Key takeaway
To find the base (whole) from a percent and its part, rearrange the formula: whole = part ÷ (percent ÷ 100). If 30 is 20% of something, the whole is 30 ÷ 0.20 = 150. The trap is dividing by the percent without first turning it into a decimal.
Scope and assumptions
This is the reverse of the standard part-of-whole formula. Domain rules (tax-inclusive vs exclusive) may change the setup. This site is informational and not official.
Short answer: To find the base (whole) from a percent and its part, rearrange the formula: whole = part ÷ (percent ÷ 100). If 30 is 20% of something, the whole is 30 ÷ 0.20 = 150. The trap is dividing by the percent without first turning it into a decimal.
Quick navigation
- The formula → percentage-formula
- How to calculate → how-to-calculate-percentage
- Reverse a decrease → reverse-a-percentage-decrease
- The calculator → Percentage Calculator
1. Start from the basic formula
The part-of-whole formula is:
P% = (part ÷ whole) × 100
You know P% and part, and want the whole. Solve for whole:
whole = part ÷ (P% ÷ 100)
2. Worked example
30 is 20% of what number?
- part = 30, percent = 20 → decimal 0.20
- whole = 30 ÷ 0.20 = 150
Check: 30 of 150 = (30 ÷ 150) × 100 = 20%. ✓
3. Reverse table
| Part | Percent | Whole = part ÷ (P/100) |
|---|---|---|
| 30 | 20% | 30 ÷ 0.20 = 150 |
| 45 | 15% | 45 ÷ 0.15 = 300 |
| 12 | 8% | 12 ÷ 0.08 = 150 |
| 90 | 75% | 90 ÷ 0.75 = 120 |
4. A shopping example
A jacket is $24 off, and that discount is 30% of the original price. Find the original:
whole = 24 ÷ 0.30 = $80
So the original price was $80; the sale price is $56. Same math, different label.
5. Common mistakes
- Dividing by the percent, not its decimal — 30 ÷ 20 = 1.5 is wrong; use 0.20.
- Dividing the wrong way — whole = part ÷ decimal, not decimal ÷ part.
- Using the slice as the whole — when percent < 100, the base is always bigger than the part.
- Confusing with reverse-change — this finds the base from a straight percent; a "before a 20% rise" problem needs reverse-a-percentage-decrease logic instead.
6. More base examples
| Part | Percent | Whole = part ÷ (P/100) |
|---|---|---|
| 18 | 12% | 18 ÷ 0.12 = 150 |
| 54 | 45% | 54 ÷ 0.45 = 120 |
| 9 | 5% | 9 ÷ 0.05 = 180 |
| 66 | 33% | 66 ÷ 0.33 = 200 |
| 120 | 80% | 120 ÷ 0.80 = 150 |
The base is always larger than the part when the percent is under 100, because the part is only a slice of the total you are recovering. Once the percent passes 100, the base becomes smaller than the part — a useful sanity check as you calculate.
7. Real situations that need the base
Finding the base shows up whenever a discount or a share is known but the original is not. A $24 markdown at 30% off means the original was 24 ÷ 0.30 = $80. A survey where 90 people are 75% of respondents means 90 ÷ 0.75 = 120 people total. In both, you know the slice and the rate, and you recover the whole by dividing — never by subtracting the percent from the slice.
The base is the number the percent was taken from, and that is exactly what the formula gives back. When a question gives you "X is P% of what", the word "of" is your clue that the missing piece is the whole, so divide the part by the decimal form of the percent. This reverses the usual "percent of a number" direction on purpose.
FAQ
Q: How do I find the whole from a percent? whole = part ÷ (percent ÷ 100). If 30 is 20%, the whole is 30 ÷ 0.20 = 150.
Q: What if I know the percent and the part? That is exactly the base case — divide the part by the decimal form of the percent.
Q: Why convert the percent to decimal first? Otherwise 30 ÷ 20 = 1.5, which is wrong; the percent must become 0.20 before dividing.
Q: Is this the same as reverse percentage? Yes — finding the base from a percent is one form of reverse percentage. For a before-increase value see reverse-a-percentage-decrease.
Q: Can the base be larger than the part? Always, when the percent is under 100 — the part is only a slice of the whole.
Sources: Khan Academy percentage basics; Maths is Fun — Introduction to Percentages.
Disclaimer: This is the reverse of the standard part-of-whole formula. Domain rules (tax-inclusive vs exclusive) may change the setup. This site is informational and not official.
Sources and references
- Khan Academy percentage basicshttps://www.khanacademy.org/math/pre-algebra/x31947de81e3f6453:percentages
- Maths is Fun — Introduction to Percentageshttps://www.mathsisfun.com/percentage.html