Percentages: What They Are, How to Calculate Them, and Solved Exercises

Learning how to calculate percentages is not simply about memorizing a formula. It is necessary to understand what the percentage represents and identify which value is known and which one needs to be found. This interpretation allows us to solve everything from simple calculations to problems involving increases, discounts, taxes, grades, and percentage changes.

Key idea: A percentage tells us how many parts out of 100 a quantity represents. For example, 25% means 25 out of every 100, or 25/100 = 0.25.

What Is a Percentage?

The word percentage refers to a quantity expressed in relation to one hundred. The symbol used to represent it is %.

For example, saying that 20% of a group of students has a particular characteristic means that, for every 100 students, 20 meet that condition. Mathematically, we can express 20% as both a fraction and a decimal number.

20% = 20 / 100 = 0.20

In this way, a percentage can be interpreted as a fraction with a denominator of 100 or as a decimal number.

Percentage	Fraction	Decimal
10%	10/100	0.10
25%	25/100	0.25
50%	50/100	0.50
75%	75/100	0.75
100%	100/100	1

How Do You Calculate a Percentage?

To calculate a percentage of a quantity, we can first convert the percentage into a decimal and then multiply it by the quantity.

Percentage of a quantity = quantity × (percentage ÷ 100)

For example, to calculate 30% of 200, we first convert 30% into a decimal:

30% = 30 ÷ 100 = 0.30

Then we multiply the quantity by the decimal:

200 × 0.30 = 60

Therefore, 30% of 200 is 60.

Step-by-Step Procedure

Identify the quantity for which you want to calculate the percentage.
Identify the requested percentage.
Divide the percentage by 100.
Multiply the result by the quantity.
Check that the result is reasonable.

How to Convert a Percentage to a Decimal and a Fraction

An important skill when solving percentage problems is being able to move between different representations of the same quantity.

From Percentage to Decimal

To convert a percentage to a decimal, divide it by 100. In practice, this is equivalent to moving the decimal point two places to the left.

45% = 45 ÷ 100 = 0.45

8% = 8 ÷ 100 = 0.08

125% = 125 ÷ 100 = 1.25

Notice that a percentage can be greater than 100%. This occurs when the quantity represents more than one whole of the reference amount.

From Percentage to Fraction

To convert a percentage into a fraction, place the number over 100 and simplify whenever possible.

25% = 25/100 = 1/4

50% = 50/100 = 1/2

75% = 75/100 = 3/4

Do not confuse a percentage with a decimal number. 25% and 0.25 represent the same quantity, but they are written in different forms.

How to Calculate What Percentage One Quantity Represents

We do not always know the percentage. In some problems, we know a part and the total and need to determine what percentage the part represents.

To solve this type of problem, we use:

Percentage = (part ÷ total) × 100

Example: What Percentage Is 30 Out of 120?

We want to determine what percentage 30 represents out of a total of 120.

Percentage = (30 ÷ 120) × 100

Percentage = 0.25 × 100 = 25%

Therefore, 30 represents 25% of 120.

Example with Students

There are 40 students in a group, and 28 passed an exam. What percentage of the group passed?

The part is 28 and the total is 40.

Percentage = (28 ÷ 40) × 100

Percentage = 0.70 × 100 = 70%

70% of the students passed the exam.

How to Calculate Percentage Increases

Percentages also allow us to express how much a quantity increases relative to its original value. To calculate a percentage increase, we first find the amount of the increase and then add it to the initial quantity.

Increase = original amount × (percentage ÷ 100)

New amount = original amount + increase

Example of a Percentage Increase

A product costs $800 and its price increases by 15%. What will its new price be?

First, calculate the increase:

800 × 0.15 = 120

Then add the increase to the original price:

800 + 120 = 920

The new price is $920.

A Faster Method

We can also multiply the original amount directly by 1 + the percentage expressed as a decimal.

800 × 1.15 = 920

The number 1 represents 100% of the original amount, while 0.15 represents the 15% increase. Therefore, 1.15 represents 115%.

How to Calculate Percentage Discounts

A percentage discount works similarly, but instead of increasing the amount, we decrease it. First, we calculate the amount represented by the discount and then subtract it from the original price.

Discount = original price × (percentage ÷ 100)

Final price = original price − discount

Example of a Discount

A backpack costs $1,200 and has a 25% discount. What is the final price?

First, calculate the discount:

1200 × 0.25 = 300

Now subtract the discount:

1200 − 300 = 900

The final price of the backpack is $900.

A Quick Way to Calculate the Final Price

If a product has a 25% discount, it means we will pay 75% of its original price.

100% − 25% = 75%

1200 × 0.75 = 900

This strategy is especially useful when we need to solve discount problems quickly.

Successive Percentages: Increases and Discounts

A situation that often causes confusion occurs when a quantity changes more than once. In these cases, it is not always correct to simply add or subtract the percentages.

For example, if a price increases by 20% and then decreases by 20%, the result does not necessarily return to the original price.

Suppose a product initially costs $1,000.

First, it increases by 20%:

1000 × 1.20 = 1200

Then it decreases by 20% based on the new price:

1200 × 0.80 = 960

The final price is $960, not $1,000.

The reason is that the second 20% is calculated from a different amount. The first percentage was applied to $1,000, while the second was applied to $1,200.

Key idea: When successive percentages are applied, each percentage may have a different base. Therefore, we must perform the calculations in the order in which the changes occur.

Percentage Problems: Solved Exercises

Exercise 1: Percentage of a Quantity

What is 18% of 250?

250 × 0.18 = 45

Answer: 45.

Exercise 2: Finding the Percentage

In an assessment, a student answered 36 out of 45 questions correctly. What percentage of correct answers did the student get?

