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How to use this calculator#
- Put the earlier figure in the Old boxThe old value is the denominator in every percentage change. Last month's revenue, last year's price, the reading before the adjustment — whichever came first goes on the bottom.
- Read the sign, not just the numberA negative result is a decrease. The formula is identical either way; there is no separate ‘percentage decrease formula’ to remember, only a minus sign that falls out of the subtraction.
- Switch modes to apply a change instead of measuring oneThe second mode multiplies by (1 + P ÷ 100) to add a percentage or (1 − P ÷ 100) to take one off, which is what you want for price rises, discounts and pay increases.
- Check the reverse before you rely on itIf you need to undo a change, do not reuse the same percentage. Divide by the multiplier instead: undoing a 20% cut takes a 25% rise, and undoing a 50% cut takes a 100% rise.
The formula#
Percentage change
Change % = ((New − Old) ÷ Old) × 100 • New = Old × (1 + P ÷ 100) • Reverse change % = (1 ÷ (1 + P ÷ 100) − 1) × 100
- Old
- The starting or earlier value — always the denominator
- New
- The finishing or later value
- P
- The percentage change, positive for an increase and negative for a decrease
- Change %
- The result: how much the value moved relative to where it started
- Reverse change %
- The percentage needed to get back to the original value
Percentage change is undefined when Old is zero, because you would be dividing by zero — report the absolute movement instead. When Old is negative the sign of the result becomes misleading, so most analysts switch to absolute change or to a symmetric measure there too.
The percentage increase formula#
Percentage increase = ((New − Old) ÷ Old) × 100. The old value is always the denominator, because you are measuring growth relative to where you started. If a subscription rises from $40 to $55, the change is $15, and 15 ÷ 40 = 0.375, so the price went up 37.5%. The most common mistake is dividing by the new value, which understates every increase.
Percentage decrease, and why it is not symmetric#
The same formula handles decreases — the result simply comes out negative. Dropping from 80 to 60 gives (60 − 80) ÷ 80 = −0.25, a 25% decrease. Note that increases and decreases do not cancel out: a 25% fall from 80 lands at 60, but a 25% rise from 60 only reaches 75. You would need a 33.3% rise to get back to 80.
Applying a percentage to a number#
To add X% to a value, multiply by (1 + X ÷ 100); to subtract it, multiply by (1 − X ÷ 100). A $250 invoice with 12% added becomes 250 × 1.12 = $280, and the same invoice with 12% off becomes 250 × 0.88 = $220. Applying two percentages back to back compounds rather than adds — 10% then 10% is a 21% rise, not 20%.
Worked examples#
A rent increase, and what three of them compound to
Rent rises from $1,450 to $1,595 a month.
- Change = 1,595 − 1,450 = 145
- 145 ÷ 1,450 = 0.10
- 0.10 × 100 = 10%
- Three annual rises of 10%: 1,450 × 1.10 × 1.10 × 1.10 = 1,450 × 1.331 = 1,929.95
A 10% increase. Repeated three times it is a 33.1% rise in total, not 30%, because each rise applies to the already-increased rent.
Why a fall and a rise of the same percent do not cancel
A holding worth $80 drops to $60, then has to climb back to $80.
- The fall: (60 − 80) ÷ 80 = −0.25, a 25% decrease
- The recovery: (80 − 60) ÷ 60 = 0.3333, a 33.33% increase
- Check: 60 × 1.3333 = 80 ✓
- The two bases differ — the fall is measured against 80, the recovery against 60
You need a 33.33% gain to undo a 25% loss. A 50% loss needs a 100% gain, and an 80% loss needs a 400% gain.
