How the result is calculated
Per cent means "for each hundred", so 15% is 15 hundredths, or 0.15. Nearly every percentage question is one of a handful of rearrangements of the same relationship between a part, a whole and a rate, and the calculator has a mode for each.
partwhole × p ÷ 100
- part
- the amount the percentage describes
- whole
- the number the percentage is taken of: the base
- p
- the percentage
A percentage change treats the starting value as the whole and the difference as the part, so the change from A to B is (B − A) ÷ A × 100. Working back from a figure after a change runs the multiplication in reverse: a rise of p% multiplied the original by (1 + p ÷ 100), so dividing by that factor recovers it. When both numbers are already percentages, such as two interest rates, the plain subtraction of one from the other is a number of percentage points, and the calculator gives the relative change beside it.
Worked example
| Item | Amount | How it is obtained |
|---|---|---|
| Percentage as a decimal | 0.15 | 15 ÷ 100 |
| 15% of 200 | 30 | 200 × 0.15 |
| 200 plus 15% | 230 | 200 × 1.15 |
| 200 minus 15% | 170 | 200 × 0.85 |
A quick check without a calculator: 10% of 200 is 20, and 5% is half of that, 10, so 15% is 30.
| Item | Amount | How it is obtained |
|---|---|---|
| Price including VAT | 120 | Figure after the change |
| Multiplier the VAT applied | 1.2 | 1 + 20 ÷ 100 |
| Price before VAT | 100 | 120 ÷ 1.2 |
| VAT in the price | 20 | 120 − 100, or one sixth of 120 |
| Taking 20% off 120 instead | 96 | wrong by 4 |
The 20% was charged on 100, not on 120, so taking 20% of the higher figure removes too much. HMRC states the same rule as the VAT fraction: the rate divided by 100 plus the rate, 20 ÷ 120, which is one sixth.
Reading the result
The headline is whichever figure you asked for, and the three figures beside it answer the question people usually ask next. In the first mode they are the number with the percentage added and taken away. For a share, they are what is left and the comparison turned round. For a change, they are the raw difference, the change it would take to get back, and the percent difference, which divides the gap by the average of the two values when neither of them is the starting point.
The change back is worth reading every time. From 80 to 100 is a rise of 25%, but from 100 back to 80 is a fall of 20%, because the second change is measured on a bigger number. This is why a share that falls 50% needs to double, a 100% rise, to get back to where it was.
What changes the result
The base changes everything. "Prices rose 10%" and "prices are 10% higher than they would have been" can describe very different amounts depending on what the 10% is taken of, and a percentage quoted without its base is not a complete statement. When a change is small relative to its base, the rise and the fall that undoes it are close; when it is large, they move far apart.
Percentages of percentages need care. An interest rate moving from 4% to 5% is a rise of 1 percentage point and a rise of 25% in the rate, and the ONS writes "percentage points" in full in its statistics precisely so readers can tell the two apart. Repeated changes multiply rather than add: two rises of 10% are a rise of 21%, and a rise of 10% followed by a fall of 10% leaves you 1% down.
What is not included
The calculator works on the numbers you give it and does not round them to a currency or to a shop's or tax authority's convention, so a figure can show more decimals than a receipt would. It does not chain several changes for you: for a run of yearly rises, multiply the factors or use the compound interest calculator.
It also says nothing about whether a change matters. A rise from 2 cases to 4 is a 100% increase, and on a base that small it can easily be chance. Percentages are a way of comparing, and they compare best when the base is large and stated.
Using the result
Use the share mode to put two different-sized things on one footing: a bill as a share of income, a deposit as a share of a price. Use the change mode for prices, salaries and anything measured before and after, and read the change back alongside it. Use the reverse mode whenever a figure already has a percentage built in: a price including VAT or sales tax, a salary after a raise, a sale price after a discount. For a discount on a real purchase, the discount calculator adds stacked offers and tax to the same arithmetic.