Percentage Formulas and Basic Percentage Calculations

Percentages are one of the most useful parts of mathematics in everyday life. We use percentages when comparing prices, calculating discounts, understanding marks and scores, analyzing statistics, finding increases or decreases, calculating profits and losses, and interpreting information. Even when a problem looks complicated, many percentage questions can be solved using a few basic formulas.

The word “percent” means “per hundred.” Therefore, a percentage expresses a quantity as a part of 100. For example, 25% means 25 parts out of 100, while 75% means 75 parts out of 100. Understanding this simple idea makes it much easier to work with percentage formulas.

This article explains the basic percentage formulas and shows how to use them in common calculations. It also covers how to find a percentage, calculate a percentage of a number, find the original value, and calculate percentage increase and decrease.

What Is a Percentage?

A percentage is a way of expressing a number as a fraction of 100. The symbol used for percentage is %.

For example:

• 10% means 10 out of 100.

• 25% means 25 out of 100.

• 50% means 50 out of 100.

• 100% means the complete amount.

The basic relationship between a percentage, a part, and a whole is:

Percentage = (Part ÷ Whole) × 100

Here, the part is the amount being considered, while the whole is the total amount.

For example, if 20 students out of a class of 50 passed an examination:

Percentage = (20 ÷ 50) × 100

Percentage = 40%

Therefore, 40% of the students passed.

Basic Percentage Formula

The most important percentage formula is:

Percentage = (Part ÷ Whole) × 100

This formula can be rearranged to find the part or the whole.

To Find the Part

Part = (Percentage ÷ 100) × Whole

For example, find 30% of 200.

Part = (30 ÷ 100) × 200

Part = 60

Therefore, 30% of 200 is 60.

To Find the Whole

Whole = Part ÷ (Percentage ÷ 100)

For example, 40 is 20% of what number?

Whole = 40 ÷ (20 ÷ 100)

Whole = 40 ÷ 0.2

Whole = 200

Therefore, 40 is 20% of 200.

Converting a Percentage into a Fraction

A percentage can be converted into a fraction by placing the percentage number over 100.

Percentage = Percentage Number ÷ 100

For example:

25% = 25/100

After simplifying:

25% = 1/4

Similarly:

50% = 50/100 = 1/2

75% = 75/100 = 3/4

20% = 20/100 = 1/5

This conversion is useful when solving percentage problems involving fractions.

Converting a Percentage into a Decimal

To convert a percentage into a decimal, divide it by 100.

Decimal = Percentage ÷ 100

For example:

10% = 10 ÷ 100 = 0.10

25% = 25 ÷ 100 = 0.25

50% = 50 ÷ 100 = 0.50

75% = 75 ÷ 100 = 0.75

120% = 120 ÷ 100 = 1.20

A percentage greater than 100% becomes a decimal greater than 1.

Converting a Decimal into a Percentage

To convert a decimal into a percentage, multiply it by 100 and add the percentage symbol.

Percentage = Decimal × 100

For example:

0.25 × 100 = 25%

0.6 × 100 = 60%

0.75 × 100 = 75%

1.2 × 100 = 120%

This conversion is especially useful when a calculation produces a decimal and the answer needs to be expressed as a percentage.

Finding the Percentage of a Number

One of the most common percentage calculations is finding a certain percentage of a number.

The formula is:

Percentage of a Number = (Percentage ÷ 100) × Number

For example, find 15% of 300.

15% of 300 = (15 ÷ 100) × 300

= 0.15 × 300

= 45

Therefore, 15% of 300 is 45.

Another example:

Find 8% of 500.

8% of 500 = (8 ÷ 100) × 500

= 0.08 × 500

= 40

Therefore, 8% of 500 is 40.

Finding What Percentage One Number Is of Another

Sometimes the part and whole are known, but the percentage is unknown.

Use:

Percentage = (Part ÷ Whole) × 100

For example, a student scores 72 marks out of 90. What percentage did the student score?

Percentage = (72 ÷ 90) × 100

Percentage = 0.8 × 100

Percentage = 80%

Therefore, the score is 80%.

Another example:

A shop has 200 products, and 50 are sold. What percentage of the products were sold?

Percentage = (50 ÷ 200) × 100

Percentage = 25%

Therefore, 25% of the products were sold.

