Nearly every percentage mistake comes from one source: forgetting what the percentage is a percentage of. Percent change compares a new value with the old one, so the same gap gives different percentages depending on direction. A rise from 40 to 50 is 25 percent. A fall from 50 to 40 is 20 percent. Keep the base in mind, and the rest follows. We checked every number in this article by calculation.
The three basic questions
A percentage is a fraction out of 100. Most everyday problems are one of three forms, and each is one line of arithmetic.
What is X% of Y? Multiply.
15% of 200 = 0.15 × 200 = 30X is what percent of Y? Divide, then multiply by 100.
45 is what percent of 180? 45 ÷ 180 × 100 = 25%What’s the percent change from A to B? Take the difference, and divide by the starting value.
From 80 to 100: (100 − 80) ÷ 80 × 100 = 25% increaseOur Percentage Calculator handles each of these. Its percent change mode won’t accept 0 as the starting value, for a reason covered below.
Percent change is not symmetrical
The base changes with direction, so going up and coming back down use different percentages.
| Move | Change | Percent |
|---|---|---|
| 40 to 50 | +10 | +25% |
| 50 to 40 | −10 | −20% |
| 200 to 250 | +50 | +25% |
| 250 to 200 | −50 | −20% |
| 120 to 90 | −30 | −25% |
Going down by 25 percent needs a 33.3 percent rise to get back. Losses and recoveries are not equal, and it gets worse as the numbers grow:
| Fall | Rise needed to recover |
|---|---|
| 20% | 25% |
| 25% | 33.3% |
| 50% | 100% |
The formula is 1 ÷ (1 − fall) − 1. This matters for investments: a fund that loses half its value must then double just to break even.
Percent versus percentage points
If an interest rate moves from 2 percent to 3 percent, there are two correct descriptions and they use different words:
- It rose by 1 percentage point, the plain subtraction of 3 − 2.
- It rose by 50 percent, since (3 − 2) ÷ 2 × 100 = 50.
Saying “up 1 percent” is wrong. A 1 percent increase on 2 percent gives 2.02 percent.
This confusion can be accidental or deliberate, so watch for it in headlines:
Risk. Say a side effect goes from 1 in 1,000 people to 2 in 1,000. “Risk doubles” and “100 percent increase” are both true. So is “risk rises by 0.1 percentage points”. They feel very different. Always ask for the absolute numbers when a percentage looks dramatic.
Elections and polls. A party going from 40 percent to 44 percent gained 4 points of vote share. That’s a 10 percent increase in its votes if the turnout stayed flat. Reports usually mean points.
Margins. A profit margin moving from 10 percent to 12 percent improved by 2 points, and profit per unit of revenue rose by 20 percent. A company can truthfully pick whichever sounds better.
Compound changes multiply
Percentages applied one after another multiply. They don’t add. This is the rule that catches people most often.
Up 50 percent, then down 50 percent. Starting from 100:
100 × 1.5 = 150
150 × 0.5 = 75It’s a 25 percent loss, not zero. The order doesn’t matter, as 100 × 0.5 × 1.5 also gives 75.
Up 10, down 10. 1.1 × 0.9 = 0.99, so you lose 1 percent.
Up 20, down 20. 1.2 × 0.8 = 0.96, a loss of 4 percent. The larger the swing, the bigger the shortfall.
Stacked discounts. A shop offers 20 percent off, and then a further 10 percent off. On a $150 item:
150 × 0.80 = 120
120 × 0.90 = 108That’s $42 off, or 28 percent overall, not 30 percent. Thirty percent off would give $105, so read the fine print on any promotion that says its discounts stack.
Growth over several periods. Sales grow 8 percent one year and 12 percent the next. Total growth isn’t 20 percent:
1.08 × 1.12 = 1.2096, so 20.96%Average growth rate. The arithmetic mean misleads here. Take a stock that gains 50 percent and then loses 50 percent. The simple average of those changes is 0 percent, yet the true result is −25 percent. For an accurate per-period average, use the geometric mean, or compound annual growth rate:
CAGR = (end ÷ start)^(1 ÷ periods) − 1
From 1,000 to 1,500 over 3 years:
(1.5)^(1/3) − 1 = 14.47% per yearCheck: 1,000 × 1.1447³ ≈ 1,500. A simple average would say 16.67 percent per year, and that would overstate it.
Simple versus compound interest. $1,000 at 5 percent for 10 years gives $1,500 with simple interest, and $1,628.89 with annual compounding. The extra $128.89 is interest earned on interest. Our Compound Interest Calculator handles the longer versions of this sum, including different compounding frequencies.
The rule of 72. Divide 72 by the rate to estimate years to double. At 6 percent, it says 12 years. The exact answer is 11.9, so it’s close for ordinary rates.
Reversing a percentage
Here’s a mistake that shows up in invoices and spreadsheets every day. A price of 120 includes 20 percent VAT. What’s the price before tax?
Wrong: 120 × 0.80 = 96
Right: 120 ÷ 1.20 = 100The 20 percent was added to the base price and not to the final figure, so you must divide. Check by going forward: 100 × 1.2 = 120, whereas 96 × 1.2 = 115.20.
The same logic applies to discounts. A sale price of 80, after 20 percent off, came from 80 ÷ 0.8 = 100. Adding 20 percent to 80 gives 96, and that’s not the original.
Margin versus markup
Both compare profit to something else, and they use different bases. An item costs 60 and sells for 100:
Margin (profit ÷ selling price): 40 ÷ 100 = 40%
Markup (profit ÷ cost): 40 ÷ 60 = 66.7%To reach a 40 percent margin from a cost of 60, divide 60 ÷ (1 − 0.4), which gives 100. Adding 40 percent to cost gives only 84, for a margin of 28.6 percent. Mixing them up means selling for less than you planned.
Averaging percentages
You can’t average percentages with different bases. Two stores report conversion rates:
| Store | Visitors | Rate |
|---|---|---|
| A | 100 | 10% |
| B | 300 | 20% |
The simple average is 15 percent. The real rate is total conversions over total visitors:
(10 + 60) ÷ 400 = 17.5%Weight by the base, or better, add up the raw counts. Spreadsheet summaries that average a percentage column get this wrong all the time.
Starting from zero, and other edge cases
A change from zero is undefined. From 0 to 50 is not a percentage increase, since you can’t divide by 0. Report the absolute change, or write “new”. Be suspicious of any “infinite growth” claim.
Negative starting values. A move from −20 to −10 improved, but the formula gives 10 ÷ −20 = −50%, which suggests a decline. If a figure such as profit crosses or sits below zero, describe it in absolute terms. Using |start| as the base is a common convention, so state it if you do.
Percentages over 100. A rise from 50 to 150 is a 200 percent increase, and the new figure is 300 percent of the old. “200 percent more” and “300 percent of” are the same thing. “Twice as much” is 100 percent more, so be careful with phrases such as “three times more”, which people use in different ways.
Small bases. Going from 2 customers to 4 is a 100 percent increase. It’s accurate, and it’s not useful. Show the counts when they’re small.
A quick habit for checking your work
Before you accept a percentage, ask two questions: percent of what, and what are the raw numbers? Then test the result by going backward. If you’ve calculated a 20 percent discount took 150 to 120, confirm that 120 ÷ 0.8 returns 150. For mixed fractions and ratios, the Fraction Calculator is useful. For anything that compounds, multiply the factors rather than adding percentages.