A Free Calculator · Three Modes · All Formulas Shown · Updated 2026
Percentage calculator — three questions, one tool
Most percentage questions are one of three things: finding P% of a number, figuring out
what percent one number is of another, or measuring how much something changed. Pick the
mode that matches your question, enter two numbers, and the answer appears instantly —
with the worked arithmetic shown so you can check it or replicate it by hand.
P% of a number·X is what % of Y?·Percent change A to B
What this calculator does
Strictly arithmetic — percents, ratios, and rates of change. It does not touch
financial products, taxes, loan interest, or investment returns. Every formula
is shown in the formulas section below; nothing is a black box.
Choose which percentage question you want to answer, fill in the two numbers, and the result updates as you type.
What do you want to find?
The input labels below change to match the selected question.
%
The percentage to apply.
Result
Verification
The formulas, in full
These are the exact calculations the tool runs — the same arithmetic you could do
with a pencil. Every mode maps to one formula. The only unknowns are the numbers
you supply.
How each mode is derived
Mode 1 — What is P% of N?
result = P ÷ PERCENT_DIVISOR × N
= P ÷ 100 × N
Example: 15% of 80 → 15 ÷ 100 × 80 = 0.15 × 80 = 12
Mode 2 — X is what percent of Y?
result = (X ÷ Y) × PERCENT_DIVISOR
= (X ÷ Y) × 100 [Y must not be zero]
Example: 12 is what % of 80? → (12 ÷ 80) × 100 = 0.15 × 100 = 15%
Mode 3 — Percent change from A to B
result = ((B − A) ÷ A) × PERCENT_DIVISOR
= ((B − A) ÷ A) × 100 [A must not be zero]
Example (increase): A = 80, B = 100 → ((100 − 80) ÷ 80) × 100 = 25% increase
Example (decrease): A = 100, B = 80 → ((80 − 100) ÷ 100) × 100 = −20% decrease
Common percentages — quick reference
Fraction equivalents and P% of round numbers at a glance. Useful for mental-math
checks and for verifying calculator output on familiar values. All figures are exact.
Percent
Fraction
of 20
of 50
of 80
of 200
5%
1/20
1
2.5
4
10
10%
1/10
2
5
8
20
15%
3/20
3
7.5
12
30
20%
1/5
4
10
16
40
25%
1/4
5
12.5
20
50
33.3%
1/3 (approx.)
6.67
16.67
26.67
66.67
50%
1/2
10
25
40
100
75%
3/4
15
37.5
60
150
100%
1/1
20
50
80
200
The 1/3 row is rounded to two decimal places; the true value is a repeating decimal.
All other rows are exact. Use this table to sanity-check a calculator result —
if 20% of 80 comes out to anything other than 16, something went wrong.
Where percentages trip people up
The arithmetic is simple. The conceptual errors are surprisingly easy to make —
even people comfortable with numbers fall into these.
A 25% increase followed by a 20% decrease does not return to the start
Percent change always applies to the current base. Start at 100. A 25% increase gives 125. A 20% decrease from 125 gives 100 — but that only works here by coincidence. In general: 100 × 1.25 × 0.80 = 100. However, a 20% increase followed by a 20% decrease gives 100 × 1.20 × 0.80 = 96, not 100. The order and the amounts both matter because the denominator shifts each time.
Percent points and percent change are not the same thing
If a completion rate rises from 10% to 15%, it increased by 5 percentage points — but that is a 50% increase relative to the original 10%. Journalists and policy documents often write "rose by 5%" when they mean "rose by 5 percentage points," which are very different claims. The formula (B − A) ÷ A × 100 gives the relative percent change; B − A gives the percentage-point change. Know which one you need.
Reversing a percent change requires a different percent
A 50% decrease takes you from 100 to 50. Getting back to 100 from 50 requires a 100% increase, not another 50%. To reverse a P% decrease, apply a [P ÷ (100 − P) × 100]% increase. To reverse a P% increase, apply a [P ÷ (100 + P) × 100]% decrease. The calculator's Mode 3 tells you how large the change was; knowing how to reverse it is a separate step the formulas section above makes clear.
