TL;DR: If a price is 20% more than original and now $120, original is 120 ÷ 1.20, not 120 × 0.80 and not 120 − 20. Reverse percent undoes a growth factor. Subtracting the percent as if it were dollars is how $120 “20% off original” becomes a fanfiction original.
Write the forward sentence first
Forward: original × 1.20 = 120 if “20% increase.” Therefore original = 120 / 1.20 = 100. Forward: original × 0.85 = 51 if “15% decrease.” Original = 51 / 0.85 = 60.
If you cannot write the forward sentence, you cannot reverse it. “Now $120 after a 20% discount” is original × 0.80 = 120, original = 150. Discount and increase use different factors. The words after “after a” pick the factor.
Why 120 minus 20% is a trap
20% of 120 is $24, 120−24 = 96, which would be the original only if 20% had been taken of the new price. That is a different tax. The 20% in a typical markup problem was taken of the old price. Different bases. If a shopkeeper marked up 20% from cost to 120, cost is 100, not 96.
Sales that say “pay 80% of original” are already in reverse-friendly language: divide by 0.80. Sales that say “20% off, now $120” mean the same as pay 80%.
Worked pair
Shirt is $48 after 20% off. Original = 48 / 0.80 = $60. Check: 60 × 0.80 = 48.
Population is 120% of last year’s. This year 960. Last year = 960 / 1.20 = 800. Check: 800 × 1.20 = 960.
If checks fail, you multiplied when you should have divided, or you used 1.20 on a discount.
Taxes added after a discount
Reverse gets uglier if two percents stacked. If you only know the final register total, you may not recover original without knowing the order. Homework usually tells the order. Real life prints a receipt. Use the receipt. Do not reverse two unknown percents from a single total and expect uniqueness. Math class problems are built to be unique. Receipts are built to be itemized.
Calculator as division by a factor
If the tool has a reverse or “original amount” mode, still write ÷ 1.20 on paper. If it does not, use any division. The intelligence is choosing 1.20 vs 0.80, which no tool can hear from a microphone. You have to read “increase” vs “off.”
Estimating so you do not reverse the wrong way
If the new value is after an increase, original should be smaller. If your reverse produced a bigger original for an increase, you used 0.80. If the new value is after a discount, original should be bigger. If you got a smaller original, you used 1.20. Direction is a free check. Use it before you box the answer.
When a problem gives “increased by 20” (dollars) not “increased by 20%,” that is subtraction, not reverse percent. Percents need the % sign or the word percent. Missing signs are English traps, not reverse-formula traps.
Tax-inclusive prices
If a shelf says $120 including 20% VAT, the pre-tax amount is 120 / 1.20, not 120 × 0.80. Including tax is an increase-style reverse. People use 0.80 because “take off 20%” feels like VAT. VAT was added to the pre-tax amount, not subtracted from the shelf. Different movie, same 20% prop.
Countries print tax-inclusive and tax-exclusive differently. Homework will say “including” or “plus tax.” Circle the word. Then pick 1.20 vs 0.80. The arithmetic is one division either way.
Successive reverse
Final $81 after 10% off then 10% off: remaining factors 0.9 × 0.9 = 0.81, original = 81 / 0.81 = 100. If you reverse only one 10% you get 90, which was an intermediate, not the start. Reverse in the opposite order of the discounts if they were not the same percent; divide by the last factor first. Same percents commute. Different percents do not confuse the product, but they do confuse people who subtract 20.
If you only know the final and “two discounts happened” without the percents, you cannot recover the original. Insufficient data is an answer. Inventing 20% because 20 is a popular number is not.
When the percent is of a different item
“Commission is 20% of wholesale, retail is $120” does not tell you wholesale unless retail is defined as wholesale plus commission or similar. Hidden relationships need a second sentence. Reverse percent is not a spell that recovers any missing number attached to a 20%. It recovers the base of that specific 20%.
Draw an arrow from the percent to its base. Reverse along that arrow. If the arrow does not point at the unknown, you need another equation. Systems of equations masquerading as “just reverse it” eat exam minutes.
Check with a tiny analog
If 20% increase to 120 feels abstract, use 20% increase to 12: original 10. Scale back up. Analog numbers keep the factor 1.20 visible. If your analog original is smaller than the analog new for an increase, you are on the right planet. Then divide the real 120 by 1.20.
I still write the factor even when using the reverse-friendly calculator. The factor is the proof. The output is the arithmetic. Proof first keeps 0.80 from sneaking into a markup problem because last night’s homework was a coupon.
A raise already baked into the offer letter
The letter says the new salary is $62,400 and that this is 4% above last year. Last year is not 62400 minus 4, and it is not 62400 times 0.96. Forward sentence: old times 1.04 equals 62400. Old equals 62400 divided by 1.04, which is $60,000. The raise dollars are $2,400. People who subtract 4% of the new number take 0.04 times 62400 = $2,496 off and land on $59,904. That is close enough to fool a tired reader and wrong enough to fail a payroll audit. Four percent of the new pay is a different base than four percent of the old pay.
Say the forward line out loud before you divide. If you cannot hear “old times 1.04,” you are about to reverse a discount instead of a raise. The words “above last year” pick the factor 1.04. The words “off last year” would have picked 0.96, and the letter would be quoting a cut.
Menu prices that already include 8.25% tax
A food truck prints $10.83 on the board and the footnote says tax included at 8.25%. The board price is pretax times 1.0825. Pretax is 10.83 divided by 1.0825, about $10.00, and the tax pennies are the rest. Multiplying 10.83 by 0.0825 answers “what is 8.25% of the tax-inclusive price,” which is about $0.89, not the tax that was actually added. The tax that was added is about $0.83. Cashiers who back out tax with the wrong factor short the drawer by a nickel per item and then blame the percent key.
Keep the factor next to the sentence. Included means divide by one-plus. Added-on means multiply by the decimal and then add. The board cannot be both unless two different menus are taped to the same window.