(36 ÷ 45) × 100 = 80%

Answer: The student got 80% correct.

Exercise 3: Percentage Increase

An amount of $600 increases by 12%. What is its new value?

600 × 0.12 = 72

600 + 72 = 672

Answer: The new value is $672.

Exercise 4: Percentage Discount

An item costs $2,500 and has an 18% discount. What is the final price?

2500 × 0.18 = 450

2500 − 450 = 2050

Answer: The final price is $2,050.

Exercise 5: Finding an Amount from a Percentage

30% of an amount is 75. What is the total amount?

We can set up the problem as:

0.30 × x = 75

Solve for x:

x = 75 ÷ 0.30

x = 250

Answer: The total amount is 250.

Common Mistakes When Calculating Percentages

Percentage problems may seem simple, but there are several common mistakes that can completely change the answer.

1. Forgetting to Divide by 100

If we want to calculate 20% of 300, we should not directly multiply 300 × 20. The percentage must first be converted into a decimal or used as a fraction over 100.

20% = 20/100 = 0.20

2. Confusing the Part with the Total

When determining what percentage a quantity represents, we must correctly identify which value is the part and which is the total.

Percentage = (part ÷ total) × 100

3. Adding Percentages Without Analyzing the Situation

When dealing with successive changes, each percentage may be applied to a different amount. Therefore, we cannot always simply add or subtract the percentages.

4. Confusing Percentages with Percentage Points

If a rate increases from 20% to 25%, the increase is 5 percentage points. However, the relative increase compared with 20% is 25%.

(25 − 20) ÷ 20 × 100 = 25%

This distinction is especially important when interpreting statistical data, academic results, or economic indicators.

Relationship Between Percentages, Fractions, and Decimals

Percentages are closely related to fractions and decimal numbers. Understanding these equivalences makes it easier to solve problems.

Percentage	Equivalent Fraction	Decimal
20%	1/5	0.20
25%	1/4	0.25
40%	2/5	0.40
50%	1/2	0.50
75%	3/4	0.75

These equivalences also allow us to connect percentages with previously studied arithmetic concepts. For example:

1/2 = 0.5 = 50%

1/4 = 0.25 = 25%

3/4 = 0.75 = 75%

Where Are Percentages Used?

Percentages appear in many contexts because they allow us to compare a quantity using a common base of 100. This makes information easier to interpret and compare.

Discounts and promotions.
Price increases.
School grades.
Survey results.
Statistics.
Interest and financial transactions.
Taxes and price changes.
Economic and social indicators.
Probability and data analysis.
Comparison of academic results.

In all these cases, the meaning of a percentage depends on the reference amount. Therefore, correctly identifying the base on which the percentage is calculated is just as important as performing the operation.

Frequently Asked Questions About Percentages

What Is a Percentage?

A percentage is a way of expressing a quantity as a proportion of every 100 units. It is represented by the symbol %.

How Do You Calculate a Percentage of a Quantity?

Multiply the quantity by the percentage expressed as a decimal. For example, to calculate 15% of 200:

200 × 0.15 = 30

How Do You Convert a Percentage to a Decimal?

Divide the percentage by 100. For example:

35% = 35 ÷ 100 = 0.35

How Do You Find What Percentage a Quantity Represents?

Divide the part by the total and multiply the result by 100:

Percentage = (part ÷ total) × 100

How Do You Calculate a Discount?

First, calculate the discount by multiplying the original price by the percentage expressed as a decimal. Then subtract the discount from the original price.

How Do You Calculate a Percentage Increase?

First, calculate the increase and then add it to the original amount. You can also multiply the original amount directly by 1 plus the percentage expressed as a decimal.

Can a Percentage Be Greater Than 100%?

Yes. A percentage greater than 100% represents an amount greater than the reference total. For example, 150% is equivalent to 1.5 times the reference amount.

Is Increasing by 20% and Then Decreasing by 20% the Same?

No. The second percentage is applied to a different amount. For example, $1,000 increased by 20% becomes $1,200; subsequently decreasing $1,200 by 20% results in $960.

Conclusion: Understanding Percentages Is More Important Than Memorizing the Formula

Percentages allow us to express relationships between quantities using a common base of 100. This seemingly simple structure makes them an essential tool for interpreting mathematical, academic, economic, and everyday information.

To correctly solve a percentage problem, the first step is to identify what the total represents, what the part represents, and which quantity is unknown. We can then choose the appropriate procedure: calculating a percentage of a quantity, determining what percentage a part represents, finding an amount from a percentage, or analyzing a percentage increase or discount.

True understanding comes when we stop viewing a percentage as an isolated operation and begin relating it to fractions, decimals, ratios, and proportions. In this way, percentages become more than a formula to memorize—they become a tool for interpreting and solving problems.

Academic Resource Information

Collection

Mathematics

Knowledge Center

Arithmetic

Knowledge Area

Percentages

Topic

Percentages: What They Are, How to Calculate Them, and Solved Exercises

Level

Basic–Intermediate

Estimated Reading Time

12–15 minutes

Resource Type

Master Article

Skills Developed

Understand the concept of a percentage and its relationship to a reference amount.
Convert percentages between fractional and decimal representations.
Calculate percentages of a quantity.
Determine what percentage a part represents in relation to a total.
Calculate percentage increases.
Calculate percentage discounts.
Solve problems involving successive percentages.
Find an unknown quantity from a given percentage.
Identify common errors when calculating and interpreting percentages.
Distinguish between percentages and percentage points.
Relate percentages to fractions, decimals, ratios, and proportions.
Apply percentages to academic, economic, and everyday situations.

Path Within the Library

Digital Library → Mathematics Collection → Knowledge Center: Arithmetic → Knowledge Area: Percentages