Why use this calculator
Percentages look simple and are quietly the most error prone arithmetic most people do. Three mistakes account for almost all of it. The first is reversing the base: a jump from 40 to 50 is a 25 per cent rise, but the fall from 50 back to 40 is only 20 per cent, because the denominator changed.
The second is adding percentages that should be multiplied, which is how a 40 per cent discount plus a further 20 per cent becomes 52 per cent rather than 60. The third is confusing a percentage change with a percentage point change, which is the difference between an interest rate rising from 4 to 5 per cent and rising by 25 per cent. This calculator shows the working for each answer so the base is always visible.
How to use it
- Start by choosing the calculation type, because the meaning of the two inputs changes with it and this is where most wrong answers begin.
- In the first mode, X is the percentage and Y is the whole, so entering 25 and 200 asks what 25 per cent of 200 is.
- In the second mode, X is the part and Y is the whole, so 15 and 60 asks what share 15 is of 60.
- In the change mode, X is the old or starting value and Y is the new one, and the order matters because the old value is the denominator.
- In the increase and decrease modes, X is the percentage to apply and Y is the amount you are applying it to.
- Read the working line to confirm the calculator used the base you intended.
What each result means
- Result
- The answer to the question you selected, expressed as a number or a percentage depending on the mode.
- Working
- The arithmetic actually performed, shown so you can confirm the calculator used the base you intended.
- Reverse check
- The same relationship computed backwards. If it does not return your input, you have selected the wrong mode.
- Percentage change
- The movement from the old value to the new one, expressed against the old value as the base.
- Absolute change
- The plain difference between the two values, which is often the more honest figure on small bases.
- Direction
- Whether the value rose, fell or stayed the same, stated in words so a negative sign cannot be missed.
- Multiplier
- The factor you multiply the old value by to reach the new one. A multiplier of 1.25 is a 25 per cent rise.
- Change amount
- How much was added or removed in absolute terms by applying the percentage to the base.
The formula
Four expressions cover every mode.
A percentage of a number
R = Y × (X ÷ 100)
- X
- the rate
- Y
- the whole
One number as a share of another
R = (X ÷ Y) × 100
- X
- the part
- Y
- the whole
Percentage change
R = ((Y − X) ÷ |X|) × 100
An increase or decrease
R = Y × (1 ± X ÷ 100)
How this number is used in practice
The asymmetry of percentage change is the most costly misunderstanding in everyday finance. An investment that falls 50 per cent needs a 100 per cent gain to get back to where it started, because the recovery is measured against the smaller balance. The same asymmetry explains why a shop can discount by 40 per cent and still advertise honestly that prices have been marked up by 67 per cent from cost, and why a salary cut of ten per cent followed by a ten per cent rise leaves you one per cent behind.
Percentage points are a separate unit and mixing them with percentages produces headlines that are technically false. If a central bank moves a policy rate from four per cent to five per cent, that is a rise of one percentage point and a rise of twenty five per cent, and both statements are true about different things.
Statistical agencies including the Australian Bureau of Statistics and the Office for National Statistics are strict about this distinction precisely because the two figures are so easy to swap.
Stacked percentages multiply rather than add, which matters for discounts, commissions and tax. Two successive twenty per cent discounts leave 0.8 times 0.8, which is 64 per cent of the original price, a 36 per cent total discount rather than 40. The same maths runs the other way for compounding fees: a platform charging one per cent a year for twenty years does not take twenty per cent, it takes about eighteen per cent of the balance path, because each year's fee is charged on what remains.
Worked examples
What is 25% of 200?
50.00, with the working shown as 200.00 × 25.00 ÷ 100.
15 is what percentage of 60?
25.00%, because 15 divided by 60 is one quarter.
Percentage change from 80 to 100
25.00% increase, an absolute change of 20.00 and a multiplier of 1.2500x.
A $64,000 salary increased by 3.5%
66,240.00, an increase of 2,240.00.
When people use it
- Checking a discount, a markup or a commission before you agree to it
- Converting a raw change into a percentage for a report or a board paper
- Working out a pay rise, a rent increase or a price adjustment
- Reversing a percentage to find an original price or a pre-tax amount
- Sanity checking a statistic quoted in the news before repeating it
Tips
- Percentage change always uses the original value as the denominator. Swapping the two numbers gives a different answer, not the same one with the opposite sign.
