Reverse Percentage: How to Find the Original Number

A shirt on the rack is marked $45, down from a price the tag no longer shows. A payslip reads $4,400 after a 12% raise, and you want to know the old salary. A restaurant bill of $132 already includes a 10% service charge. Three different moments, one identical question: what was the original number?

Quick answer: To find the original number, divide the final value by (1 ± the rate). For a price reduced by 20%, divide by 0.80. For a value grown by 15%, divide by 1.15. Example: $80 after a 20% discount means the original price was $80 ÷ 0.80 = $100.

The one formula that covers every case

Every reverse-percentage question is the same question wearing different clothes. A percentage change turns an original number into a final number by multiplying it: the original times (1 − rate) for a decrease, or times (1 + rate) for an increase. To travel backwards, you undo that multiplication — which means dividing by the same multiplier.

So the universal rule is: original = final ÷ multiplier, where the multiplier is (1 − r) after a decrease and (1 + r) after an increase. Convert the rate to a decimal first — 20% becomes 0.20 — build the multiplier, then divide. That is genuinely all there is. Here are the multipliers you will reach for most often:

Change	Multiplier	Operation
10% off	0.90	Final ÷ 0.90
20% off	0.80	Final ÷ 0.80
25% off	0.75	Final ÷ 0.75
40% off	0.60	Final ÷ 0.60
10% more	1.10	Final ÷ 1.10
15% more	1.15	Final ÷ 1.15
50% more	1.50	Final ÷ 1.50

Why can’t you just add the percentage back on?

This is where almost everyone goes wrong, so it deserves its own section. If a $100 jacket is marked down 20%, you pay $80. Intuition says adding 20% to $80 gets you back to $100 — but $80 plus 20% is $96, not $100. The intuition fails because the two percentages apply to different bases: 20% of $100 is $20, but 20% of $80 is only $16.

To climb from $80 back to $100, you need to add $20 — and $20 is 25% of $80. So the return journey needs a 25% increase, not a 20% one: $80 × 1.25 = $100. Going down and coming back up are never symmetric, because each percentage is measured against a different starting line. Remember this whenever someone quotes you a “recover the loss” figure — for instance, after an investment drops 50%, it must rise 100% to break even.

Three everyday scenarios, fully worked

Scenario 1: the sale price. A jacket rings up at $80 after a 20% discount. The multiplier for 20% off is 0.80, so the original price was $80 ÷ 0.80 = $100. Check it forwards: 20% of $100 is $20, and $100 − $20 = $80. Correct.

Scenario 2: the marked-up price. A laptop sells for $1,080 after the store adds a 35% markup over its cost. The multiplier for a 35% increase is 1.35, so the original cost was $1,080 ÷ 1.35 = $800. Forwards check: 35% of $800 is $280, and $800 + $280 = $1,080. Correct.

Scenario 3: the raise. Your new salary is $66,000 after a 10% raise. The multiplier is 1.10, so your old salary was $66,000 ÷ 1.10 = $60,000. Forwards check: 10% of $60,000 is $6,000, and $60,000 + $6,000 = $66,000. Correct. If you want to double-check any of these on the fly, the percentage calculator handles the division for you.

Stripping tax and tips out of a total

The same division trick works whenever a percentage has already been folded into a number you see. A $108 receipt in an area with 8% sales tax means the pre-tax price was $108 ÷ 1.08 = $100. A bill of $74.80 with a 15% tip already added means the food itself cost $74.80 ÷ 1.15 = $65.04 (check: 15% of $65.04 is $9.76, and $65.04 + $9.76 = $74.80).

This also works for commissions, VAT, and service charges — anything described as “price plus X%” is a price that was multiplied by (1 + X%), so dividing by (1 + X%) peels it back off. The one requirement is knowing the rate that was added. If a bill bundles tax and tip together as a single 23%, divide by 1.23: you get the pre-everything price, though you can no longer split the two additions apart.

The multiply-back check: verify any answer in seconds

Here is a habit that makes reverse percentages nearly mistake-proof. After you divide and get your “original” number, immediately multiply it forward by the percentage change and confirm you land back on the number you started with. Say you calculate that $13,200 after a 10% gain came from $13,200 ÷ 1.10 = $12,000. Multiply back: 10% of $12,000 is $1,200, and $12,000 + $1,200 = $13,200. It matches, so the answer is right.

This check catches the two classic errors in one move: dividing by the rate itself instead of the multiplier (the “$80 ÷ 0.20” mistake), and adding the percentage back on symmetrically (the “$80 plus 20%” mistake). If the forward trip does not return you to the final number, your original is wrong — and you will know before it matters, not after.

How do I find the original price before a discount?

Divide the sale price by (1 − the discount rate). A 25%-off price of $75 becomes $75 ÷ 0.75 = $100. The mistake to avoid is subtracting the percentage from the sale price — the discount was taken from the original number, so the original is what you divide by the remaining share.

How do I reverse a percentage increase?

Divide the new amount by (1 + the rate). If a $60,000 salary followed a 10% raise, the old salary was $60,000 ÷ 1.10 = $54,545.45. The raise percentage was calculated from the starting salary, so you must divide — subtracting 10% of the new figure gives a wrong answer.

Why doesn’t adding the same percentage back restore the original?

Because each percentage applies to a different base. Cutting $100 by 20% leaves $80, and raising $80 by 20% gives $96 — not $100. The missing $20 is 25% of $80, so restoring the original needs a 25% increase. Going down and back up are never symmetric.

How do I remove sales tax from a total price?

Divide the total by (1 + the tax rate). A $108 receipt where tax is 8% means the pre-tax price was $108 ÷ 1.08 = $100. The same trick strips VAT, service charges, or any percentage already folded into a total — as long as you know the rate that was added.

Is there one formula for every reverse percentage?

Yes: original = final ÷ multiplier, where the multiplier is (1 − r) for a decrease and (1 + r) for an increase. Convert the rate to a decimal (20% = 0.20), build the multiplier, then divide. It works for discounts, markups, raises, taxes, tips, and population growth alike.

Reverse percentages look intimidating only because the direction feels unnatural — every percentage you meet in daily life is described going forward. Flip it around: identify the final number, build the (1 ± r) multiplier, and divide. Run the multiply-back check, and you will never be fooled by a missing price tag again.