The word percentage comes from the idea of “per hundred.” Whenever you see the term “percent” or the symbol %, it simply means “out of 100.” For example, 30% means 30 parts out of 100 parts in total. Percentages are a quick and easy way to compare quantities, even if the actual totals are different.

It’s important to note that when the % sign is used with a number or a value, we read it as “percent” (for example, 50 percent), not “percentage.”

In this guide, we will explain the main concepts of percentages in a simple way. You will also learn different types of percentage problems, along with useful tips and shortcuts. We have included solved examples so you can see how to apply these methods in practice. Going through these will help you understand percentages better and prepare for exams with confidence.

What is the Percentage of a Number?

In mathematics calculations, a percentage is a numeral or ratio that can be defined as a fraction of 100. In other words, we can say that the percentage is specified as a given fraction or part in every hundred. This implies that it is a fraction with 100 as the denominator and is commonly symbolised by the symbol “%” symbol.
For example, if we have to estimate the percent of a number, then divide the number by total and multiply it by 100. In a live test, Savita scored 45% marks, which means that she scored 45 marks out of 100.

Learn about Percentages in this video

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Percentage Formula

To hold a better command on percentage calculation we need to know all percentage formulas. The basic formula used to calculate the percentage is equivalent to the ratio of actual value to the complete value multiplied by 100. The formula of the percentages is expressed as:

\(\text{Percentage formula}=\left(\frac{\text{Actual value}}{\text{Total value}}\right)\times100\)
For Example: \(\frac{2}{4}\times100=0.5\times100=50\%\)

Percentage Difference Formula

The percentage difference can be understood as the change in the value of an amount over some time in terms of percentage. If there are two values and we need to determine the percentage difference between the given two values, then this can be calculated by the below steps:
Step 1: Compute the difference (i.e. subtract one value from the other) skip any negative sign if obtained.
Step 2: Estimate the average of the two values (add the values, then divide by 2).
Step 3: Finally divide the difference by the average obtained.
Step 4: Transform the obtained answer to a percentage for the result to be in percentages.

PercentageDifferenceFormula=\(\left|\frac{\text{FirstValue}−\text{SecondValue}}{\left(\frac{\text{First Value}+\text{Second Value}}{2}\right)}\right|\times100\%\)

The modulus symbols represent absolute value so that any negative outcome becomes positive.

Learn about Profit and Loss

Percentage Increase and Decrease Formula

Two cases might appear while computing percentage difference namely:
Percentage increase.
Percentage decrease.
Let us learn how to calculate both through the formula:
The percentage increase is equivalent to subtracting the original number from the new number and dividing the obtained answer by the original number. Multiply the final answer by 100 for the answer to be in percentage.

Percentage Increase=\(\frac{\text{Rise in the Number}}{\text{Original Number}}\times100\%\)
Rise in the Value= New number – Original number
Likewise, percentage decrease is comparable to subtracting the new number from the original numeral and dividing the obtained answer by the original number. Multiply the final answer by 100 for the answer to be in percentage.

Percentage Decrease=\(\frac{\text{Decrease in the Number}}{\text{Original Number}}\times100\%\)

Decrease in the Number=Original number – New number
We should remember that when the new value/number is greater than the old number/value, it is a percentage increase, otherwise, it is a decreasing percentage.

Percentage Formula in terms of a Fraction

To transform a fraction into a percentage divide the top/numerator number by the bottom/denominator number and lastly multiply the result by 100%.

\(\frac{\text{Numerator Value}}{\text{Denominator Value}}\times100\%\)

Percentage Change Formula

Sometimes when it is required to get the increase or decrease in any quantity as percentages, which is also directed to as percentage change is given by the formula:

Percentage Change=\(\frac{\text{New value – Original value}}{\text{Original value}}\times100\)

How to Find Percentage?

To estimate the percentage of any value/ data/ number, we can apply the various formulas as discussed above as per the condition applied. Let us learn the basic method to find the percentage.
A% of a data = B
Here B is the necessary percentage.
If we wish to remove the % sign, then the formula is expressed as:
A/100 * given data = B
For example:
How to calculate 20% of 60.
Let 20% of 60 = Y
20/100 * 60 = Y
Y = 12

Similarly; 8% means 8 out of every 100, or in fraction we write 8/100.
In the same way, 50% can be composed as a fraction, 1/2, or a decimal, 0.5.

Percentage Table

A percentage table shows the equivalent values of common fractions expressed as percentages. It helps quickly understand how much a fraction represents out of 100. This is useful in math, finance, and everyday calculations to compare parts of a whole easily.

