Reverse Percentages

A reverse percentage starts from the end. You know what the number became after someone added or removed a percent, and you want the number from before. The forward formulas multiply. The reverse formulas divide by the same factor. Almost every error in this family is a subtraction that felt like an undo key and was not.

After an increase

If a value grew by p percent, the result equals original × (1 + p/100). So:

original = result / (1 + p/100)

A bill is €115 after a 15% service charge. Original = 115 / 1.15 = €100. Check: 15% of 100 is 15.

Subtracting 15% of 115 instead: 115 × 0.15 = 17.25, and 115 − 17.25 = 97.75. Then 97.75 × 1.15 ≈ 112.41, not 115. You overshot. The percent you subtracted was a percent of the wrong base — the enlarged one.

A salary is €48,000 after a 4% raise. Previous = 48000 / 1.04 ≈ €46,153.85. The raise itself was about €1,846, which is 4% of the old salary, not 4% of the new one (4% of 48,000 is €1,920, a common wrong answer that then fails the check: 46,080 × 1.04 = 47,923, not 48,000).

After a decrease

original = result / (1 − p/100)

The divisor must stay positive, so p cannot be 100 or more. A 100% decrease leaves zero, and you cannot uniquely recover a starting value from zero.

You paid €70 after 30% off. original = 70 / 0.70 = €100.

A stock index is at 8,000 after a 20% drop. Previous = 8000 / 0.80 = 10,000. People say “it needs to rise 20% to recover.” It needs to rise 2,000 on a base of 8,000, which is 25%. The reverse of a percent drop is not the same percent rise. That asymmetry is the heart of common percentage mistakes.

VAT and tax included

A gross price is a reverse-percentage problem in work clothes. Net = gross / (1 + rate/100). At 23%, divide by 1.23. At 20%, divide by 1.20. The VAT calculator extract mode is this division with the tax line shown separately. Doing it by hand matters when you want to see why “23% of the gross” is the wrong tax. Full examples live in how to calculate VAT.

Two steps, still reversible

A price was increased 10%, then discounted 10%. The combined factor is 1.10 × 0.90 = 0.99. If you pay €99, the original was 99 / 0.99 = €100. You are not back by coincidence of “+10 and −10”; the factors multiply to 0.99, so you are 1% down, and the reverse divides by 0.99.

If you know only the final price and only one of the percents, you cannot uniquely split the history. Reverse percentages need the rate. They do not infer it. If the rate is what you are missing, you need both the before and the after, which is ordinary percent change, not a reverse.

A worksheet of undos

Read the verb before you pick the row. “20% smaller than last year” is a decrease, divisor 0.80. “20% of last year” would be a different sentence and a different original (if 48 is 20% of last year, last year was 240). The word than usually signals a change. The word of usually signals a share. Reverse percentages are for the change case, when the rate is known and the start is not.

The marks row is easy to misread in the other direction. If a teacher added 10 percentage points to a percent score, you subtract 10 points; you do not divide by 1.10. “Added 10% of the raw score” is a relative bump and does divide. Percentage points is the vocabulary. This table assumes the relative reading because that is the one division undoes.

Fees that are not a pure percent

A €100 item with 20% off and then a €5 shipping fee is not a reverse-percent puzzle until you remove the fee. Sale price of the item = (amount you paid − shipping) only if shipping was added after the discount and was not itself discounted. If you paid €85 all-in and shipping was €5, the discounted item was €80, and the pre-discount price at 20% off was 80 / 0.80 = €100. If you divide 85 by 0.80 you get €106.25 and you have treated the shipping as if it had been discounted too. Peel off flat fees first. Then divide once.

Successive percents still use one combined divisor. Ten percent off and then another ten percent off is a pay factor of 0.81, so a €81 tag came from €100, not from 81 / 0.90 / 0.90 done in the wrong order — those two orders happen to match, which is comforting, and a fixed fee in the middle of them would not.

When to use a calculator

Use the forward tools to check: take your recovered original, apply the percent, and demand the result you started with. If it misses, you subtracted instead of dividing. Use reverse division whenever a tag, a payslip, or a tax-inclusive total is the only number printed and the rate is stated next to it. If the rate is not stated, stop. Inventing “probably 20%” is not a reverse percentage. It is a guess with extra steps. The discount calculator checks the forward direction. The VAT calculator extract mode is the tax-shaped version of the same division. Neither tool can infer a rate you do not have.