How to calculate percentages

A percentage is a way of expressing a number as a fraction of 100. "45%" means 45 out of every 100, or the fraction 45/100, or the decimal 0.45. Every percentage calculation, no matter how it's phrased, comes down to one of three question types: finding a percentage of a number, finding the percent change between two numbers, or working backward from a percentage to find the original number.

Converting between percentages, decimals, and fractions

Before doing any calculation, it helps to be fluent in moving between the three forms of the same value:

• To turn a percentage into a decimal, divide by 100 (or move the decimal point two places left): 45% = 0.45.
• To turn a decimal into a percentage, multiply by 100 (move the decimal point two places right): 0.6 = 60%.
• To turn a percentage into a fraction, write it over 100 and simplify: 45% = 45/100 = 9/20.

Finding a percentage of a number

This is the most common type: "what is 45% of 80?" Convert the percentage to a decimal, then multiply.

Worked example

Find 45% of 80.

1. Convert 45% to a decimal: 45 / 100 = 0.45.
2. Multiply: 0.45 × 80 = 36.

A quick mental-math shortcut: 10% of any number is that number moved one decimal place left (10% of 80 is 8), so 45% is close to 4.5 times that (8 × 4.5 = 36), a fast way to sanity-check the exact answer.

Worked example: a percentage over 100

Find 150% of 40.

1. Convert 150% to a decimal: 1.5.
2. Multiply: 1.5 × 40 = 60.

150% of a number is one and a half times that number: the whole amount plus half again. That matches the result, since 60 is 40 plus 20 (half of 40).

Finding what percentage one number is of another

This flips the first question around: "18 out of 24 students passed. What percentage is that?" Divide the part by the whole, then multiply by 100.

Worked example

What percentage is 18 out of 24?

1. Divide: 18 / 24 = 0.75.
2. Convert to a percentage: 0.75 × 100 = 75%.

Percent increase and decrease

Percent change measures how much a value has grown or shrunk, relative to where it started. The formula is the same whether the value went up or down:

Percent change = (new value - original value) / original value × 100

A positive result means an increase; a negative result means a decrease.

Worked example: percent increase

A shirt's price rises from $50 to $65. What's the percent increase?

1. Find the difference: 65 - 50 = 15.
2. Divide by the original value: 15 / 50 = 0.3.
3. Convert to a percentage: 0.3 × 100 = 30%.

Worked example: percent decrease

A laptop's price drops from $800 to $680. What's the percent decrease?

1. Find the difference: 680 - 800 = -120.
2. Divide by the original value: -120 / 800 = -0.15.
3. Convert to a percentage: -0.15 × 100 = -15%, meaning a 15% decrease.

The key detail people miss most often here: the denominator is always the originalvalue, never the new one. Dividing by the wrong value produces a number that looks reasonable but doesn't answer the question actually being asked.

Reverse percentages: finding the original value

Sometimes you know a percentage of a number and need to find the number itself. "15% of a number is 30. What's the number?" This is solved with division, not multiplication.

Worked example

15% of a number is 30. Find the number.

1. Convert 15% to a decimal: 0.15.
2. Divide the known amount by that decimal: 30 / 0.15 = 200.
3. Check: 0.15 × 200 = 30. Correct.

Worked example: reverse percentage after a discount

A jacket is on sale for $68 after a 20% discount. What was the original price?

1. If the price dropped by 20%, the sale price represents 100% - 20% = 80% of the original price.
2. Convert 80% to a decimal: 0.8.
3. Divide the sale price by that decimal: 68 / 0.8 = 85.
4. Check: 85 - (0.2 × 85) = 85 - 17 = 68. Correct.

A common mistake here is taking 20% of the sale price and adding it back, which gives $81.60, the wrong answer. The 20% discount was always calculated against the original price, so reversing it means dividing by 80%, not adding 20% of the discounted price.

Percentage points versus percent change

These two ideas sound similar but describe different things. Suppose an interest rate moves from 5% to 8%. As a percentage point change, that's a 3 percentage point increase, a simple subtraction: 8 - 5 = 3. As a percent change, it's (8 - 5) / 5 × 100 = 60%, because the 3-point jump is 60% of the original 5% rate. News headlines and financial reports sometimes blur these two, so it's worth checking which one is actually being reported before drawing a conclusion.

Common mistakes

• Forgetting to convert the percentage to a decimal before multiplying.Multiplying by 45 instead of 0.45 gives an answer 100 times too large.
• Dividing by the new value instead of the original in percent change. The denominator in a percent change calculation is always the starting value.
• Reversing a discount by adding a percentage of the sale price. Reverse a percentage decrease by dividing by (100% - discount%), not by adding back a percentage of the already-discounted price.
• Confusing percentage points with percent change. A move from 5% to 8% is 3 percentage points, but a 60% relative increase. Use the term that matches what's actually being asked.
• Rounding too early. Rounding a decimal partway through a multi-step percentage problem can shift the final answer noticeably; keep full precision until the last step.

Where percentages show up every day

Sales tax, restaurant tips, loan interest, test scores, batting averages, population growth, and nutrition labels are all percentages in disguise. Because the underlying calculation is always one of the three types covered above, the same three formulas cover essentially every everyday percentage question you'll run into, from splitting a 20% tip on a $54 bill to figuring out how much a retirement account grew in a year.

Practice it yourself

Try the percentage calculator for a quick answer, or head topercentage practice for an unlimited set of "percent of," percent change, and reverse percentage problems with step-by-step explanations.