Reverse Percentage Calculator
This reverse percentage calculator works back from a number that has already changed to the number it started at. Enter the amount after an increase, after a decrease, or a figure you know is a given percentage of something, and it returns the original.
How to use the reverse percentage calculator
- Pick the mode. After an increase for prices that went up, salaries after a raise, or totals that include tax. After a decrease for sale prices, discounts and reductions. X% of what? for the plain question “45 is 15% of which number?”.
- Enter the number you have. This is the figure after the change, not before. It is the price you paid, the salary you earn now, the total on the receipt.
- Enter the percentage that was applied. Just the percentage itself — 20 for a 20% change, not 1.20 or 0.80.
- Read the original. The result shows the starting value, the size of the change in units, and the multiplier used, so you can see the working rather than just the answer.
Everything here is one operation: dividing by a multiplier. The three modes exist because people arrive at that division from three different directions.
The reverse percentage formula
Forward, a percentage change multiplies. A 20% increase multiplies by 1.20; a 20% decrease multiplies by 0.80. Going backwards is the inverse operation, and the inverse of multiplying is dividing. That is the whole method.
- After a rise of p%:
original = current ÷ (1 + p ÷ 100) - After a fall of p%:
original = current ÷ (1 − p ÷ 100) - “A is p% of what?”:
whole = A ÷ (p ÷ 100)
The third one is the same division with a bare multiplier instead of one built from 1. If 45 is 15% of a number, then 45 ÷ 0.15 = 300. Converting the percentage to that decimal is the one step people skip; the percent to decimal calculator does it in isolation if you want to see it separately.
The multiplier table
Learn to read a percentage change as a single number to multiply by, and reversing it becomes obvious. Here are the multipliers you meet most often, with what each does to $200 in both directions.
| Change | Multiplier | $200 becomes | To reverse, divide by |
|---|---|---|---|
| up 5% | 1.05 | $210.00 | 1.05 |
| up 8% | 1.08 | $216.00 | 1.08 |
| up 10% | 1.10 | $220.00 | 1.10 |
| up 20% | 1.20 | $240.00 | 1.20 |
| up 25% | 1.25 | $250.00 | 1.25 |
| up 50% | 1.50 | $300.00 | 1.50 |
| down 10% | 0.90 | $180.00 | 0.90 |
| down 15% | 0.85 | $170.00 | 0.85 |
| down 20% | 0.80 | $160.00 | 0.80 |
| down 25% | 0.75 | $150.00 | 0.75 |
| down 30% | 0.70 | $140.00 | 0.70 |
| down 50% | 0.50 | $100.00 | 0.50 |
Two facts fall straight out of the right-hand column. Dividing by a multiplier greater than 1 makes the number smaller, which is what you want when undoing a rise. Dividing by a multiplier less than 1 makes the number bigger, which is what you want when undoing a fall. If your answer moved the wrong way, you multiplied when you should have divided.
The mistake almost everyone makes
The instinct is to take 20% of $120, which is $24, and add it back. That gives $144. Test it: 20% off $144 is 144 × 0.80 = $115.20, not $120. The answer is wrong by $6.
The reason is that the 20% was never a percentage of $120. It was a percentage of the original price, which is the number you are trying to find. Taking 20% of the reduced price measures the discount against the wrong quantity — a smaller quantity — so you recover too little.
Example: the same problem done correctly
The original was $150 and the discount was $30. That $30 is 20% of $150 and 25% of $120, which is exactly why the two approaches disagree.
The same asymmetry runs the other way. If something rose 20% to $120, the original was 120 ÷ 1.20 = $100, not $96. A rise of 20% and a fall of 20% are not inverses of each other: up 20% then down 20% takes $100 to $120 to $96, an overall 4% loss. If you are working the forward direction on a sale price, the percent off calculator handles it, including stacked discounts.
The always-safe check is to run your answer forward. Apply the original percentage change to the number you got. If it lands on the figure you started with, you are right.
Removing sales tax or VAT from a total
A tax-inclusive total is a percentage increase, so removing tax is a reverse percentage. Divide the gross figure by 1 + rate. Do not take the tax rate off the gross amount.
tax = gross − net
Example: pulling 8% sales tax out of $86.40
The pre-tax price is $80 and the tax is $6.40. Taking 8% of $86.40 instead gives $6.91, which would leave a net of $79.49 — and 8% of $79.49 is $6.36, not $6.91, so it fails the check.
Example: 20% VAT inside a $126 total
A useful shortcut for 20% VAT: the tax is the gross divided by 6. 126 ÷ 6 = 21. For a 5% rate the tax is the gross divided by 21, and for 25% it is the gross divided by 5.
The error here is worth quantifying because it is systematic. At an 8% rate, taking the percentage off the gross rather than dividing understates the net by about 0.6%. At a 20% rate the gap is 4%, which on a large invoice is real money.
