How to calculate percentages

Three percentage problems, one set of tools

Almost every percentage question you run into day to day is one of three types: finding a percentage of a number, finding what percentage one number is of another, or finding how much a number changed in percentage terms. A tip, a test score, and a rent increase all use different formulas, even though they all involve the word percent.

The trick is telling them apart before you calculate, since the wrong formula on the right numbers gives a confident, wrong answer. Each type below has its formula and a real worked example.

Finding X percent of Y

This is the most common case: you know a percentage and a total, and you want the piece that percentage represents. Convert the percent to a decimal by dividing by 100, then multiply by the total.

Example: you want to tip 18% on a $64 restaurant bill. Convert 18% to 0.18, then multiply: 0.18 x 64 = $11.52. That is your tip amount, and $64 + $11.52 = $75.52 is the total you would pay.

• Formula: Result = (X / 100) x Y
• X is the percentage, Y is the total or base amount
• Example: 18% of $64 = (18 / 100) x 64 = $11.52

Finding what percent X is of Y

This flips the first case around: you know a piece and a total, and you want to express the piece as a percentage of the total. Divide the piece by the total, then multiply by 100 to convert the resulting decimal into a percent.

Example: you answered 42 questions correctly out of 50 on a test. Divide 42 by 50 to get 0.84, then multiply by 100: your score is 84%.

• Formula: Percent = (X / Y) x 100
• X is the part, Y is the whole it is being compared to
• Example: 42 out of 50 = (42 / 50) x 100 = 84%

Finding percent change: increases and decreases

Percent change measures how much a number moved, relative to where it started. Subtract the old value from the new value, divide by the old value, and multiply by 100. A positive result is a percent increase; a negative result is a percent decrease.

Example, an increase: rent goes from $1,200 to $1,350 a month. Change is $1,350 minus $1,200, which is $150. Divide by the old value: 150 / 1,200 = 0.125. Multiply by 100 for a 12.5% increase.

Example, a decrease: a jacket drops from $80 to $60. Change is $60 minus $80, which is negative $20. Divide by the old value: -20 / 80 = -0.25. Multiply by 100 for a result of -25%, meaning the price decreased by 25%.

• Formula: Percent change = ((New - Old) / Old) x 100
• The old, starting value is always the denominator, never the new one
• A negative result means a decrease; drop the sign and call it a percent decrease

Percentage points versus percent, and why changes don't add

Two mix-ups cause more wrong answers than any calculation mistake. The first is confusing percentage points with percent. If a savings rate moves from 4% to 6%, that is a 2 percentage point increase, a simple difference of the two figures. But expressed as a relative percent change, it is a 50% increase, since (6 - 4) / 4 x 100 = 50%. Both descriptions are correct; they answer different questions, and swapping one for the other changes the meaning of the number.

The second mix-up is treating percent changes as if they add and subtract like plain numbers. A 20% increase followed by a 20% decrease does not return you to where you started. Start at $100: increase by 20% to reach $120. Decrease that $120 by 20%, which is $24, to land at $96, four dollars short of the original $100. Each percent change applies to whatever the current value is, not the original one, so consecutive changes compound instead of canceling out.

Common mistakes

Most percentage errors come from a handful of repeatable slips. Watch for these.

• Using the wrong base for percent change: the denominator is always the starting value, never the ending one, or the answer will be off.
• Forgetting to multiply by 100: dividing 42 by 50 correctly gives 0.84, but reporting that as the answer instead of 84% misstates the result by a factor of 100.
• Adding percent changes across different bases: '10% off, then an extra 10% off' is not 20% off the original price; the second discount applies to the already-reduced price, so the combined discount is 19%.
• Swapping percentage points for percent, or the reverse: a rate rising 2 percentage points and a rate rising 50 percent can describe the same move, so state which one you mean.
• Rounding too early in a multi-step problem: round only the final answer, since rounding partway through a calculation compounds small errors into a visibly wrong result.