the sum of sample sizes. (50+80=130).
- Divide the total number obtained in Step 3 by the number obtained in Step 4. So, 31/130=0.24. This decimal number represents 24% which is the required average percenatge.

### How Do we Calculate Percentage?

Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. The formula used to calculate percentage is: (value/total value)×100%.

### What is Percentage of a Number?

Percentage of a number is the value of the number out of 100. For example, in a class there are 26 girls and 24 boys. So, the percentage of girls in the class is 52%, which means out of 100, 52 are girls.

### What is Percentage Change?

Percentage change is the change in percentage from the old value to the new value. It is calculated using the following formula: Percentage change= (difference between old and new values/old value)×100%

### What are Real Life Examples of Percentage?

Some real life examples of percentages are listed below:

- Your phone’s or laptop’s battery percentage.
- Percentage of nutrients on a food packet.
- Composition of oxygen, carbon-dioxide, nitrogen etc in air.
- Percentage of your marks in a test.
- Comparison of number of patients recovered from Covid between two or more cities is done in percentage etc.

### Can Percent be More Than 100?

Yes, percentage can be more than 100 when we have the value that is larger than the total value.

### What is the Formula for Percent into Decimal?

To convert percent to decimal, drop the percent symbol (%), divide it by 100, and write the decimal form of the fraction thus obtained.

### What do you mean by percentage?

In maths, a percentage is a value or ratio that shows a fraction of 100. Percent means per 100. It does not have any unit.

### What is the symbol of percentage?

Percentage is denoted by ‘%’ symbol. It is also termed as per cent.

### What is the percentage formula?

The formula to calculate percentage of a number out of another number is:

Percentage = (Original number/Another number) x 100

### What is the percentage of 45 out of 150?

(45/150) x 100 = 30%

### What is 40% of 120?

40% of 120

= 40/100 x 120

= 48

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