Once you’ve mastered the Percentages concept, Also, learn more about Partnership concepts in depth!

How to Convert Fractions to Percentages?

A fraction can be represented by; \(\frac{x}{y}\) and can be converted into percentage by the below formula:

\(\frac{x}{y}\times 100\)

Therefore by the formula, it is clear that we can convert fraction to percentage merely by multiplying the given fraction by 100.

Note:

To convert percentages into fraction, divide it by 100.

Example: 25% = 25/100 = ¼

To convert a fraction into percentage, multiply it by 100.

Example: ⅕ = ⅕ x 100 = 20%

Finding Percentages

Finding percentages means working out how much a part is compared to the whole, usually out of 100. It helps to quickly compare quantities or understand proportions in an easy way.

1. When the Total is 100

If the total of all values adds up to 100, the percentage for each value is simply the number itself.
This is because "percent" means per hundred", so no extra calculation is needed.

Example:
Sally bought tiles in three colors for her house.

Here, since the total is 100, the number of tiles directly gives the percentage.

2. When the Total is NOT 100

If the total is not equal to 100, we need to calculate the percentage using the formula:

Percentage = (Part ÷ Total) × 100

Example:
Emma has a bracelet with:

• 8 red beads
• 12 blue beads

Total beads = 8 + 12 = 20

Now, we find the percentages:

• Red beads = (8 ÷ 20) × 100 = 40%
• Blue beads = (12 ÷ 20) × 100 = 60%

Percentage Calculator

Learn how to calculate a percentage through our percent calculator.

Difference between Percentage and Percent

The words percentage and percent are nearly related to one another. The tradition for operating percent and percentage is as specified. The word percent (or the symbol %) accompanies a specific number, on the other hand, the word percentage is used without a number.

An example of Percent:
More than 65% of the country’s population have been vaccinated with the first dose of Covid-19.
An example of Percentage:
A very large percentage of the world’s population has been exposed to Covid-19 pandemic.

Important Definitions related to Percentages

Let us know some of the important definitions related to the percentage.

Important Points About Percentages

1. How to Find a Percentage

To find what percent one number is of another, just divide the part by the whole and multiply by 100.

Example: To find what percent 20 is of 200 → (20 ÷ 200) × 100 = 10%.

2. Percentage Change

When a number increases or decreases, we can show that change using percentages.

This is called percentage increase or decrease.

3. Fractions and Percentages

• To change a fraction to a percentage, multiply it by 100.

Example: ½ × 100 = 50%

• To change a percentage to a fraction, divide it by 100.

Example: 75% = 75 ÷ 100 = ¾

4. Percentages Work Both Ways
If you take 50% of 60, you get 30.
If you take 60% of 50, you also get 30.
So, percentages are reversible in this way.

Marks Percentage

Marks obtained by students in various exams during school and colleges are mostly out of 100. These marks are calculated in terms of percent. For example, consider if a student has scored X marks out of total marks. And, if we have to decide the percentage score; then we divide the scored mark from total marks and multiply the result by 100.

How to Calculate Percentage of Marks

To find out how much percentage of marks a student got, use this simple formula:

Percentage = (Marks Obtained ÷ Total Marks) × 100

Let’s look at some examples:

1. Got 30 out of 100:
   Percentage = (30 ÷ 100) × 100 = 30%
2. Got 10 out of 20:
   Percentage = (10 ÷ 20) × 100 = 50%
3. Got 23 out of 50:
   Percentage = (23 ÷ 50) × 100 = 46%
4. Got 13 out of 40:
   Percentage = (13 ÷ 40) × 100 = 32.5%
5. Got 90 out of 120:
   Percentage = (90 ÷ 120) × 100 = 75%

Understanding Ratios, Fractions, Percents, and Decimals

A ratio shows how two numbers compare.
A fraction shows part of a whole.
A percent tells us how many parts out of 100.
A decimal is another way to write a fraction.

Here’s a simple table to help you understand the connection between them when the first number is 1

Tips and Tricks to solve Percentage based Questions Faster

Candidates can find different tips and tricks from below for solving the questions related to percentage.

Tip # 1: Candidates need to make sure that they know all the important formulas of percentage which are mentioned below.

• Profit % = profit x 100 / cost price
• Loss % = loss x 100 / cost price
• Markup % = (markup / cost price) x 100
• Discount % = (discount / market price) x 100

Tip # 2: Successive Percentage Change: We can use successive percentage change formulas to solve percentage related problems where the product of two quantities equal the third quantity. For example,

⇒Length x Breadth = Area

⇒Price x Quantity purchased = Expenditure

⇒If any quantity is increased by x%, then y% and later on z%, the overall or effective percentage increase is:

⇒[(100 + x) / 100) (100 + y) / 100) (100 + z / 100) -1] x 100

Also check Pipe and Cistern concepts here once you are through with Percentages concepts!

