How to calculate percentage increase, step by step
To calculate percentage increase, subtract the old value from the new one, divide by the old value, and multiply by 100. That is the whole method, and everything below is just that sentence done carefully. A percentage increase answers one question: how big is the change, compared with where it started? The change itself is in dollars, or in whatever unit the numbers carry; the percentage turns that change into a share of the starting value, so a $25 rise on $50 is 50.00% and a $175 rise on $25 is 700.00% — the starting value decides the percentage, not the size of the change.
The worked example: $80 to $100
The worked example. Step 1 — find the change: 100 − 80 is $20.00. Step 2 — divide by the starting value: 20 ÷ 80 is 0.25. Step 3 — multiply by 100: 0.25 × 100 is 25.00%. Step 4 — read the sign: the new value is bigger than the old one, so it is an increase, and the answer is +25.00%. The change is $20.00 and the starting value is $80 — and those two numbers are the whole calculation. If the new value had been smaller, the same steps would give a negative answer, which is a decrease, and the formula does not change.
How to calculate percentage decrease
Percentage decrease uses exactly the same formula, and the sign does the work. From $100 to $80, step 1 gives a change of 80 − 100, which is −$20.00. Step 2 divides by the starting value, 100. Step 3 multiplies by 100, which gives −20.00%. So the decrease is 20%, not 25% — even though the dollars are the same $20.00 as the increase the other way. The reason is the base: the increase was measured against 80, and the decrease is measured against 100, and $20 is a quarter of 80 but only a fifth of 100. A rise and a fall of the same dollars are never the same percent, and that asymmetry is the single most useful thing to remember about percentage change.
Percentage increase vs percentage points
Percentage points are the plain difference between two percentages, and they are not the same as a percentage increase. A rate that moves from 10% to 12% rose by 2 percentage points — 12 minus 10 — but that is a 20% increase in the rate itself, because 2 ÷ 10 × 100 is 20%. News reports mix them up constantly, usually by calling a 2-point rise a rise of two percent, which understates it tenfold. The rule is simple: when the thing changing is itself a percentage — a tax rate, an interest rate, a poll share — say how many points it moved, and say what percent that is of the starting rate if you want the relative size.
Reversing a percentage increase
Sometimes you know the final value and the percentage, and you need the original. A price ends at $100 after a 25% increase: what did it cost before? Divide the final value by 1 plus the percentage as a decimal — 100 ÷ 1.25 is $80.00. The 1.25 is the original 100% plus the 25% increase, written as a multiplier. For a decrease, divide by 1 minus the percentage: a price ends at $80 after a 20% decrease, so it started at $100.00, because 80 ÷ 0.8 is 100.
A second example: 50 to 75
Take a second example to make the pattern stick. Step 1 — the change: 75 − 50 is $25.00. Step 2 — divide by the starting value: 25 ÷ 50. Step 3 — multiply by 100, which gives 50.00%. So from 50 to 75 is a 50.00% increase — the value grew by half of itself. And the reverse: 75 after a 50% increase started at 50, because 50 plus $25.00 is 75. A percentage over 100% is perfectly ordinary — 25 to 200 is a 700.00% increase, because the value grew by seven times itself — and it usually means the starting value was small.
Checking your answer
Whatever method you used, check it with the numbers you already have. First, check the sign: if the new value is bigger than the old one, the answer must be positive, and if it is smaller, the answer must be negative. A positive answer on a fall means the old and new values got swapped. Second, check the size: a 25% increase on $80 is $20.00, and $80 plus $20.00 is $100 — if the change you found does not add back to the new value, the division went wrong somewhere. Third, check the extreme cases: no change at all is 0%, and losing everything is a 100% decrease, so an answer outside −100% on the downside has no meaning for a positive starting value. Run your numbers through the percentage-increase-calculator and compare them with the worked examples above.
Common mistakes
The first mistake is dividing by the new value instead of the old one: (100 − 80) ÷ 100 gives 20%, which is the decrease the other way round, not the increase you were asked for. The second is forgetting to multiply by 100, which leaves the answer as 0.25 instead of 25% — the decimal is right and the presentation is wrong, and it reads as a quarter of a percent. The third is treating a rise followed by an equal fall as a round trip: a price that rises 25% and then falls 25% does not come back to where it started — it ends at 93.75, because the 25% fall is taken from the higher price. And the fourth is the percentage-points mix-up: a rate moving from 10% to 12% rose 2 percentage points, not two percent, and calling it two percent understates the change tenfold.
A quick check you can do in your head
You can sanity-check any percentage increase without a calculator. First, find the change and add it back: a 25% increase on $80 is $20.00, and $80 plus $20.00 is $100, which is where you started. Second, check the size against a half: a 50% increase on $50 adds $25.00, so 50 becomes 75 — if your change is bigger than the starting value itself, the percentage is over 100%. Third, check the direction: an increase ends higher than it started, and a decrease ends lower. If a calculator says the increase from $80 to $100 is anything but 25.00%, the numbers are wrong.