How to do it by hand
Mental-math methods you can use when a calculator is not handy. Start with the
10% anchor — it is the fastest building block for most percent questions.
Find 10% by moving the decimal
10% of any number is that number divided by 10 — just shift the decimal one place left. 10% of 80 = 8. 10% of 350 = 35. 10% of 6 = 0.6. This is the universal first step for nearly every mental-math percentage problem.
Build your target percent from 10%
5% = half of 10%. 20% = double 10%. 15% = 10% + 5%. 25% = divide by 4 (or 10% + 10% + 5%). Example: 15% of 80 — 10% is 8, 5% is 4, total is 12. Same answer as P ÷ 100 × N = 15 ÷ 100 × 80 = 12. The mental-math shortcut and the formula always agree.
Find "X is what percent of Y" by long division
Divide X by Y to get a decimal, then multiply by 100 to convert to a percent. Example: 12 out of 80 — 12 ÷ 80 = 0.15, and 0.15 × 100 = 15%. Alternatively, scale both numbers to make the denominator 100: 12/80 = 15/100 = 15%. This scaling trick works cleanly when Y is a factor of 100 (e.g., 25, 50, 20, 10).
Estimate percent change with rounding
Round both A and B to easy numbers, compute (B − A) ÷ A × 100, then adjust. A price from $47 to $56: round to $50 → $55. Change ≈ $5 on $50 = 10%. Actual: ($56 − $47) ÷ $47 × 100 ≈ 19.1%. The rounding got you in the right ballpark, not the right answer — use the calculator for anything where the number matters.
Where to buy
Got your numbers? Here's where to pick up what you need:
The vocabulary that comes up in percent problems — defined plainly so you can
match the right formula to the right question.
Percent
Literally "per hundred." A percent expresses a ratio relative to 100. 15% means 15 out of every 100, or the fraction 15/100, or the decimal 0.15. The three representations are interchangeable and the calculator uses the decimal form internally.
Base (or whole)
The reference quantity you are taking a percent of. In "15% of 80," the base is 80. In "X is what percent of Y," Y is the base. In percent change, the starting value A is the base. Confusing the base is the most common source of wrong answers — always identify it first.
Part (or portion)
The result of applying a percent to a base: P% of N = part. In "12 is what percent of 80," 12 is the part and 80 is the base. The relationship is always: percent = part ÷ base × 100.
Percent change
How much a value shifted relative to its starting point: (B − A) ÷ A × 100. Positive = increase; negative = decrease. The starting value A is the denominator, so the same absolute change (say, +10) produces a different percent change depending on where you started.
Percentage point
The arithmetic difference between two percentages. If a rate moves from 10% to 15%, it increased by 5 percentage points — but by 50% in relative terms. Percentage points are an absolute unit; percent change is relative. The two are only equal when the starting value is 100.
Decimal multiplier
A percent converted to decimal form for multiplication. 15% becomes 0.15 (divide by 100). A 20% increase multiplier is 1.20 (1 + 0.20); a 20% decrease multiplier is 0.80 (1 − 0.20). Multipliers let you chain percent changes: 1.25 × 0.80 = 1.00, meaning a 25% increase then a 20% decrease brings you back to the original.
Ratio
A comparison of two quantities: X to Y, or X/Y. A percent is a ratio expressed as parts per hundred. Ratios and percents answer the same question ("how do these two numbers compare?") in different forms — ratios are often more natural for fractions, percents for quick communication.
Frequently asked
To find P% of N, multiply N by P and divide by 100: result = P ÷ 100 × N. For example, 15% of 80 = 15 ÷ 100 × 80 = 0.15 × 80 = 12. The fraction P/100 converts the percent into a decimal multiplier. Mentally: find 10% first (move the decimal left one place: 10% of 80 = 8), then scale — 5% is half of that (4), so 15% = 8 + 4 = 12.