- A percentage point is not a percentage. Moving from 4% to 5% is one percentage point and a 25% rise, and the two are not interchangeable.
- Percentages stack multiplicatively. Two 20% discounts give 36% off, not 40%, because the second applies to the already reduced price.
- To reverse a discount, divide rather than add back. A price of $75 after 25% off was $75 ÷ 0.75, which is $100, not $75 plus 25%.
- Be wary of percentage change on small bases. A rise from two cases to three is a 50% increase and almost always meaningless.
- A 50% fall needs a 100% rise to recover. Asymmetry is the rule, not an edge case, whenever the base changes.
- When a figure can be negative, quote the absolute change as well. Percentage change across a sign change is misleading in every direction.
What it assumes, and where it stops being accurate
- Percentage change from a starting value of zero is undefined, so the calculator asks for the absolute change instead rather than printing an infinite result.
- Expressing a number as a percentage of zero is also undefined and is blocked for the same reason.
- Results are shown to two decimal places, so very small percentages on very large bases are rounded in the display.
- It treats the inputs as pure numbers with no units. It will happily compute a percentage of two incompatible quantities if you ask it to.
- Percentage change across a change of sign, such as from a loss to a profit, is reported but is not a meaningful statistic.
- Compounding is not applied. Applying an increase twice in this tool is two separate single step calculations.
Terms used on this page
- Percentage point
- The arithmetic difference between two percentages. A move from 4% to 5% is one percentage point.
- Base
- The number a percentage is calculated against. Changing the base changes the answer even when the amounts do not.
- Relative change
- A change expressed as a proportion of the starting value, which is what a percentage change reports.
- Multiplier
- The factor that converts the old value into the new one. A 15% rise is a multiplier of 1.15.
Questions people ask
- How do I calculate a percentage increase?
- Subtract the old value from the new one, divide by the old value, then multiply by one hundred. Going from 80 to 100 gives twenty divided by eighty, which is 0.25, so a 25 per cent increase. The old value is always the denominator, which is why the reverse trip from 100 back to 80 is a 20 per cent decrease rather than 25.
- What is the difference between a percentage and a percentage point?
- A percentage point is the arithmetic difference between two percentages, while a percentage change is the relative difference. A rate moving from four per cent to five per cent has risen by one percentage point and by twenty five per cent. Statistical agencies insist on the distinction because reporting the larger number without the unit turns a modest move into a dramatic one.
- Why do two 20% discounts not add up to 40%?
- Because the second discount applies to the already reduced price rather than the original. Twenty per cent off leaves 0.8 of the price, and twenty per cent off that leaves 0.8 of 0.8, which is 0.64. You pay 64 per cent of the original, so the combined discount is 36 per cent. Stacked percentages always multiply rather than add.
- How do I find the original number from a percentage?
- Divide rather than adding back. If an item costs seventy five dollars after a twenty five per cent discount, the original was seventy five divided by 0.75, which is one hundred. If fifty is twenty five per cent of something, that something is fifty divided by 0.25, which is two hundred. Adding the percentage back always undershoots.
- Can a percentage be more than 100?
- Yes, whenever the part is larger than the whole it is being compared with. A business whose revenue trebles has grown by two hundred per cent, and a loan whose total interest exceeds the amount borrowed has an interest cost above one hundred per cent of principal. Only proportions of a fixed total, such as a share of votes cast, are capped at one hundred.
- Why is percentage change from zero undefined?
- Because the calculation divides by the starting value, and dividing by zero has no answer. Going from zero sales to ten sales is an infinite increase in relative terms, which tells a reader nothing. Report the absolute change instead, or quote the new figure alongside a note that the base period was zero.
Use this with
Percentage Calculator answers one part of the question. These pick up where it stops: Percentage Change Calculator, Fraction Calculator, Ratio Calculator, Markup Calculator and Discount Calculator.
Where these figures come from
This calculation is jurisdiction-neutral arithmetic. No country-specific rules are applied.
- NIST Special Publication 811 (use of the SI). Correct use of percent, percentage point and derived units
- Australian Bureau of Statistics. Statistical language, including percentage change conventions
- UK Office for National Statistics. Published guidance on reporting rates and percentage change
Percentage Calculator produces an estimate for general information. It is not financial, tax, medical, legal or engineering advice. read the full disclaimer. Spotted a wrong number? Report it and it gets fixed. The form picks up this calculator and the inputs you used.