Finding the Original Number

Sometimes you know the percentage and the resulting part, but you need to find the original whole amount.

The formula is:

Whole = Part ÷ (Percentage ÷ 100)

For example, 30% of a number is 90. Find the number.

Whole = 90 ÷ (30 ÷ 100)

Whole = 90 ÷ 0.3

Whole = 300

Therefore, the original number is 300.

This method is useful when working backward from a percentage result.

Percentage Increase

A percentage increase describes how much a quantity has increased compared with its original value.

First find the increase:

Increase = New Value − Original Value

Then:

Percentage Increase = (Increase ÷ Original Value) × 100

For example, a product’s price increases from $200 to $250.

Increase = 250 − 200

Increase = 50

Percentage Increase = (50 ÷ 200) × 100

Percentage Increase = 25%

Therefore, the price increased by 25%.

The original value is used as the denominator when calculating percentage increase.

Percentage Decrease

A percentage decrease describes how much a quantity has decreased compared with its original value.

First find the decrease:

Decrease = Original Value − New Value

Then:

Percentage Decrease = (Decrease ÷ Original Value) × 100

For example, a price decreases from $500 to $400.

Decrease = 500 − 400

Decrease = 100

Percentage Decrease = (100 ÷ 500) × 100

Percentage Decrease = 20%

Therefore, the price decreased by 20%.

Finding the New Value After a Percentage Increase

When a number increases by a certain percentage, you can calculate the new value directly.

New Value = Original Value × (1 + Percentage ÷ 100)

For example, increase 400 by 15%.

New Value = 400 × (1 + 15 ÷ 100)

New Value = 400 × 1.15

New Value = 460

Therefore, the new value is 460.

Another way is to calculate the increase first:

15% of 400 = 60

Then:

400 + 60 = 460

Both methods give the same result.

Finding the New Value After a Percentage Decrease

For a percentage decrease, use:

New Value = Original Value × (1 − Percentage ÷ 100)

For example, decrease 800 by 25%.

New Value = 800 × (1 − 25 ÷ 100)

New Value = 800 × 0.75

New Value = 600

Therefore, the new value is 600.

The same result can be found by calculating 25% of 800:

25% of 800 = 200

Then:

800 − 200 = 600

Percentage Change

Percentage change is commonly used when comparing an original value with a new value.

The general formula is:

Percentage Change = (Change ÷ Original Value) × 100

The change can be an increase or a decrease.

For an increase:

Percentage Change = [(New Value − Original Value) ÷ Original Value] × 100

For a decrease:

Percentage Change = [(Original Value − New Value) ÷ Original Value] × 100

For example, a quantity changes from 250 to 300.

Change = 300 − 250 = 50

Percentage Change = (50 ÷ 250) × 100

= 20%

Therefore, the quantity increased by 20%.

Percentage Difference

Percentage difference is used when comparing two values without treating one value as the original or starting value.

The formula is:

Percentage Difference = [|Value 1 − Value 2| ÷ ((Value 1 + Value 2) ÷ 2)] × 100

The vertical bars indicate that the absolute difference is used, so the difference is treated as a positive value.

For example, two measurements are 80 and 100.

Difference = |80 − 100| = 20

Average = (80 + 100) ÷ 2 = 90

Percentage Difference = (20 ÷ 90) × 100

≈ 22.22%

Therefore, the percentage difference is approximately 22.22%.

Percentage difference and percentage change are not always interchangeable. Percentage change normally uses a specific original value, while percentage difference uses the average of the two values as the reference.

Using Percentages in Discounts

Percentages are frequently used to calculate discounts.

The discount amount is:

Discount = (Discount Percentage ÷ 100) × Original Price

The final price is:

Final Price = Original Price − Discount

For example, a shirt costs $800 and has a 20% discount.

Discount = (20 ÷ 100) × 800

= 160

Final Price = 800 − 160

= 640

Therefore, the discounted price is $640.

A direct formula can also be used:

Final Price = Original Price × (1 − Discount Percentage ÷ 100)

So:

Final Price = 800 × 0.80

= 640

Using Percentages in Profit and Loss

Percentages are also important when calculating profit and loss.

If the cost price is less than the selling price, there is a profit.

