Percentage - How to Calculate, Formula, Tricks and Calculator

Percentage is a very important part of Mathematics. Understanding percentages or calculating percentages is crucial for interpreting data in various fields and making informed decisions in personal and professional life.

Percentage is a way to express a number as a fraction of 100. The word “Percent” comes from the Latin word “Per centum”, which means “By the 100”. It is denoted by the symbol “%”. To calculate the percentage, we have to divide the part of the number by the whole number and then multiply it by 100.

Example: 25% or $\frac{25}{100}=\frac{1}{4}$, 50% of the class means half of the class, etc.

Percentages can also be represented in decimal or fraction form like 0.25%, 0.6%, etc.

In this chapter, we will learn how to find percentages in various fields, how a percentage calculator works, the percentage formula, how to calculate percentages and many more.

How to calculate the percentage?

If we want to calculate x% of y, then it is given by $\frac{\text{x} \times \text{y}}{100}$. This is the formula to calculate the percentage.
Let’s take an example to understand it better.

What is 25 percent of 200?

Here, x = 25 and y = 200. To calculate the percentage, we have to divide 25 by 100, and then multiply it by 200.

$\therefore$ Required percentage = $\frac{25}{100}×200$ = 150.

So, 25 is 12.5% of 200.

Concept of multiplying factor to calculate percentages

Multiplying factors to calculate percentages is an integral part of Percentages.

If we divide the percentage by 100, we will get the multiplying factor. Then we can multiply the number with the multiplying factor to get the desired result.

Let’s take an example to understand it better.

What is 30% of 300?

To get the multiplying factor, we will divide 30 by 100. Then multiply it by 300 to get the desired result.

Multiplying factor = $\frac{30}{100}=0.3$

$\therefore$ Required result = 300 × 0.3 = 90

So, 30% of 300 is 90.

A few examples are listed below:

Concept of fractional equivalent for calculating percentages

In mathematics, calculating percentages using fractional equivalents can provide us with a straightforward and often more intuitive way to handle percentage problems.

If we convert fractions into percentages, it will simplify many problems and it will be often easy to calculate.

To convert fractions into percentages, we have to multiply the fraction by 100. Then we will get the fractional equivalent percentage.

To do the other way around, divide the percentage by 100 to get the percentage equivalent to a fraction.

Let’s take two examples to understand both ways better.

What percentage is equivalent to $\frac{1}{25}$?

Here, to find the percentage, we just have to multiply the fraction by 100.

$\therefore$ Required percentage = $\frac{1}{25}×100$ = 4%

Now, we look at an example of the vice-versa situation.

Convert 30% into fraction.

Here, to find the required fraction, we have to divide the given percentage by 100.

$\therefore$ Required fraction = $\frac{30}{100}=\frac{3}{10}$

Here is a list of some very important fractions equivalent to percentages.

Types of Percentages Questions:

Type 1: Finding the percentage

To calculate x% of y, we have to divide x by 100 and multiply it with y to get the result.

The general formula is: Required value = $\frac{x}{100}×y$

Let’s take an example to understand it better.

Find 10% of 120.

Using the formula, we get,

Require value = $\frac{10}{100}×120$ = 12

Also, x% of y = y% of x

Let’s prove this.

So, $\frac{x}{100}×y = \frac{y}{100}×x$

⇒ $\frac{xy}{100}=\frac{xy}{100}$

Therefore, the above assumption is true.

To understand better let’s check if 120% of 10 is equal to 10% of 120.

10% of 120 = 12 [Calculated above]

Now, 120% of 10 = $\frac{120}{100}×10$ = 1.2 × 10 = 12

So, 120% of 10 = 10% of 120

Type 2: Comparing two values in terms of percentages

To find x is what percentage of y, we have to divide x by y and multiply it by 100.

General formula = $\frac{x}{y}×100$

Let’s take an example to understand it better.

Find out 50 is what percentage of 500?

First, we will divide 50 by 500, then multiply by 100 to get the result.

$\therefore$ Required percentage = $\frac{50}{500}×100$ = 10%

Type 3: Percentage change

To determine whether x is what percentage more or less than y, we have to first subtract y from x. Then divide it by y and multiply it by 100 to get the required percentage.

General formula is: Required percentage = $\frac{x-y}{y}×100$

There will be two situations.

1. Where x > y, in that case, x will be more percentage than y.

2. Where x < y, in that case, x will be less percentage than y

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Let’s take two examples to understand it better.