Reference tables#
| Change | Multiply by | Change needed to reverse it |
|---|---|---|
| −50% | 0.50 | +100% |
| −33.33% | 0.6667 | +50% |
| −25% | 0.75 | +33.33% |
| −20% | 0.80 | +25% |
| −10% | 0.90 | +11.11% |
| −5% | 0.95 | +5.26% |
| +5% | 1.05 | −4.76% |
| +10% | 1.10 | −9.09% |
| +20% | 1.20 | −16.67% |
| +25% | 1.25 | −20% |
| +50% | 1.50 | −33.33% |
| +100% | 2.00 | −50% |
The reverse change is always 1 ÷ multiplier − 1. Reversing gets disproportionately harder as losses grow, which is the whole reason drawdowns matter in investing.
| Sequence | Combined multiplier | Actual net change |
|---|---|---|
| +10% then +10% | 1.10 × 1.10 = 1.21 | +21% |
| +10% then −10% | 1.10 × 0.90 = 0.99 | −1% |
| +20% then −20% | 1.20 × 0.80 = 0.96 | −4% |
| −30% then +30% | 0.70 × 1.30 = 0.91 | −9% |
| +50% then −50% | 1.50 × 0.50 = 0.75 | −25% |
| −20% then −20% | 0.80 × 0.80 = 0.64 | −36% |
| +25% then +25% | 1.25 × 1.25 = 1.5625 | +56.25% |
An up-then-down pair of the same size always ends below where it started. The shortfall is P² ÷ 100 percent, so ±20% loses 4% and ±50% loses 25%.
| Annual rate | After 5 years | After 10 years | After 20 years |
|---|---|---|---|
| 2% | +10.41% | +21.90% | +48.59% |
| 3% | +15.93% | +34.39% | +80.61% |
| 5% | +27.63% | +62.89% | +165.33% |
| 7% | +40.26% | +96.72% | +286.97% |
| 10% | +61.05% | +159.37% | +572.75% |
Read the reverse of this table as the rule of 72: at 7% a value roughly doubles in ten years, which the +96.72% figure confirms.
Common mistakes#
- Dividing by the new valueFrom 40 to 55, dividing 15 by 55 gives 27.3% instead of 37.5%. The denominator must be the value you started from, otherwise you are answering a different question — what fraction of the new total the change represents.
- Reporting percentage points as percentagesA conversion rate moving from 2% to 3% is one percentage point, but a 50% increase in conversions. Quoting the wrong one in a report either flatters the result absurdly or buries a genuinely large improvement.
- Averaging percentage changes arithmeticallyTwo periods of +50% and −50% average to 0% arithmetically, but the money is down 25%. The correct answer uses the geometric mean: √(1.50 × 0.50) = 0.866, so −13.4% per period. Growth rates must be multiplied, never averaged.
- Treating a change from a tiny base as meaningfulGoing from 1 sale to 3 is a 200% increase, which sounds enormous and tells you almost nothing. When the base is small, report the absolute numbers alongside the percentage or the figure will mislead every reader.
Frequently asked questions#
What is the formula for percentage increase?
Subtract the old value from the new value, divide the result by the old value, then multiply by 100. Written out: ((New − Old) ÷ Old) × 100.
Can a percentage increase be more than 100%?
Yes. Whenever the new value is more than double the old one, the increase exceeds 100%. Going from 20 to 60 is a 200% increase, because the change of 40 is twice the original 20.
Why can't I use the same percentage to reverse a change?
Because the base you apply it to has changed. A 20% cut from 100 gives 80, but a 20% rise from 80 gives 96. To undo a 20% cut you need a 25% increase.
What if the old value is zero?
Percentage change is undefined, since you would be dividing by zero. Report the movement as an absolute change instead.
Key terms#
- Percentage change
- The difference between two values expressed as a proportion of the earlier one: ((New − Old) ÷ Old) × 100.
- Percentage point
- The plain subtraction of one percentage from another. Use it whenever both figures are already percentages.
- Multiplier (growth factor)
- 1 + P ÷ 100. Chains by multiplication across periods, which is what makes compounding calculations short.
- CAGR
- Compound annual growth rate — the single yearly rate that turns a starting value into an ending value over n years: (End ÷ Start)^(1/n) − 1.
- Base effect
- The distortion caused when the starting value is unusually small or large, making the resulting percentage change look dramatic without much real movement.
- Relative vs absolute change
- Absolute change is New − Old in the original units; relative change scales that by the old value to give a percentage.
Sources#
- Prealgebra 2e — percent increase and decrease — OpenStax, Rice University
- Guide for the Use of the International System of Units (SP 811) — NIST
- Algebra review — preliminaries — Paul's Online Notes, Lamar University
Figures last checked .
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