Calculators & everyday math
Percentage Formulas Explained: Part, Whole, and Rate
Almost every everyday percentage question is one of three questions in disguise, all built from the same relationship between a part, a whole, and a rate. Once you can tell which one you’re being asked, the formula follows. This guide lays out all three, plus the formula for change over time.
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One relationship behind them all
part = whole × rate ÷ 100
In “12 is 15% of 80”, 80 is the whole, 15 is the rate (the percentage), and 12 is the part. Every basic percentage question gives you two of these and asks for the third.
The three questions
| You know | You want | Formula | Example |
|---|---|---|---|
| Rate and whole | Part | part = whole × rate ÷ 100 | 15% of 80 = 80 × 15 ÷ 100 = 12 |
| Part and whole | Rate | rate = part ÷ whole × 100 | 12 out of 80 = 12 ÷ 80 × 100 = 15% |
| Part and rate | Whole | whole = part ÷ rate × 100 | 12 is 15% of 12 ÷ 15 × 100 = 80 |
Telling the questions apart
- “What is 15% of 80?” — the percentage is given and “of” points to the whole. You want the part.
- “What percent of 80 is 12?” or “12 out of 80 is what percent?” — you want the rate.
- “12 is 15% of what?” — the whole is missing. This is the one people get wrong most often, because it means dividing by the percentage rather than multiplying.
The fourth formula: change over time
When something moves from an old value to a new one, the rate you want is measured from the old value: percentage change = (new − old) ÷ old × 100.
It is the rate formula in disguise — the part is the change and the whole is the starting value. From 80 to 92 the change is 12, and 12 ÷ 80 × 100 = 15%.
Percentage Formulas FAQ
- How do I find what percent one number is of another?
- Divide the part by the whole and multiply by 100. 18 out of 24 is 18 ÷ 24 × 100 = 75%.
- How do I find the whole when I know a percentage of it?
- Divide the part by the percentage and multiply by 100. If 30 is 40% of a number, the number is 30 ÷ 40 × 100 = 75.
- Why multiply by 100?
- A rate like 12 ÷ 80 = 0.15 is a fraction of 1. Percent means “per hundred”, so multiplying by 100 turns 0.15 into 15%.
- Does “of” always mean multiply?
- In percentage questions, yes: “15% of 80” means 0.15 × 80.
- Which value is the whole in a percentage change?
- The starting value. That is why a rise from 80 to 92 is 15%, while a fall from 92 back to 80 is about 13.04%.
Related guides
- Percentage of a NumberThe one method that always works for finding a percentage of a number, plus mental shortcuts for 10%, 15%, 25%, and more, with worked examples.
- Calculate Percentage ChangeThe percentage change formula with worked examples, how to read positive and negative results, a spreadsheet formula, and checks that catch errors.
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