Percentage Calculator

What is a percentage?

A percentage is a number expressed as a fraction of 100. The word comes from the Latin per centum, meaning by the hundred, and the sign % is a stylized way of writing it. Saying 45% means 45 out of every 100, which is the same as the fraction 45/100 or the decimal 0.45.

Percentages make quantities of different sizes easy to compare. A test score of 42 out of 50 and one of 68 out of 80 are hard to compare as fractions, but as percentages they are 84% and 85%. They appear everywhere: interest rates, discounts, tax rates, test scores, statistics and nutrition labels.

To turn a fraction into a percentage, divide the top by the bottom and multiply by 100: 3/8 = 0.375 = 37.5%. To turn a percentage into a decimal, divide by 100 by moving the decimal point two places left: 7.5% = 0.075. Percentages above 100 are valid too; 150% of a value is one and a half times that value.

Percentage formula

Most everyday percentage problems are versions of one relationship between a percentage and two values. Write the percentage as a decimal, and solve for whichever piece is missing.

What is 20% of 150? Multiply: 0.20 × 150 = 30. 30 is what percent of 150? Divide: 30 ÷ 150 = 0.20, or 20%. 30 is 20% of what number? Divide by the decimal: 30 ÷ 0.20 = 150. The three common phrases in the calculator correspond to these three rearrangements.

Increasing or decreasing a value by a percentage is a single multiplication. To add 20% to 150, multiply by 1.20 to get 180. To take 20% off, multiply by 0.80 to get 120. This is how sales tax, markups and discounts are usually calculated.

P × V₁ = V₂

P

the percentage written as a decimal (20% = 0.20)

V₁

the starting value

V₂

the result

Any one of the three can be found from the other two: V₂ = P × V₁, P = V₂ ÷ V₁, or V₁ = V₂ ÷ P.

Percentage difference formula

Percentage difference compares two values when neither is the obvious starting point, such as the prices of the same item at two stores or two measurements of the same quantity. It divides the absolute difference by the average of the two values, so the answer is the same whichever order you enter them.

Percentage difference = |V₁ − V₂| ÷ [(V₁ + V₂) ÷ 2] × 100. For 150 and 200, the difference is 50 and the average is 175, so the percentage difference is 50 ÷ 175 × 100 ≈ 28.57%.

Percentage change formula

Percentage change measures how much a value has grown or shrunk relative to where it started. It is the right tool when there is a clear before and after, such as last year's and this year's sales or an old and a new price.

Percentage change = (new value − old value) ÷ |old value| × 100. A positive result is an increase; a negative result is a decrease. Going from 150 to 200 is an increase of 50 ÷ 150 × 100 ≈ 33.33%. Going back from 200 to 150 is a decrease of 50 ÷ 200 × 100 = 25%. Because the base changes, a percentage increase followed by the same percentage decrease does not return you to the starting value: raising 100 by 50% gives 150, and cutting 150 by 50% gives 75.

Be careful not to confuse percentage change with a change in percentage points. If an interest rate rises from 4% to 5%, it has risen by one percentage point, which is a 25% increase in the rate.

Frequently asked questions