What a percentage is — and the five calculations on this page
A percentage is a number written out of 100, so 15% means 15/100, or 0.15. Every mode below is that one idea rearranged: multiply by it, divide by it, or use it to compare two numbers. The answer always comes with the formula that produced it, with your own numbers substituted, so you can check the working or repeat it by hand.
- X% of Y — the amount itself: Y × X ÷ 100. 15% of 200 is 30.
- X is what % of Y — the share: X ÷ Y × 100. 30 out of 200 is 15%.
- % change — how far a number moved from a starting value: (new − old) ÷ |old| × 100.
- % difference — how far apart two numbers are when neither is the baseline: |a − b| ÷ ((|a| + |b|) ÷ 2) × 100.
- Add or subtract % — the number after the change: X × (1 + Y ÷ 100). A negative percentage subtracts.
Results are rounded to six decimal places. That is more than enough for money and for exam answers, and it keeps repeating results such as 1 ÷ 3 readable instead of filling the line with digits.
Three worked examples
A 200,000 item with 15% off: 200,000 × 15 ÷ 100 = 30,000 saved, so you pay 170,000. The Add or subtract mode does both steps in one go — type 200000 and −15 and it returns 170,000, with the 30,000 shown underneath as the amount subtracted.
You scored 42 marks out of 60: 42 ÷ 60 × 100 = 70%. The same mode answers budget questions — a 1,250 line inside an 8,000 budget is 15.625% of it, and the six-decimal rounding means you see the exact figure rather than a truncated 15.6%.
Rent moved from 950 to 1,045: (1,045 − 950) ÷ 950 × 100 = +10%, with the absolute change of 95 shown beside it. Swap the two fields and the same pair of numbers gives −9.090909%. That asymmetry is not a bug, and the next section explains when it matters.
Percent change, percent difference and percentage points
Percent change has a direction, because it divides by the starting value. Going 950 → 1,045 is +10%; coming back 1,045 → 950 is −9.090909%. Whenever one number came first — last month, last year, the original price — this is the calculation you want, and getting the From and To fields the right way round is the whole job.
Percent difference is symmetric: it divides by the average of the two, so 950 and 1,045 are 9.523809% apart whichever order you type them in. Use it when neither number is a baseline — two lab measurements of the same quantity, two suppliers’ quotes, this year’s figure for two different regions.
A percentage point is the third thing, and it is where news reports go wrong most often. An interest rate moving from 3% to 4% has risen by 1 percentage point — but that is a 33.33% increase in the rate. Say “points” when you subtract two percentages, and “percent” only when you divide.
Discounts, mark-ups and working backwards
Add or subtract % handles both directions from one field. Enter 50000 and 10 to add Korea’s 10% VAT and get 55,000; enter −20 instead and you get the sale price. The extra line underneath tells you how much was actually added or taken off, which is the number you usually need for a receipt.
Percentages do not cancel out. Take 20% off 100 and you have 80; add 20% back and you get 96, not 100, because the second 20% is taken of the smaller number. For the same reason two consecutive 10% rises come to +21%, not +20%.
To undo a discount you divide, not add. A price of 36,000 after 20% off is 80% of the original, so the original was 36,000 ÷ 0.8 = 45,000. The general rule is sale price ÷ (1 − discount ÷ 100), and the same shape works for tax: a VAT-inclusive 55,000 came from 55,000 ÷ 1.1 = 50,000.
Common mistakes and what the error messages mean
- The wrong base. “20% of what?” decides the answer — 20% of the pre-tax bill and 20% of the post-tax bill are different amounts.
- Averaging percentages. 50% of 10 and 50% of 1,000 cannot be averaged into anything meaningful; average the amounts first, then take the percentage.
- Adding successive changes together. Each change is taken of the number that existed at that moment, so they multiply rather than add.
- Zero as the baseline. Percent change from 0 has no meaning and the calculator refuses it, and percent difference between 0 and 0 is undefined for the same reason.
Negative numbers are allowed everywhere. Percent change divides by the absolute value of the starting number, so a loss shrinking from −80 to −40 reads as a +50% change rather than flipping sign unexpectedly.
Which mode to use, by situation
Shopping and bills: “X% of Y” for a discount amount, “Add or subtract %” for the price you actually pay, and the reverse formula above when you only know the sale price. For restaurant bills the tip calculator adds the split and the rounding on top.
Work and study: “X is what % of Y” for a share — marks, completion rates, a line item inside a budget — and “% change” for month-on-month or year-on-year movement in a KPI. Both are the calculations spreadsheets get wrong when a cell is formatted as a percentage but stores a raw number.
Money over time needs a different tool. A price that rose 50% over three years is not “50% a year” — spreading a total return across a holding period is the ROI calculator’s annualized figure, and money that compounds belongs in the compound interest calculator.