Percentage Formula: How to Calculate Percentages

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Percentages appear in discounts, test scores, interest rates, tax, statistics and science, and almost every percentage question is one of a handful of types. Also, understanding percentages helps you interpret data, make informed financial decisions, and analyze trends in various fields. This guide shows each type with its formula and a worked example, providing clear steps and practical applications.

Understanding the “Whole”

Every percentage calculation relies on identifying the “whole” or “base” value. This is the reference point against which you compare a “part. Clearly defining the whole prevents common calculation errors.

Percent, fraction and decimal conversions

You often need to convert between percents, decimals, and fractions to perform calculations or to express values in a different format. Each form represents a portion of a whole, but they serve different purposes in mathematics and daily communication.

Percent	Decimal	Fraction
1%	0.01	( frac{1}{100} )
10%	0.1	( frac{1}{10} )
12.5%	0.125	( frac18 )
20%	0.2	( frac15 )
25%	0.25	( frac14 )
( 33frac13% )	0.333…	( frac13 )
50%	0.5	( frac12 )
75%	0.75	( frac34 )
150%	1.5	( frac32 )

Converting Percent to Decimal

To convert a percent to a decimal, divide the percentage by 100. This is equivalent to moving the decimal point two places to the left.

1. Remove the percent symbol (%).
2. Divide the number by 100.

( 75 div 100 = 0.75 ).

( 3.5 div 100 = 0.035 ).

Converting Decimal to Percent

To convert a decimal to a percent, multiply the decimal by 100 and add the percent symbol. This is equivalent to moving the decimal point two places to the right.

1. Multiply the decimal by 100.
2. Add the percent symbol (%).

Convert 0.4 to a percent.

Convert 1.25 to a percent.

Converting Fraction to Percent

To convert a fraction to a percent, first convert the fraction to a decimal by dividing the numerator by the denominator, then convert the resulting decimal to a percent.

1. Divide the numerator by the denominator to get a decimal.
2. Multiply the decimal by 100 and add the percent symbol (%).

Convert ( frac18 ) to a percent. ( 1 div 8 = 0.125 ).

Convert ( frac67 ) to a percent. ( 6 div 7 approx 0.85714 ). You may need to round repeating decimals.

Type 1: Percent of a number (Finding the Part)

This is one of the most common percentage calculations. You know the whole amount and a percentage, and you want to find the specific part that percentage represents.

$$text{part} = frac{text{percent}}{100} times text{whole}$$

Steps to calculate a percent of a number

1. Convert the percentage to a decimal by dividing by 100.
2. Multiply the decimal by the whole number.

Example 1.

• Convert 15% to a decimal: ( 15 div 100 = 0.15 ).
• Multiply the decimal by the whole number: ( 0.15 times 80 = 12 ).

Example 2: Calculating a discount. A shirt costs $45. How much is the discount?

• Convert 20% to a decimal: ( 20 div 100 = 0.20 ).
• Multiply the decimal by the original price: ( 0.20 times $45 = $9 ).

The discount is $9. The shirt would cost ( $45 – $9 = $36 ).

Mental shortcut. For simple percentages, you can often do this in your head.

Type 2: What percent one number is of another (Finding the Percentage)

Here, you have a part and a whole, and you want to express the part as a percentage of the whole. This is useful for calculating grades, market share, or comparing quantities.

Steps to find what percent one number is of another

1. Divide the part by the whole.
2. Multiply the result by 100 and add the percent symbol (%).

Example 3. You score 45 out of 60 on a test. What is your percentage score?

• Divide the part (your score) by the whole (total score): ( 45 div 60 = 0.75 ).
• Multiply by 100 and add %: ( 0.75 times 100 = 75% ).

Example 4. What percent of 250 is 40?

• Divide the part (40) by the whole (250): ( 40 div 250 = 0.16 ).
• Multiply by 100 and add %: ( 0.16 times 100 = 16% ).

Type 3: Finding the whole (Reverse Percentage)

Sometimes you know a part of a number and the percentage it represents, and you need to find the original whole number. This is often called a reverse percentage calculation.

$$text{whole} = frac{text{part}}{text{percent}/100}$$

Steps to find the whole number

1. Convert the percentage to a decimal by dividing by 100.
2. Divide the given part by this decimal.

Example 5.

• Convert 20% to a decimal: ( 20 div 100 = 0.20 ).
• Divide the part (30) by the decimal: ( 30 div 0.20 = 150 ).

Example 6: Original price before a discount. What was its original price?

• If the jacket received a 15% discount, you are paying 100% – 15% = 85% of the original price.
• Convert 85% to a decimal: ( 85 div 100 = 0.85 ).
• Divide the discounted price ($68) by the decimal: ( $68 div 0.85 = $80 ).

The original price of the jacket was $80.)

Type 4: Percent Increase and Decrease (Percent Change)

Percent change measures how much a quantity has changed relative to its original value. This is crucial for tracking growth, inflation, price fluctuations, or performance metrics.

A positive answer indicates a percent increase; a negative answer indicates a percent decrease.

Steps to calculate percent change

1. Subtract the old value from the new value to find the amount of change.
2. Divide the amount of change by the old value.
3. Multiply the result by 100 and add the percent symbol (%).

Example 7. A price rises from $50 to $65. Calculate the percent increase.

• Find the change: ( $65 – $50 = $15 ).
• Divide by the old value: ( $15 div $50 = 0.3 ).
• Multiply by 100 and add %: ( 0.3 times 100 = 30% ).

