Sometimes you know a part and its percent, but not the whole. "12 is 40% of what number?" You can still use what you know about percents of a number: work backward to 100%.
Key idea
If a part is $p\%$ of the whole, use the part to find 10% (or 1%, or another easy percent), then multiply up to 100%.
Why it works
The part and the whole are in the same ratio as the percent and 100: $12 : \text{whole} = 40 : 100$. Equivalent ratios do the rest: divide by 4 to get "3 is 10%", then multiply by 10 to get "30 is 100%".
| amount | 12 | 3 | 30 |
|---|---|---|---|
| percent | 40% | 10% | 100% |
Worked example: Work back to 100%
12 is 40% of what number?
- 40% is 4 groups of 10%, so 10% is $12 \div 4 = 3$.
- 100% is 10 groups of 10%: $3 \times 10 = 30$.
- Check: 40% of 30 is $4 \times 3 = 12$.
Answer: 30
Worked example: Saving for a bike
Ana has saved \$45. That is 60% of the price of a bike. What is the price?
- 60% is 3 groups of 20%, so 20% is $\$45 \div 3 = \$15$.
- 100% is 5 groups of 20%: $5 \times \$15 = \$75$.
- Check: 60% of \$75 is \$45.
Answer: \$75
Common mistake
Multiplying the part by the percent. $12 \times 40\% = 4.8$ answers a different question: "What is 40% of 12?" Here 12 is the part, not the whole. Quick check: when the percent is less than 100%, the whole must be bigger than the part. 4.8 is smaller than 12, so it cannot be the whole.