To find what percent X is of Y, divide X by Y and multiply by 100: result = (X ÷ Y) × 100. For example, 12 is what percent of 80? (12 ÷ 80) × 100 = 0.15 × 100 = 15%. This requires Y to be nonzero — you cannot divide by zero. Think of it as asking: out of every 100 parts of Y, how many parts is X?
Percent change from A to B = (B − A) ÷ A × 100. A positive result is an increase; negative is a decrease. Example: from 80 to 100 → (100 − 80) ÷ 80 × 100 = 25% increase. From 100 to 80 → (80 − 100) ÷ 100 × 100 = −20% decrease. The starting value A must be nonzero. Note that a 25% increase followed by a 20% decrease does not return to the original — each step recalculates from the current base.
A percentage is a value relative to a whole (30% of 200 = 60). A percentage point is an absolute arithmetic difference between two percentages. If a rate rises from 10% to 15%, it increased by 5 percentage points, but by 50% relative to the original 10%. Confusing the two is a common error in news reporting: "interest rates rose 2%" might mean 2 percentage points (from 4% to 6%) or a 2% relative increase (from 4% to 4.08%). The context — or the formula — tells you which one applies.
Percent change is always relative to the starting (base) value, which changes depending on direction. Going from 50 to 100 is a 100% increase, but going from 100 back to 50 is only a 50% decrease — because the denominator is different in each case. This asymmetry is not a math error; it is built into the formula (B − A) ÷ A × 100. For symmetric comparisons, some fields use a midpoint formula — but this calculator uses the standard percent change formula, which is what most everyday contexts expect.
If you know the result after a P% increase or decrease and want the original, rearrange the percent-of formula. After an increase: original = result ÷ (1 + P/100). After a decrease: original = result ÷ (1 − P/100). For example, if a price after a 20% increase is $120, the original was $120 ÷ 1.20 = $100. A common error is to subtract P% from the result directly — that gives the wrong answer because you would be taking P% of the already-changed value, not the original.
Start with 10% — move the decimal one place left. 10% of $64 = $6.40. Then scale: 20% = double the 10% figure ($12.80); 5% = half of 10% ($3.20); 15% = 10% + 5% ($9.60); 25% = divide by 4 ($16.00); 50% = divide by 2 ($32.00). For tips: 15% of a $47 bill — round to $50, 10% = $5, 5% = $2.50, so 15% ≈ $7.50. These shortcuts are exact for round numbers and close enough for quick estimates on any number.
Use "what percent of Y is X?" (X ÷ Y × 100) when you want to express X as a portion of a total — a test score, a discount rate, a completion percentage. Use percent change ((B − A) ÷ A × 100) when you are comparing a value before and after some event — a price shift, a measurement over time. The key difference: "percent of" expresses a part-whole relationship at one moment; "percent change" expresses movement between two moments.
Common mistakes
Percentage errors almost always trace back to using the wrong base number. The four patterns below cover the mistakes that produce the largest wrong answers.
Using the wrong base when calculating a percent change
Percent change = (new − old) ÷ old × 100. The base is always the old (starting) value. A price that rises from $80 to $100 is a 25% increase (20 ÷ 80), not a 20% increase (20 ÷ 100). Dividing the difference by the new value is the single most common percentage error.
Assuming percent increases and decreases cancel out symmetrically
A 25% increase followed by a 20% decrease does not return to the original value. After +25%, the base is 125; a 20% reduction of 125 is 25, leaving 100 — which only works by coincidence here. In most cases, the two percentages operate on different bases and do not cancel. The correct approach is to chain the multipliers: original × 1.25 × 0.80.
Treating "percent of" and "percent more than" as the same question
"A is 120% of B" means A = 1.20 × B. "A is 20% more than B" means the same thing. But "A is 120% more than B" means A = 2.20 × B — the 120% is added on top of the original 100%. The phrase "more than" signals addition to the base; the word "of" alone does not.
Applying a discount percent to the discounted price instead of the original
Two successive discounts of 20% and 10% are not equivalent to a single 30% discount. The second discount applies to the already-reduced price: 100 × 0.80 × 0.90 = 72, a combined 28% off — not 30%. Each discount is always calculated on whatever the current price is, not the original.