Important Points About Percentages

1. How to Calculate a Percentage
   To find what percentage a number is out of a total, use this formula:
   Percentage = (Number ÷ Total) × 100
   Example: If you scored 45 marks out of 50,
   (45 ÷ 50) × 100 = 90%
2. Percentage Change (Increase or Decrease)
   When something increases or decreases, we can show the change as a percentage.
   Example: If a pen’s price increases from ₹20 to ₹25,
   Increase = ₹5
   (5 ÷ 20) × 100 = 25% increase
3. Converting Fractions and Percentages
   • Fraction to Percentage: Multiply the fraction by 100.
     Example: (3 ÷ 4) × 100 = 75%
   • Percentage to Fraction: Divide by 100.
     Example: 80% = 80 ÷ 100 = 4/5
4. Percentages Can Be Reversible
   X% of Y is the same as Y% of X.
   Example: 50% of 60 = 60% of 50 = 30

Percentage Solved Example for Competitive Exams

Question 1: 20 gram is what percentage of 1 kg?

Solution : Here, quantity 1 = 20 grams and quantity 2 = 1kg = 1000 grams

⇒Hence, required percentage = 20/1000 × 100 = 2%

Question 2: If the price of sugar is increased by 10%, then by how much percent consumption should be reduced so that the expenditure remains the same?

Solution : Let the price be Rs. x /kg Consumption be y kg

⇒Hence, expenditure = price × consumption ⇒ Expenditure = xy

⇒Price of sugar is increased by 10% Hence, new price of sugar = 1.1x per kg

⇒Let new consumption be z kg

⇒Hence, new expenditure = (1.1x) × z Now, new expenditure = old expenditure

⇒ (1.1x) × z = x × y ⇒ z = y/1.1

Reduction in consumption = (y – z) = y – (y/1.1) = y/11

∴ Percentage reduction in consumption = [(y/11)/y] × 100 = 100/11 = 9.09%

Question 3: The population of a town 2 years ago was 245,000. It increased by 15% in the first year and then increased by 20% in the second year. What is the current population of the town?

Solution : The population of a town 2 years ago was 245000 It increased by 15% in the first year

∴ The population after first year will be = (115 / 100) x 245000 = 281750

⇒The population then increased by 20% in the second year.

∴ The population after second year will be = (120 / 100) x 281750 = 338100

Question 4: An electric bully was bought at Rs. 4100. Its value depreciates at the rate of 7% per annum. Its value after one year will be:

Solution: Actual price of the electric bully = Rs. 4100

⇒ Depreciation rate = 7%

∴ Value after 1 year = 4100 – 7% of 4100 = 4100 – 4100 × (7/100) = Rs. 3813

Question 5: If A’s income is 40% more than the income of B, then what percentage of B’s income is less than income of A?

Solution: Let the income of B be 100

∴ Income of A = 140

⇒B’s income is less than income of A by (140 – 100) = 40

⇒Required percentage = (40 / 140) x 100 = 200 / 7 = 28 (4/7) %

Question 6: If A is 40% less than B, then B is how much percentage more than A?

Solution: Given, A is 40% less than B Let B be 100

⇒A = B – 40% of B = 100 – 40% of 100 = 100 – 40 = 60

∴ Required % = {(100 – 60)/60} × 100 = (40/60) × 100 = 66.66%

⇒When you’ve finished with Percentage, you can read about Number Series concepts in depth here!

Word Problems on Percentages

1. A shopkeeper had some mangoes. He sold 30% of them and still had 280 mangoes left. How many mangoes did he have in the beginning?

Solution:
Let the total number of mangoes be N.

After selling 30%, he has 70% left. So,
(70/100) × N = 280

N = (280 × 100) / 70
N = 400

Answer: He originally had 400 mangoes.

2. Out of two numbers, 50% of the bigger number is equal to 80% of the smaller number. If the sum of the two numbers is 180, what is the bigger number?

Solution:
Let the bigger number be X.

Then, the smaller number = 180 – X

According to the question,
(50 × X) / 100 = (80 × (180 – X)) / 100

50X = 80(180 – X)
50X = 14400 – 80X
50X + 80X = 14400
130X = 14400
X = 110.77 (approx)

Answer: The bigger number is about 111.

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