Example 8. Sales fall from 80 units to 60 units. What is the percent decrease?

• Find the change: ( 60 – 80 = -20 ).
• Divide by the old value: ( -20 div 80 = -0.25 ).
• Multiply by 100 and add %: ( -0.25 times 100 = -25% ).

Applying a percent change

To increase a number by p%, multiply it by ( 1 + frac{p}{100} ). Conversely, to decrease a number by p%, multiply it by ( 1 – frac{p}{100} ).

The total cost is ( $200 times (1 + 0.08) = $200 times 1.08 = $216 ).

The new population is ( 10,000 times (1 – 0.05) = 10,000 times 0.95 = 9,500 ).

Why changes don’t cancel

Applying successive percentage changes does not simply add or subtract the percentages.

Type 5: Percent error and Percent Difference

These calculations quantify the discrepancy between values. Percent error measures accuracy against a known true value, while percent difference compares two values when neither is considered “true.”

Steps to calculate percent error

1. Find the absolute difference between the measured value and the true (accepted) value.
2. Divide this difference by the absolute true value.
3. Multiply by 100 and add the percent symbol (%).

Example 9. You measure the acceleration due to gravity (g) as 9.6 m/s²; the accepted true value is 9.81 m/s². Calculate the percent error.

• Find the absolute difference: ( lvert 9.6 – 9.81 rvert = lvert -0.21 rvert = 0.21 ).
• Divide by the true value: ( 0.21 div 9.81 approx 0.021406 ).
• Multiply by 100 and add %: ( 0.021406 times 100 approx 2.14% ).

Percent difference

Percent difference compares two measurements when neither is the “true” one. You divide the absolute difference between the two values by their average, then multiply by 100.

Example 10. Two students measure a length as 40 cm and 60 cm. Find the percent difference between their measurements.

• Find the absolute difference: ( lvert 40 – 60 rvert = lvert -20 rvert = 20 ).
• Find the average of the two values: ( (40 + 60) div 2 = 100 div 2 = 50 ).
• Divide the difference by the average: ( 20 div 50 = 0.4 ).
• Multiply by 100 and add %: ( 0.4 times 100 = 40% ).

Percentage calculator

Choose the question type and enter the numbers to check your work.

The full-page percentage calculator covers all five types.

Percentage points vs percent

It is crucial to distinguish between “percentage points” and “percent. However, it rises by News reports often blur the two, but the difference matters significantly for understanding the magnitude of change, especially in financial or statistical contexts.

Repeated percentage changes

When you apply the same percent change multiple times, you are dealing with exponential growth or decay, not simple addition or subtraction. For instance, a population growing 3% a year for 10 years does not grow by 30%. Instead, each year’s growth is applied to the *new* increased population. This multiplies the population by a factor of ( 1 + 0.03 ) = 1.03 each year. Over 10 years, the total growth factor is ( 1.03^{10} approx 1.344 ), meaning a 34.4% increase, not 30%. This is how interest compounds; see the compound interest formula and exponential functions for more details.

Common mistakes and how to fix them

• Dividing by the new value in percent change. Always divide by the *original* (old) value. The original value serves as the base for comparison.
• Adding percentages of different amounts. You cannot simply add percentages if they refer to different wholes. For example, 10% of $100 plus 20% of $50 is not 30% of $150. Calculate each part separately: (0.10 * $100) + (0.20 * $50) = $10 + $10 = $20.
• Reversing a discount by adding the same percent back. As shown in Example 6, a 15% discount means you pay 85% of the original. To find the original, however, you must divide by 0.85, not add 15% to the discounted price.
• Forgetting to convert the percent to a decimal before multiplying. Always convert percentages (e.g., 25% to 0.25) when performing multiplication or division in formulas. The “%” symbol indicates “per hundred” and implies division by 100.
• Misinterpreting the “whole” or base. Ensure you correctly identify which number represents the total or original amount for your calculation. This is the denominator in most percentage formulas.

Who this advice suits (and who it doesn’t)

This guide suits anyone needing to understand and apply basic percentage calculations. For example, this includes students learning fundamental math, individuals managing personal finances (budgets, discounts, interest), professionals interpreting data or sales figures, and anyone wanting to make sense of percentage-based information in daily life. The formulas and examples provide a solid foundation for practical applications.

This advice may not fully suit those requiring advanced statistical analysis involving weighted percentages, probability distributions, or complex financial models that incorporate percentages within larger algorithms. While the core concepts remain, these specialized fields often involve additional mathematical tools and considerations beyond the scope of this introductory guide.

Practice problems

1. Find 35% of 240.
2. 18 is what percent of 72?
3. 45 is 30% of what number?
4. A laptop drops from $900 to $720. What is the percent decrease?
5. A $60 meal has a 15% tip. What is the total?
6. A measured length is 52 cm; the true length is 50 cm. Find the percent error.

Answers:

Frequently asked questions

What is the formula for percentage?

How do you calculate a percentage of a number?

How do you calculate percent increase?

Subtract the old value from the new value, divide the result by the old value, and then multiply by 100.

How do you find the original amount before a percentage change?

If you know the new amount and the percentage change, divide the new amount by the appropriate multiplier.

What is 1/8 as a percent?

Further reading

• Percentage (Wikipedia) — definitions, conversions and common errors.
• Relative change and difference (Wikipedia) — percent change, percent difference and percent error in detail.