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Percentage Calculator

Q: Why doesn't a 20% rise then a 20% fall return the original?

Because each percentage is applied to a different base. Increasing 100 by 20 percent gives 120. Decreasing 120 by 20 percent removes 24, not 20, leaving 96. The rise is calculated on 100 and the fall on 120, so they do not cancel. To reverse a 20 percent increase you divide by 1.2, which removes 16.67 percent.

Q: Percentage change or percentage difference?

Percentage change has a direction: it measures movement from an old value to a new one, so the starting value is the base. Percentage difference compares two values with no before and after, using their average as the base. Change from 40 to 50 is 25 percent; the percentage difference between 40 and 50 is 22.2 percent.

Q: How do I find the original price before a discount?

Divide rather than subtract. If an item costs 80 after a 20 percent discount, the sale price represents 80 percent of the original, so divide 80 by 0.8 to get 100. Subtracting 20 percent from 80 gives 64, which is wrong, because the discount applied to the original price and not to the reduced one.

Q: Can a percentage be more than 100?

Yes. Anything above 100 percent simply means more than the whole. If sales go from 50 to 150, that is a 200 percent increase, because the gain is twice the original amount. Percentages over 100 are only impossible when the quantity is bounded by definition, such as the share of a single total.

Four percentage questions, one page. Pick the one you're asking, and the working is shown underneath the answer.

What is P% of a value?

Percentage

Of this value

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Step-by-step workingCheck by hand

How percentages actually work

A percentage is just a fraction with 100 fixed as the denominator. "Per cent" means per hundred, so 15% is another way of writing 15/100, or 0.15. Every percentage calculation is that same conversion followed by ordinary arithmetic — which is why the working below always starts by dividing by 100.

The confusion almost never comes from the arithmetic. It comes from losing track of what the percentage is being taken of. Change the base and the same percentage produces a different number, which is the root of nearly every percentage mistake people make.

The three bases people mix up

Percent vs percentage points

A rate moving 4% → 5% is one percentage point, but a 25% increase. Percentage points measure the gap; percent measures relative change.

Increases don't reverse

100 up 20% is 120. 120 down 20% is 96, not 100 — the rise was taken on 100 and the fall on 120. To undo a 20% rise, divide by 1.2.

Change vs difference

Change has a direction and uses the starting value as its base. Difference has no before-and-after and uses the average of the two. 40 → 50 is 25% change but 22.2% difference.

The four formulas

P% of V = V × (P ÷ 100)
X is what % of Y = (X ÷ Y) × 100
change = ((new − old) ÷ |old|) × 100
V adjusted by P% = V × (1 ± P ÷ 100)

Worked example

A jacket is marked down 30% to $63. What did it cost before? The instinct is to add 30% to 63, giving $81.90 — and that is wrong. The 30% came off the original price, so $63 represents 70% of it. Divide instead: 63 ÷ 0.70 = $90. Check it: 30% of 90 is 27, and 90 − 27 = 63. ✓

That single move — dividing to reverse a percentage rather than subtracting — is worth more than any other percentage rule. It applies to discounts, tax-inclusive prices, and anywhere you know the result and want the starting figure.

Frequently asked questions

What's the difference between percent and percentage points?

They measure different things. If an interest rate moves from 4% to 5%, that is a rise of one percentage point — but a 25% increase, because 1 ÷ 4 = 0.25. Percentage points describe the gap between two percentages; percent describes relative change. Mixing them up is the commonest percentage error in news reporting, and it usually makes a change sound far larger or smaller than it is.

Why doesn't a 20% rise then a 20% fall return the original?

Because each percentage applies to a different base. 100 increased by 20% is 120. 120 decreased by 20% removes 24 — not 20 — leaving 96. The rise was calculated on 100 and the fall on 120, so they cannot cancel. To reverse a 20% increase you divide by 1.2, which is a decrease of 16.67%.

Percentage change or percentage difference?

Use change when there is a before and an after — the starting value is the base. Use difference when comparing two values with no direction, such as two products' prices; it uses their average as the base. From 40 to 50 is a 25% change, but the percentage difference between 40 and 50 is 22.2%.

How do I find the original price before a discount?

Divide, don't subtract. If something costs $80 after 20% off, that $80 is 80% of the original, so 80 ÷ 0.8 = $100. Subtracting 20% from 80 gives 64, which is wrong — the discount applied to the original price, not the reduced one. The same logic recovers a pre-tax figure from a tax-inclusive total.

Can a percentage be more than 100?

Yes. Anything above 100% simply means more than the whole. If sales go from 50 to 150, that is a 200% increase — the gain is twice the original amount. Percentages over 100 are only impossible when the quantity is bounded by definition, such as the share of a single total.

Sources and method

OpenStax, Prealgebra 2e — Solve General Applications of Percent

Last reviewed: 19 Sep 2026 Sources last verified: 19 Sep 2026 Results use the assumptions explained on this page. Report an error How we check calculations

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