Working back to a salary or a price before a rise
Anything quoted after a percentage increase reverses the same way: rent after an annual uplift, a fare after a fuel surcharge, a subscription after a price change, a salary after a raise.
Example: a salary after a 6% raise
You earn $58,300 now and the raise was 6%. What was it before?
Subtracting 6% of $58,300 instead would give $54,802, and a 6% raise on that is $58,090 — $210 short.
Example: two increases in a row
Rent went up 4% one year and 6% the next, and is now $1,653.60. Divide by both multipliers, in either order.
The starting rent was $1,500.00. Note that the two rises combine to 10.24%, not 10%, because the second applies to a base that already includes the first.
The “X is Y% of what?” case
This mode is for questions with no before-and-after at all: a part is given as a percentage of a whole and the whole is missing. If a deposit of $4,750 is 19% of a purchase price, then 4,750 ÷ 0.19 = $25,000. If 63 students are 45% of a year group, the year group is 63 ÷ 0.45 = 140. Setting a price against a target percentage from the other side is what the markup calculator is for.
Common mistakes
- Adding the percentage back on. $120 after 20% off is $150, not $144. Divide by 0.80.
- Taking the tax rate off the gross. An 8% tax on a $86.40 total is $6.40, found by dividing by 1.08, not the $6.91 you get by taking 8% of $86.40.
- Using the wrong multiplier direction. A rise divides by a number above 1; a fall divides by a number below 1. Check which way your answer moved.
- Assuming a rise and a fall of the same percentage cancel. Up 20% then down 20% leaves you 4% below where you started.
- Adding successive percentages. Two rises of 4% and 6% multiply to 10.24%, not 10%. Divide by each multiplier in turn.
- Confusing “30% of” with “30% more than”. The first divides by 0.30, the second by 1.30.
- Trying to reverse a 100% discount. The multiplier is zero and dividing by zero has no answer. Any original price would give a final price of $0.
- Skipping the forward check. Apply the change to your answer. If it does not return the number you started with, redo it.
Frequently asked questions
A jacket cost $120 after 20% off. What was the original price?
$150. The multiplier for 20% off is 0.80, so 120 ÷ 0.80 = 150. Check it forward: 150 × 0.80 = 120. The discount was $30. The common wrong answer is $144, from adding 20% of $120 back on, but 20% off $144 is $115.20, so it fails the check.
How do I remove 8% sales tax from a total of $86.40?
Divide by 1.08. 86.40 ÷ 1.08 = $80.00, so the tax was $6.40. Taking 8% of the $86.40 total gives $6.91, which is too much, because the tax was charged on the $80 net figure rather than on the gross. Use the “after an increase” mode with 8 as the percentage.
45 is 15% of what number?
300. Convert the percentage to a decimal and divide: 45 ÷ 0.15 = 300. Check it by going forward: 300 × 0.15 = 45. The wrong move is multiplying, which turns a part into something even smaller than the whole it came from. Whenever the number you have is a percentage of something, divide by that percentage in decimal form.
My salary is $58,300 after a 6% raise. What was it before?
$55,000. Divide by 1.06: 58,300 ÷ 1.06 = 55,000, and the raise was worth $3,300. Subtracting 6% of the new salary instead gives $54,802, which is $198 too low. The percentage was applied to the old salary, so the old salary has to be the number you divide back to.
Why can’t I just take the percentage off the new number?
Because the percentage was a share of the original, not of the result. Taking 20% of a discounted $120 measures against a base that is already 20% smaller, so you recover too little. The fix is always division by the multiplier. Test any reverse answer by running the change forward on it.
How do I reverse two percentage changes in a row?
Divide by each multiplier, in either order. Rent that rose 4% and then 6% to $1,653.60 came from 1,653.60 ÷ 1.06 ÷ 1.04 = $1,500.00. Combining them first works too: 1.04 × 1.06 = 1.1024, so the total rise was 10.24%, not 10%. Successive percentages never simply add, because each one is applied to a base the previous one already changed.
What was the price before a 25% increase if it is now $250?
$200. Divide by the multiplier 1.25: 250 ÷ 1.25 = 200, and the increase was $50. Subtracting 25% of $250 gives $187.50, which is wrong — adding 25% to $187.50 returns $234.38, not $250. The forward check catches this every time, and it costs one multiplication.
Does a 20% rise followed by a 20% fall get me back to where I started?
No, you end 4% lower. $100 rises to $120, then a 20% fall takes 20% of $120, which is $24, leaving $96. The two percentages are measured against different bases. To return to $100 from $120 you need a fall of 16.67%, since 20 ÷ 120 = 0.1667.
What if the discount was 100%?
Then the multiplier is zero and the original price cannot be recovered. Every possible original price gives a final price of $0, so there is no unique answer, and the division is undefined. The same applies to “0 is 0% of what?”. Any percentage below 100 reverses normally.
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