If x = 150 and y = 120, then x is what percentage more than y?

We will use the above-mentioned formula to get the result.

$\therefore$ Required percentage = $\frac{150-120}{120}×100$ = 25%

So, 150 is 25% more than 120.

If x = 150 and y = 250, then x is what percentage less than y?

Again, we will use the above-mentioned formula to get the result.

$\therefore$ Required percentage = $\frac{150-250}{250}×100$ = -40%

So, 150 is 40% less than 250.

Type 4: Successive change

Successive percentage change is a very important topic in Percentage. As the name suggests, it means changing percentages consequently or one after another. Sometimes all increases, sometimes all decreases, sometimes both.

Example: His salary has increased by 10% in the first year and 20% in the second year.

His salary has increased by 10% in the first year and decreased by 5% in the second year.

Consecutive Increases

The General formula for this type of situation is:

The Multiplicating factor = $(1+\frac{x}{100})$, where x is the increase percentage

For successive times, we have to multiply the original number with the multiplicative factor multiple times.

Suppose the value “A” undergoes consecutive increases of x1, x2, x3, ......,xn, then

The final value
= A × $(1+\frac{x_1}{100})×(1+\frac{x_2}{100})×(1+\frac{x_3}{100})×......×(1+\frac{x_n}{100})$

Example:

Suppose a product is valued at 1000. Its price increased 10% first year and 20% 2nd year. So what is the final price?

Here, for the first year, multiplicating factor = $1+\frac{10}{100}$ = 1.1

For the first year, multiplicating factor = $1+\frac{20}{100}$ = 1.2

$\therefore$ Final value = 1000 × 1.1 × 1.2 = 1320

Consecutive decreases

General formula for this type of situation is:

Multiplicating factor = $(1-\frac{x}{100})$, where x is the decrease percentage

For successive times, we have to multiply the original number with the multiplicative factor multiple times.

Suppose the value “A” undergoes consecutive decreases of x1, x2, x3, ......,xn, then

The final value
= A × $(1-\frac{x_1}{100})×(1-\frac{x_2}{100})×(1-\frac{x_3}{100})×......×(1-\frac{x_n}{100})$

Example:

Suppose a product is valued at 1000. Its price decreased 10% first year and 20% 2nd year. So what is the final price?

Here, for the first year, multiplicating factor = $1-\frac{10}{100}$ = 0.9

For the first year, multiplicating factor = $1-\frac{20}{100}$ = 0.8

$\therefore$ Final value = 1000 × 0.9 × 0.8 = 720

First increase, then decrease

For this type of question, we have to find the separate multiplicative factors first, then multiply them with the original value to get the final result.

Suppose the value of A first increases x%, then decreases y%,

The final value = A × $(1+\frac{x}{100})×(1-\frac{x}{100})$

Example:

Suppose a product is valued at 1000. Its price increased 10% first year and decreased 20% 2nd year. So what is the final price?

Here, for the first year, the multiplicating factor = $1+\frac{10}{100}$ = 1.1

For the first year, multiplicating factor = $1-\frac{20}{100}$ = 0.8

$\therefore$ Final value = 1000 × 1.1 × 0.8 = 880

First decrease, then increase

For this type of question, we have to find the separate multiplicative factors first, then multiply them with the original value to get the final result.

Suppose the value of A first decreases x%, then increases y%,

The final value = A × $(1-\frac{x}{100})×(1+\frac{x}{100})$

Example:

Suppose a product is valued at 1000. Its price decreased 10% first year and increased 20% 2nd year. So what is the final price?

Here, for the first year, the multiplicating factor = $1-\frac{10}{100}$ = 0.9

For the first year, multiplicating factor = $1+\frac{20}{100}$ = 1.2

$\therefore$ Final value = 1000 × 0.9 × 1.2 = 1080

Percent point

A percentage point is a simple and precise way to express the absolute change between two percentages.

General formula is: Percent point = New percentage - Old percentage

It is different from percentage change.

Percentage change gives us a relative change of two values, as a percentage of the original value.

Percentage change formula is: $\frac{\text{New percentage - Old percentage}}{\text{Old percentage}}×100$

Let’s take an example to understand it better.

Suppose Kushal’s savings in the last month were 10%. This month he saved 13%.

So, here Kushal’s savings increase (13% - 10%) = 3 percent points

Application of percentage

• The application of percentages can be seen in elections. The winner gets more percentage of votes than the loser.

