How to Calculate Percentages

Percentages are a dialect of fractions. “Per cent” means “per hundred,” so 25% is 25/100, which is the same quantity as 1/4 and as 0.25. Once that translation is boring, the four formulas people actually need stop looking like four topics. They are two questions — take a piece, or name a ratio — plus the bookkeeping of a before-and-after.

Use the percentage calculator when you want the number. Use this page when you want to know which number you were supposed to ask for.

Turning a percent into a multiplier

The bridge between the percent sign and ordinary arithmetic is division by 100.

To apply a percent, multiply by the decimal. To remove the percent sign from a rate you will add or subtract, do the same: 8% interest as a decimal is 0.08. Forgetting this division is how “8% of 500” becomes 4,000 instead of 40. If the result is about 100 times too big, the percent never became a decimal.

What is X% of Y?

This is the piece-of-a-whole question. Result = (X / 100) × Y.

15% of 200. 15/100 = 0.15. 0.15 × 200 = 30.

7.5% of 80. 0.075 × 80 = 6. Half of 15% would be a different problem; 7.5 is already the rate, so you do not halve after multiplying.

250% of 18. 2.5 × 18 = 45. More than 100% is allowed. It means more than the whole reference amount. It does not mean “an increase of 250%” unless the sentence says increase.

A useful rearrangement: X% of Y equals Y% of X. So 8% of 25 equals 25% of 8. A quarter of 8 is 2, which is the easier product. The what is X% of Y calculator does not care which way you find easier; the identity is for your head, and it is developed further in mental math for percentages.

X is what percent of Y?

This names a ratio. Percent = (X / Y) × 100, and Y cannot be zero.

25 is what percent of 200? 25/200 = 0.125. × 100 = 12.5%.

42 out of 50. 42/50 = 0.84 = 84%. Test scores are this formula with a friendlier name. Weighted tests are several of these formulas glued together; see test-score percentages.

9 is what percent of 12? 9/12 = 0.75 = 75%.

The order is the whole difficulty. The phrase “percent of” points at the divisor. If you catch yourself dividing the larger number by the smaller one “because percents should be under 100,” you have added a rule the definition does not have. 18 is 150% of 12. That can be the right answer.

The dedicated tool is X is what percent of Y. When the percent is already baked into a final price and you need the original, you want the inverse instead, covered in reverse percentages.

Applying an increase or a decrease

You have a starting value and a rate. You want the new value, not the ratio.

Increase: new = value × (1 + p/100)

Decrease: new = value × (1 − p/100)

The change itself, separated from the new total, is value × (p/100).

100 increased by 15%. 100 × 1.15 = 115. The increase alone is 15.

100 decreased by 15%. 100 × 0.85 = 85.

€64.50 with an 18% tip. The tip is the “percent of” piece: 64.50 × 0.18 = €11.61. The total is the increase formula: 64.50 × 1.18 = €76.11. Same rate, two questions — the piece, or the piece plus the original. The tip calculator prints both because dinner arguments are usually about the total.

A 30% discount on €119. Pay 70% of the price: 119 × 0.70 = €83.30. Savings = €35.70. “Percent off” is a decrease. The number you pay is 100% minus the discount, times the price.

Watch the language “percent increase” versus “percent of.” A 200% increase on 50 is 50 × (1 + 2) = 150. 200% of 50 is 100. The formulas differ by whether the original is still in the result.

Change from an old number to a new one

Now you are not given the rate. You are given both endpoints.

Percent change = (new − old) / |old| × 100

The absolute value on the old number keeps the sign of the change in the numerator, where it belongs: positive means growth, negative means decline. Some textbooks drop the absolute value and assume the old number is positive, which is the usual case for prices and scores.

From 80 to 100. (100 − 80) / 80 × 100 = 25% increase.

From 100 to 80. (80 − 100) / 100 × 100 = 20% decrease.

Those are not mismatches. The gap is 20 either way. The base changed, so the percent changed. Undoing a 25% increase takes a 20% decrease. Common percentage mistakes keeps a table of these return trips, because this is the error that makes “we gave it all back” false.

From 50 to 0. (0 − 50) / 50 × 100 = −100%. A total loss is a 100% decrease. The reverse, from 0 to 50, divides by zero and is not a percent change. Say “it went from nothing to 50.”

The percentage change calculator is the right tool once both numbers exist. The increase and decrease tools are the right ones when you have a rate instead of a second number.

Choosing the formula on purpose

If two rows seem to fit, read the sentence again and mark the base. The calculator cannot do that marking for you. It can only multiply what you hand it.

A longer worked mix

A club had 80 members. It grows 25%, then loses 10 members, then you are asked what percent of the *original* club remains.

1. After growth: 80 × 1.25 = 100.
2. After the loss: 100 − 10 = 90. The loss was a count, not a percent. Do not turn 10 into 10% unless the sentence did.
3. 90 is what percent of the original 80? 90/80 × 100 = 112.5%.

Someone who “adds 25 and subtracts 10” and declares +15% has mixed a percent with a headcount. The steps above keep the units intact. That bookkeeping, more than any trick, is the skill.

When to use the calculators

Use them for any figure you will pay, publish, or grade. Use the matching tool rather than the general one when you want the labels (tip, discount, change) to match the story, so you are less likely to type the rate into the wrong box.

Use mental anchors — 10%, then halves and doubles — as a sanity check. If 15% of 200 comes back as 300, the anchor (10% is 20, so 15% is near 30) tells you the decimal slipped. The procedure is in mental math for percentages.

Do not average a pile of percentages until you know each one was taken of a comparable whole. Do not describe a move between two rates (8% interest to 10% interest) with the change formula until you have decided you mean relative change and not percentage points. The four formulas are enough for daily life only when the sentence is already precise.