The formula fits in one breath: difference divided by where you started. The consequences don't. The same $20 reads as a 50% jump or a rounding error depending on the base, a 50% loss demands a 100% gain to undo, and averaging two percentage changes quietly gives you the wrong answer every time. Here's the arithmetic and the traps around it.
Percentage change = (new − old) ÷ old × 100
Three pieces: the difference between the two values, the original value as the base, and a multiply by 100 to turn the ratio into a percent. A positive result is an increase, a negative result is a decrease, and the size of the base is what the whole metric hangs on. The percentage change calculator runs the formula both directions and labels the result.
One choice to make before you compute: change, or difference? Percentage change needs a before-and-after and divides by the original. Percentage difference compares two values with no starting point — it divides the gap by the average of the two, treating them symmetrically. Comparing two phone prices where neither came first? Difference. Everything else in this guide is change.
Take it step by step. A subscription climbs from $80 to $100 a year:
Working backwards is just as common. If a price after a 25% increase is $500, the original was 500 ÷ 1.25 = $400 — divide by (1 + rate), don't subtract 25% of $500, which gives $375 and is wrong. A raise works the same way: 4% on a $60,000 salary is 60,000 × 0.04 = $2,400, landing at $62,400.
Same formula; the sign flips on its own. A stock falls from $100 to $80: (80 − 100) ÷ 100 × 100 = −20%, a 20% decrease.
Notice what just happened with the numbers from the two sections. Moving from 80 to 100 is +25%, but moving from 100 to 80 is −20% — the same $20, different percentages, because each move divides by its own starting point. This isn't a quirk to memorize; it's the core behavior of the metric, and it's why "up 25%, then down 20%" leaves you exactly where you started.
Because the base is doing half the work. Watch one fixed $20 move against a series of starting values:
| Move | Difference | Percentage change |
|---|---|---|
| $40 → $60 | +$20 | +50.0% |
| $80 → $100 | +$20 | +25.0% |
| $180 → $200 | +$20 | +11.1% |
| $980 → $1,000 | +$20 | +2.0% |
| $1,980 → $2,000 | +$20 | +1.0% |
Headlines exploit this constantly. Doubling from 2 customers to 4 is a 100% jump worth nothing; a 2% gain on a billion-dollar revenue line is enormous. The defensive habit is to read every percentage with its absolute numbers attached.
Because each percentage multiplies a different amount. Lose 50% of $100 and you hold $50; a 50% gain on $50 adds only $25, leaving $75. Full recovery needs a 100% gain on the $50. The deeper the loss, the more lopsided the repair:
| Loss | Value left (from $100) | Gain needed to break even |
|---|---|---|
| 10% | $90 | 11.1% |
| 20% | $80 | 25% |
| 33.3% | $66.70 | 50% |
| 50% | $50 | 100% |
| 75% | $25 | 300% |
| 90% | $10 | 900% |
This is why drawdowns obsess portfolio managers: a 90% loss doesn't need a good year back, it needs the position to multiply tenfold. The general rule is gain = 1 ÷ (1 − loss) − 1: the required recovery grows far faster than the loss.
Chain them with multiplication, one plus each rate. A 25% rise then a 20% fall is 1.25 × 0.80 = 1.00 — flat, despite the percentages looking like they should net to +5%:
| Sequence | Chained (multiply factors) | Naive average | Actual result |
|---|---|---|---|
| +10% then −10% | 1.10 × 0.90 = 0.99 | 0% | −1.0% |
| +20% then −20% | 1.20 × 0.80 = 0.96 | 0% | −4.0% |
| +50% then −50% | 1.50 × 0.50 = 0.75 | 0% | −25.0% |
| +25% then −20% | 1.25 × 0.80 = 1.00 | +2.5% | 0.0% |
| +100% then −50% | 2.00 × 0.50 = 1.00 | +25% | 0.0% |
The pattern in the middle rows is worth internalizing: equal up-and-down percentages always end below the start, and the bigger the swing, the bigger the bite. Volatility itself has a cost — part of why "average return" and "what you actually made" diverge over time.
Points are the gap; percent is the relative change. A mortgage rate moving from 3% to 5% rises 2 percentage points, but as a relative change that's 2 ÷ 3 = 66.7% — two honest numbers describing the same move, differing by a factor of 33.
Financial news defaults to points: a Fed hike of 25 basis points is 0.25 percentage points. Marketing copy prefers percent, where a 0.25-point move on a 3% rate can parade as an 8.3% increase. When you read a claim about rates, polls, or market share moving by some percent, the first question to ask is whether the writer meant points.
Nearly every number in the news is one. Monthly inflation is the CPI's percentage change over the past twelve months. Quarterly earnings, unemployment moves, population trends, price hikes, poll swings — all (new − old) ÷ old. Even a store markdown is a percentage decrease: 30% off a $90 jacket saves $27 — the discount calculator handles the sale-price side. For the simpler building block — X% of Y, or what percent one number is of another — the percentage calculator is the tool.
Enter the original and the new value — get the difference, the percentage change, and whether it counts as an increase or decrease.
Percentage Change Calculator →Percentage change is (new − old) ÷ old × 100 — increases and decreases alike. The base decides how big the number reads, losses need disproportionately larger gains to undo, chained changes multiply rather than add, and points are not percent. Run your pairs through the percentage change calculator, sanity-check growth claims with the percentage calculator, and never let a headline quote a percentage without its dollars.
Subtract the old value from the new value, divide by the old value, and multiply by 100. A price moving from 80 to 100 dollars is (100 − 80) ÷ 80 × 100 = 25%, a 25 percent increase. The same formula covers decreases — the result just comes out negative.
Identically: (new − old) ÷ old × 100. A fall from 100 to 80 gives (80 − 100) ÷ 100 × 100 = −20%, a 20 percent decrease. Note the asymmetry: 80 to 100 is +25% while 100 to 80 is −20%, because each uses a different starting base.
Percentage change divides by the original value and needs a before-and-after. Percentage difference compares two values symmetrically by dividing their absolute difference by their average. Use change for how much did it grow, and difference for how far apart are these two when neither is the starting point.
Percentage points are the simple gap between two percentages. A rate rising from 3% to 5% climbs 2 percentage points, which is a 66.7% relative increase. Confusing the two turns a modest move into a dramatic one or vice versa, especially with interest rates and polls.
Multiply the growth factors, never the percentages. A 25% rise followed by a 20% fall is 1.25 × 0.80 = 1.00, net zero — not +5%. Average the two percentages and you get the wrong answer every time there is a change, because each was computed against a different base.