Profit = Selling Price − Cost Price

Profit Percentage = (Profit ÷ Cost Price) × 100

For example, an item is purchased for $500 and sold for $600.

Profit = 600 − 500

= 100

Profit Percentage = (100 ÷ 500) × 100

= 20%

Therefore, the profit is 20%.

If the selling price is less than the cost price, there is a loss.

Loss = Cost Price − Selling Price

Loss Percentage = (Loss ÷ Cost Price) × 100

For example, an item costs $800 and is sold for $680.

Loss = 800 − 680

= 120

Loss Percentage = (120 ÷ 800) × 100

= 15%

Therefore, the loss is 15%.

Common Percentage Values

Some percentages are particularly useful because they can be calculated quickly.

50% = 1/2

25% = 1/4

20% = 1/5

10% = 1/10

5% = 1/20

For example, to find 25% of 200, you can divide 200 by 4:

200 ÷ 4 = 50

Therefore, 25% of 200 is 50.

Similarly, 10% of 700 is 70 because dividing a number by 10 gives 10% of that number.

These relationships can make mental percentage calculations faster.

Worked Example of a Basic Percentage Problem

Suppose a library has 1,200 books. If 35% of the books are science books, how many science books are there?

Use:

Part = (Percentage ÷ 100) × Whole

Part = (35 ÷ 100) × 1,200

Part = 0.35 × 1,200

Part = 420

Therefore, the library has 420 science books.

Now suppose 84 of those 420 science books are about physics. What percentage of the science books are physics books?

Percentage = (Part ÷ Whole) × 100

Percentage = (84 ÷ 420) × 100

Percentage = 20%

Therefore, physics books make up 20% of the science books.

This example shows how the same basic formulas can be used in different ways.

Common Mistakes in Percentage Calculations

Percentage problems are usually straightforward, but a few mistakes occur frequently.

Using the Wrong Whole

The denominator in a basic percentage calculation should normally represent the relevant whole or reference amount.

For example, when calculating percentage increase, the original value is normally used as the reference.

Forgetting to Multiply by 100

If you calculate:

20 ÷ 80 = 0.25

This is a decimal, not yet a percentage.

To express it as a percentage:

0.25 × 100 = 25%

Confusing Percentage With Percentage Points

A change from 30% to 40% is an increase of 10 percentage points. The relative percentage increase is:

[(40 − 30) ÷ 30] × 100

= 33.33%

These are two different ways of describing the change.

Applying a Percentage to the Wrong Value

When a problem involves several changes, identify which amount the percentage applies to. A 20% increase followed by a 20% decrease does not return a value to its starting point because the second 20% is calculated from a different amount.

For example, start with 100:

After a 20% increase:

100 × 1.20 = 120

After a 20% decrease:

120 × 0.80 = 96

The final value is 96, not 100.

Percentage Formula Summary

The most important formulas can be summarized as follows:

Percentage = (Part ÷ Whole) × 100

Part = (Percentage ÷ 100) × Whole

Whole = Part ÷ (Percentage ÷ 100)

Percentage Increase = [(New Value − Original Value) ÷ Original Value] × 100

Percentage Decrease = [(Original Value − New Value) ÷ Original Value] × 100

New Value After Increase = Original Value × (1 + Percentage ÷ 100)

New Value After Decrease = Original Value × (1 − Percentage ÷ 100)

Discount = (Discount Percentage ÷ 100) × Original Price

Profit Percentage = (Profit ÷ Cost Price) × 100

Loss Percentage = (Loss ÷ Cost Price) × 100

Learning these formulas provides a strong foundation for solving many everyday and mathematical percentage problems.

Conclusion

Percentages provide a simple way to describe parts of a whole using 100 as the reference. The basic formula, Percentage = (Part ÷ Whole) × 100, forms the foundation for many other percentage calculations. By rearranging this relationship, you can find an unknown part, whole, or percentage.

Percentage formulas are also used to calculate increases, decreases, discounts, profits, losses, and percentage changes. The most important step is to identify the original or total value correctly before applying the formula. With regular practice, percentage calculations become easier and can often be performed mentally for simple values.

Understanding these basic percentage formulas is an important foundation for more advanced topics in mathematics, statistics, finance, and everyday quantitative reasoning.