How do I find the original value before a percentage increase?

Divide by one plus the rate as a decimal. A figure of £120 that includes a 20% increase comes from £120 ÷ 1.2 = £100. The instinct to subtract 20% instead is wrong because the 20% was calculated on the original, not on the figure you are holding — the base has changed, and division is what undoes that.

Why can I not just subtract the percentage?

Because the percentage was a share of the original, and the number in front of you is the total that includes it. Subtracting takes the same percentage of a larger figure, so it removes too much. The result is short by exactly the rate squared — 4% at a 20% rate, 9% at 30%, 25% at 50%. At small rates the gap is easy to overlook, which is precisely why the habit survives.

How do I remove VAT from a total?

Divide by 1 plus the VAT rate. At 20% that means dividing by 1.2, so £120 gross is £100 net with £20 of VAT. The faster route to the VAT itself is the fraction p ÷ (100 + p) of the gross figure — at 20% that is one sixth, so £120 ÷ 6 = £20 without calculating the net price at all. At 5% the fraction is one twenty-first.

How do I find the original price before a discount?

Divide by one minus the discount as a decimal. A £80 jacket in a 20% sale started at 80 ÷ 0.8 = £100. The mirror-image mistake here is adding 20% back to £80, which gives £96 — short by the same 4%, because the same 1 − p² factor applies whichever direction the change went.

Why does a 20% gain then a 20% loss not break even?

Because the second percentage is taken of a different base. Up 20% takes you to 1.2, and down 20% of that takes 0.24 away rather than the 0.2 you gained — landing at 0.96, or 4% down. The general result is 1 − p², which is the same quantity that measures the subtraction error, and for good reason: both come from multiplying (1 + p) by (1 − p).

What is the fraction shortcut for tax?

The tax contained in a gross figure is the rate divided by a hundred plus the rate. At 20% that is 20 ÷ 120, or exactly one sixth. At 5% it is one twenty-first, at 10% one eleventh, at 12.5% one ninth, and at 25% one fifth. Not every rate gives a memorable fraction — 7% works out to 7 ⁄ 107, which is exact and useless — but where it does, the shortcut saves a step and is easy to do mentally.

Can a 100% decrease be reversed?

No, and the arithmetic says so directly: the formula divides by 1 − 1, which is zero. A fall of 100% leaves nothing, so every possible starting value produces the same final figure and there is no way to tell which one it was. Beyond 100% the value would have had to pass through zero into negative territory, which is a different situation than a percentage decrease describes.

How is this different from a percentage change calculation?

Percentage change starts with two values and asks what the rate was. A reverse percentage starts with one value and the rate, and asks for the other value. They are the same relationship read in different directions, and the confusion usually comes from the word "of": in a change calculation you divide by the old value; in a reverse calculation you divide by the multiplier, which is one plus or minus the rate.

Does the order of two percentages matter?

No, though it feels as though it should. Multiplication commutes, so a 10% rise followed by a 20% fall gives the same result as a 20% fall followed by a 10% rise — 1.1 × 0.8 and 0.8 × 1.1 are both 0.88. What does matter is that neither sequence gives the 10% net change that adding the percentages would suggest. Percentages compose by multiplying, not by adding.

How do I check my answer?

Apply the change forwards and see whether you land on the figure you started with. If the original really was £100 and the rate really was 20%, then 100 × 1.2 must give back exactly 120. This catches the subtraction error immediately: 96 × 1.2 is 115.20, not 120, which says at once that £96 was never the original.

Reverse Percentage Calculator

A reverse percentage works backwards: you have the figure after a change and want the one before it. You divide. After an increase of p, the original is the final divided by (1 + p). After a decrease, divide by (1 − p). What you do not do is subtract, and that is the mistake almost everyone makes —...

original = final ÷ (1 + rate)

You divide — you do not subtract. Taking 20% off a price that already includes 20% VAT gives an answer 4% too low, and that shortfall is exactly the rate squared at every rate.

WHAT YOU KNOW

THE VALUE YOU HAVE

RATE (%)

symbol

THE ORIGINAL VALUE

£100.00

£120.00 ÷ 1.2000 = £100.00. The increase itself was £20.00, which is 20.00% of the original rather than of the figure you started with.

ORIGINAL

£100.00

ADDED

£20.00

FINAL

£120.00

Taking 20.00% off £120.00 would give £96.00, and that is wrong by £4.00. The size of the error is not arbitrary: it is exactly 20.00% squared, or 4.00% of the true answer. That identity holds at every rate, because the wrong method multiplies by (1 − r) where the right one divides by (1 + r), and those two differ by precisely 1 − r². At small rates it is easy to miss — 5% costs a quarter of a percent — and it grows quickly.

The shortcut worth memorising. The amount added is the rate over one hundred plus the rate — here 20.00 ÷ 120.00, which is exactly 1⁄6. So the amount added to £120.00 was £20.00, reachable in one step rather than two. At 20% that is one sixth, which is why VAT-registered businesses divide gross takings by six in their heads.

Step	Working	Result
The multiplier	1 + 20.00 ÷ 100	1.2000
Divide	£120.00 ÷ 1.2000	£100.00
The change	£120.00 − £100.00	£20.00
Check	£100.00 × 1.2000	£120.00

The same squared term explains why percentages do not undo each other. A value that rises 20.00% and then falls 20.00% does not return to where it started — it lands at 96.000% of it, short by the same 4.00%. Up and down are not symmetric operations, because the second percentage is taken of a different base than the first. It is the same algebra as the mistake above, seen from another angle.

WORKING BACKWARDS

The percentage was taken of the original, not of the final. That is the whole reason reversing needs division: the base has changed, and the figure you are holding is the one that came after.

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HOW TO USE

1
Choose whether the value went up or came down. The two are different calculations: dividing by 1.2 and dividing by 0.8 give quite different answers from the same figure.
2
Enter the value you actually have — the one after the change — and the rate that was applied to the original.
3
Read the multiplier before the answer. Everything follows from dividing by it, and seeing it written out is what makes the difference between this and subtracting obvious.
4
Use the fraction shortcut when the rate is a common one. At 20% the added amount is a sixth of the gross figure, which needs no division at all.

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Last updated: August 2, 2026 · original = final ÷ (1 ± rate) · Subtracting instead falls short by exactly the rate squared.