The secret to percentages? Start with the base.
A practical way to stop second-guessing discounts, ratios, and percentage comparisons.
Ask “a percentage of what?”
A percentage is a ratio expressed per hundred. To find what percentage one number is of another, divide the part by the whole and multiply by 100. For example, 25 out of 200 is 25 ÷ 200 × 100 = 12.5%.
The most important choice is the base, or denominator. The same 25 is 25% of 100 but only 12.5% of 200. A percentage without a clearly stated base can be misleading.
Reversing a percentage needs division
Suppose an item is $80 after a 20% discount. The discounted price is 80% of the original, so the original is $80 ÷ 0.80 = $100. Adding 20% to $80 gives $96 and does not undo the discount because the base has changed.
The same idea applies when removing included tax. A $108 total including an 8% tax has a pre-tax base of $108 ÷ 1.08 = $100. Subtracting 8% of $108 would remove too much.
Percentage points are different from percent change
A change from 20% to 25% is an increase of 5 percentage points. Relative to the original 20%, it is a 25% increase: (25 − 20) ÷ 20 × 100. These are two useful but different descriptions.
When comparing rates, state whether you mean the arithmetic difference in percentage points or a relative percentage change. If the original value is zero, the usual percentage-change formula is undefined.
Use a quick reasonableness check
For a positive whole, half is 50%, one quarter is 25%, and one tenth is 10%. If your answer conflicts with those familiar landmarks, check whether you reversed the part and whole.
The Percentage Calculator shows the denominator in both the form and the working. Enter your numbers and verify the base before sharing the result.
Make the numbers your own.
Put this guide into practice with a free calculator.