Percentage calculator

How to calculate a percentage

To find a percentage of a number, divide the percentage by 100 and multiply: 20% of 85 is 0.2 × 85 = 17. To find what percent one number is of another, divide the part by the whole and multiply by 100: 17 out of 85 is 17 ÷ 85 × 100 = 20%.

Every percentage question links three numbers: a part, a whole and a rate. Knowing any two gives the third, so first decide which one the question is missing:

• What is 20% of 85? asks for the part: 17.
• 17 is what percent of 85? asks for the rate: 20%.
• 17 is 20% of what? asks for the whole: 17 ÷ 0.2 = 85.

Percentage formulas at a glance

The % sign stands for the number 0.01 (NIST Guide to the SI, §7.10.2), so dividing by 100 turns a percentage into a decimal: 7.5% is 0.075. Each common question then has one formula:

The last two rows use the same pair of numbers and disagree because they divide by different bases: 250 for the change, and the average, 265, for the difference. Percent change, often written Δ%, has a direction; percent difference has none.

A whole of 0, a starting value of 0 or two numbers that average 0 leave nothing to divide by, so those questions have no percentage answer.

Percentage increase: find the rate or the new amount

"Percentage increase" covers two questions. When you know the old and new amounts, the rate is (new − old) ÷ old × 100: from 80 to 100 is 20 ÷ 80 × 100 = 25%. When you know the rate, the new amount is old × (1 + rate ÷ 100): 80 × 1.25 = 100.

The calculator treats these as separate modes, and X and Y change meaning between them. Type rates as percentages, 25 for 25%, not 0.25:

A decrease uses the same formulas. From 100 to 80 is (80 − 100) ÷ 100 × 100 = −20%, and 100 decreased by 20% is 100 × 0.8 = 80. Percent difference treats 80 and 100 as equals: 20 ÷ 90 × 100 = 22.22%.

How to work out a percentage by hand

A student scores 43 out of 52 on a test. To express that as a percentage:

1. Divide the part by the whole: 43 ÷ 52 = 0.826923…
2. Multiply by 100: 82.6923…%.
3. Round once, at the end: 82.7%.

Rounding in step 1 instead, to 0.83, reports 83%. That is 0.3 points too high, which can be enough to cross a grade boundary.

For mental arithmetic, build the rate out of easy pieces. Moving the decimal point one place left gives 10%, two places gives 1%, and half of 10% is 5%. A 15% tip on a bill of 84 is 8.40 (10%) plus 4.20 (5%), or 12.60.

The tip and bill split calculator does the same sum and divides it between people.

Fractions, decimals and percentages are the same number in different notation. Divide the numerator by the denominator to get the decimal, then multiply by 100 to get the percentage:

Percent change and percentage points

When the two values are themselves percentages, there are two ways to state the change. A mortgage rate moving from 2.5% to 3.0% rose by 0.5 percentage points, by plain subtraction, and by 20% in relative terms (0.5 ÷ 2.5 × 100).

In basis points, where one basis point is 0.01 percentage point, the same move is 50 basis points.

A fall and a rise of the same percentage do not cancel, because the rise is measured from the lower value. The rise needed to recover from a fall of p% is p ÷ (100 − p) × 100:

The gap widens quickly: an investment that loses half its value has to double to break even.

With a negative starting value, divide by its absolute value. From −50 to −25 is 25 ÷ 50 × 100 = +50%, a rise toward zero. A spreadsheet formula written as (new − old) ÷ old returns −50% for the same move, so check which convention a report uses.

From a starting value of 0 there is no percent change; report the absolute change instead.

Reverse percentages: finding the original value

To undo a percentage change, divide by the multiplier that produced it. After a 15% discount a jacket costs 68. The sale price is 85% of the original, so the original was 68 ÷ 0.85 = 80.

Adding 15% of 68 back on gives 68 × 1.15 = 78.20, which is 1.80 short, because the 15% was taken from 80, not from 68.

The same error appears when removing sales tax or VAT from a gross price; the GST, VAT and sales tax calculator handles that case directly.

Markup and margin are a reverse-percentage pair. An item that costs 80 and sells for 100 has a markup of 25% (20 ÷ 80) and a margin of 20% (20 ÷ 100): the same profit over different bases.

The discount and markup calculator converts between the two.

Common percentage mistakes

Successive discounts multiply. A 20% discount followed by a further 10% off leaves 0.80 × 0.90 = 0.72 of the price, a total discount of 28%, not 30%. The order of the two discounts makes no difference.

"50% off, then an extra 50% off" takes three quarters off the price, not all of it. Other errors that recur:

• Adding successive increases. Two 10% rises multiply to 1.1 × 1.1 = 1.21, a 21% increase. Interest added to a balance each period grows the same way; the compound interest calculator runs that over many periods.
• Dividing a total change by the number of years. A price that rises from 100 to 150 over five years has gone up 50%, but not 10% a year: five 10% rises compound to 1.1⁵ = 1.61, a 61% rise. The steady yearly rate is 1.5^(1/5) − 1 = 8.45%, which the scientific calculator evaluates as 1.5^(1/5) - 1.
• Confusing "percent of" with "percent more than". 150% of 40 is 60, while 150% more than 40 is 40 + 60 = 100.
• Averaging percentages with different bases. If 30% of 200 people in one town and 60% of 50 people in another answer yes, the combined share is (60 + 30) ÷ 250 = 36%. The simple average, 45%, gives the small town the same weight as the large one.
• Decreasing by more than 100%. A positive quantity cannot fall by more than 100% and stay positive: 50 decreased by 120% is −10. If a negative result is impossible in your situation, the rate or the base is wrong.
• Rounding the rate before using it. Using 8% instead of 8.45% understates five years of growth by about 3 percentage points, since 1.08⁵ = 1.469 instead of 1.5. Keep full precision and round only the figure you report.