• Percentages are used in mixture and population. To count what percentage of one element is mixed with another element or how much percentage of the population increases or decreases in a certain place we use percentages.

• Percentages are used in our day-to-day life like in our income and expenditure. To calculate what percent of our salary is spent on which purpose and at the end of the month what percent of the salary is saved, we used percentages.

• Percentages are used to represent exam scores and grades. Also to track attendance rate or to count how many percentages of boys and girls are there.

• Percentages are also used in simplification and approximation.

Tips and Tricks

• Multiply the fractions by 100 to get the percentage easily.

• During the calculation of successive percentage change, use the multiplication factor to get the result quickly.

• x% of y is always equal to y% of x.

• If you are given a part of a number and you have to calculate the percentage, then divide it by the whole number and then multiply it by 100 to get the percentage.

Solved Examples

Q1. 5% of $a = b$, then $b$% of 20 is the same as:

1. 20% of $\frac{a}{2}$

2. 50% of $\frac{a}{20}$

3. 50% of $\frac{a}{2}$

4. 20% of $\frac{a}{20}$

Hint: Use the concept: $a$% of $b$ = $b$% of $a$

Answer:

Given: 5% of $a = b$

⇒ b = $\frac{5}{100} × a = \frac{a}{20}$

Now, $b$% of 20 = 20% of $b$ = 20% of $\frac{a}{20}$

Hence, the correct answer is 20% of $\frac{a}{20}$.

Q2. 0.06% of 250% of 1600 is:

1. 24

2. 0.24

3. 0.024

4. 2.4

Hint: Use the concept: $x$% of $y$ = $\frac{x×y}{100}$

Answer:

Given: 0.06% of 250% of 1600

= $\frac{(1600×250×0.06)}{(100×100)}$

= $\frac{(16×6)}{40}$

= $\frac{24}{10}$

= 2.4

Hence, the correct answer is 2.4.

Q3. If the salary of Manoj is 40% less than that of Subhash, then by what percentage is the salary of Subhash more than that of Manoj?

1. 60%

2. $66\frac{1}{4}$%

3. $66\frac{2}{3}$%

4. 65%

Hint: Required percentage = $\frac{\text{Difference in salary}}{\text{Salary of Manoj}} × 100$

Answer:

Let Subhash's salary be Rs. 100.

The salary of Manoj is 40% less than that of Subhash

Then, Manoj's salary = (100 – 40% of 100) = Rs. 60

So, the required Percentage = $\frac{\text{Difference in salary}}{\text{Salary of Manoj}} × 100$%

= $\frac{40}{60} $× 100%

= $\frac{200}{3}$%

= $66\frac{2}{3}$%

Hence, the correct answer is $66\frac{2}{3}$%.

Q4. There are 1400 students in a school, 25% of them wear spectacles and $\frac{2}{7}$th of those wearing spectacles are boys. How many girls in the school wear spectacles?

1. 250

2. 100

3. 200

4. 300

Hint: Calculate the number of boys wearing spectacles, then subtract that from the number of students wearing spectacles.

Answer:

Total number of students = 1400

Number of students wearing spectacles $= 1400 × \frac{25}{100} = 350$

Number of boys wearing spectacles $= 350 × \frac{2}{7} = 100$

Therefore, the number of girls wearing spectacles $= 350 -100 = 250$

Hence, the correct answer is 250.

Q5. If a number is increased by 84, it becomes 107% of itself. What is the number?

1. 600

2. 900

3. 1500

4. 1200

Hint: Let y be the number. If 84 is added to y, it becomes 107% of y, i.e. y + 84 = 107% of y

Answer:

Given: If a number is increased by 84, it becomes 107% of itself.

Let the number be y.

According to the question,

y + 84 = 107% of y

⇒ y + 84 = 1.07y

⇒ 84 = 1.07y – y

⇒ 84 = 0.07y

⇒ y = 1200

Hence, the correct answer is 1200.

Q6. A man wills 40% of his wealth to his wife and the rest to his children. What percentage of the wealth willed to the wife do the children get?

1. 150%

2. 66.6%

3. 50%

4. 20%

Hint: Let the total wealth be $x$. The man wills 40% of his wealth to his wife i.e. 40% of $x$ i.e. 0.4$x$.

Answer:

Let the total wealth be $x$.

The man wills 40% of his wealth to the wife = 40% of $x$ = 0.4$x$

Wealth for the children = $x$ – 0.4$x$ = 0.6$x$

So, the required percentage $=\frac{\text{wealth for children}}{\text{wealth to the wife}} × 100 = \frac{0.6x}{0.4x} ×100 = 150$%

Hence, the correct answer is 150%.

Q7. In a motor with 120 machine parts, 5% of the parts were defective. In another motor with 80 machine parts, 10% of the parts were defective. For the two motors considered together, the percentage of defective machine parts was:

1. 8

2. 7.5

3. 6.5

4. 7

Hint: Divide the total number of defective parts by the total number of machine parts and multiply by 100.

Answer:

We have,

Defective parts in 120 machine parts = 5% of 120 = 6

Defective parts in 80 machine parts = 10% of 80 = 8

Total number of defectives parts = 6 + 8 = 14

Total number of machine parts = 120 + 80 = 200

Total percentage of defective parts in the two motors = $\frac{14}{200}$ × 100 = 7%

Hence, the correct answer is 7.

Q8. If a number is increased by 25% and the resulting number is decreased by 25%, then the percentage increase or decrease finally is:

1. decreased by $6\tfrac{1}{4}$%

2. No change

3. increased by $6\tfrac{1}{4}$%

4. increased by 6%

Hint: First, assume the number to be 100 and then solve as per given conditions to find the desired value.

Answer:

Given: A number is increased by 25%.
The resulting number is decreased by 25%.
Let the number be 100.
Increased by 25% = (100 + 25) = 125
125 is decreased by 25%, then it becomes = $125×\frac{75}{100}$ = $\frac{375}{4}$
So, the required decrease percentage

$=(\frac{100-\frac{375}{4}}{100}) ×100$%
= $\frac{25}{4}$%
= $6\tfrac{1}{4}$%

Hence, the correct answer is 'decreased by $6\tfrac{1}{4}$%'.

Q9. In a factory, the production of cycles rose to 48,400 from 40,000 in 2 years. The rate of growth per annum is:

1. 9%

2. 8%

3. 10.5%

4. 10%

Hint: Assume the rate is R. Then use this formula:

Production after 2 years = past production $(1+\frac{\text{R}}{100})^2$

Answer:

The production of cycles rose to 48400 from 40000 in 2 years.

Past production = 40000

Production after two years = 48400

Time = 2 years

Let $R$ be the rate of growth per annum.

According to the question,

Production after 2 years = past production $(1+\frac{\text{R}}{100})^2$

⇒ $48400 = 40000 (1+\frac{\text{R}}{100})^2$

⇒ $\frac{48400}{40000} = (1+\frac{\text{R}}{100})^2$

⇒ $\frac{484}{400} = (1+\frac{\text{R}}{100})^2$

⇒ $(\frac{22}{20})^2 = (1+\frac{\text{R}}{100})^2$

⇒ $\frac{22}{20} = 1+\frac{\text{R}}{100}$

⇒ $\frac{22}{20}-1 = \frac{\text{R}}{100}$

⇒ $\frac{2}{20} = \frac{\text{R}}{100}$

⇒ $R = 10$%

Hence, the correct answer is 10%.

Q10. Peter bought an item at a 20% discount on its original price. He sold it with a 40% increase on the price he bought it. The new sale price is greater than the original price (in percent) by:

1. 10%

2. 8%

3. 7.5%

4. 12%

Hint: Let the original price be Rs. 100 and use the formula,

Percentage Change = $\frac{\text{Final price – Initial price}}{\text{Initial price}}\times100$

Answer:

Let the original price be Rs. 100.

The cost price after 20% discount will be Rs. 80.

The selling price = 140% of Rs. 80 = $\frac{140}{100}\times80$ = Rs. 112

Percentage change = $\frac{112-100}{100}\times100$ = 12%

Hence, the correct answer is 12%.

Q11. A number is first decreased by 20%. The decreased number is then increased by 20%. The resulting number is less than the original number by 20. Then the original number is:

1. 200

2. 400

3. 500

4. 600

Hint: Let $x$ be the initial value then use the concept of percentage and the given information to get the required value.

Answer:

Let $x$ be the initial value.

The number becomes 0.8$x$ after being reduced by 20%.

The number becomes 1.2 × (0.8$x$) = 0.96$x$ after increasing by 20%. This is 20 less than the initial number.

According to the question,

0.96$x$ = $x$ – 20

⇒ 0.04$x$ = 20

⇒ $x$ = 500

Hence, the correct answer is 500.