Percentage change, with the direction attached
An increase and a decrease of the same size are not the same percentage, because they are measured against different starting numbers. This page keeps the base and the direction visible.
SOURCE·Arithmetic only — no external rate·REVIEWED JUL 2026
Going from 80 to 100 is a 25% increase — the change (20) divided by the starting value.
How percentage change works
- Formula: (new − old) ÷ old × 100
- 80 → 100 is a +25% increase (20 ÷ 80)
- 100 → 80 is a −20% decrease (20 ÷ 100)
- Same 20-unit gap. Different base, different answer — that is the whole trap.
How percentage change is worked out
- Subtract the old value from the new value. A positive result is an increase, a negative one a decrease — the direction is part of the answer, not a footnote.
- Divide by the old value. This is the step people get wrong: the base is always the value you started from, never the one you ended at.
- Multiply by 100. Going 80 → 100 is +25%; going 100 → 80 is −20%. Same 20-unit gap, different bases, different answers.
- Because the base moves, percentage changes multiply rather than add — and a fall needs a larger rise to undo it. Recovering a 50% drop takes a 100% rise.
change % = ((new − old) ÷ old) × 100
Full method, rounding and worked examples →
SOURCE·Arithmetic only — no external rate·REVIEWED JUL 2026
What it takes to recover a fall
A fall and the rise that undoes it are never the same percentage, because the rise is measured against the smaller number left behind. This is the table that explains why a portfolio down 50% needs to double, not gain 50%.
| Fall | Left from 100 | Rise needed | Multiplier |
|---|---|---|---|
| −5% | 95 | +5.26% | 1.0526× |
| −10% | 90 | +11.11% | 1.1111× |
| −15% | 85 | +17.65% | 1.1765× |
| −20% | 80 | +25% | 1.25× |
| −25% | 75 | +33.33% | 1.3333× |
| −30% | 70 | +42.86% | 1.4286× |
| −40% | 60 | +66.67% | 1.6667× |
| −50% | 50 | +100% | 2× |
| −60% | 40 | +150% | 2.5× |
| −70% | 30 | +233.33% | 3.3333× |
| −75% | 25 | +300% | 4× |
| −80% | 20 | +400% | 5× |
| −90% | 10 | +900% | 10× |
SOURCE·Arithmetic only — no external rate·REVIEWED JUL 2026
Direction, and why it is part of the answer
“A 20% change” is not an answer. Up or down changes both the sign and the base, and reporting one without the other is how a 20% cut gets described as recoverable by a 20% rise.
Increase: the new value is larger, the result is positive, and the base is the smaller starting number — which is why increases can exceed 100% without anything being wrong. Doubling is +100%; tripling is +200%.
Decrease: the new value is smaller, the result is negative, and the base is the larger starting number — which is why a decrease can never exceed 100%. Losing everything is −100%, and there is nothing below that.
The asymmetry, stated plainly
Because a fall shrinks the base, undoing it always takes a larger percentage than the fall itself. Drop 20% and you need +25% back. Drop 50% and you need +100% — the number doubles. Drop 80% and you need +400%. The table above is that relationship for every common fall.
Percentage points are a different unit
If a margin moves from 4% to 6%, that is 2 percentage points, and simultaneously a 50% increase in the margin. Both are true and they are not interchangeable. Where the quantity being measured is already a percentage, say which unit you mean.
The four percentage calculations, and when each one applies
Every percentage question you will ever meet is one of four shapes. Naming the shape first is what stops the classic mistakes — and it is why this site keeps the four apart instead of putting one box on the page and hoping.
P% of N. Tips, commission, deposits, interest. 15% of $200 → $30.
P% off N. Sales and discounts — compute what you pay, not what you save. 30% off $80 → $56.
A is what % of B. Test scores, market share, progress. 42 of 60 → 70%.
% change. Price rises, growth, drops. 80 → 100 is +25%; 100 → 80 is −20%.
The base is everything
Nearly every percentage error is a wrong base — dividing by the new value instead of the old one, or taking a percentage of the discounted price instead of the original. The rule that fixes all of them: the base is the number the percentage is “of”, which in change problems is always the value you started from.
It is also why percentages do not add. A 20% rise followed by another 20% rise is not 40% — it is 44%, because the second rise acts on a bigger base (1.2 × 1.2 = 1.44). Compounding is just this effect repeated.
Mental shortcuts that survive real life
- 10% is one decimal shift. 10% of $87.50 is $8.75. Nearly everything builds from this: 5% is half of it, 20% is double, 15% is one-and-a-half.
- x% of y equals y% of x. 8% of 25 feels hard; 25% of 8 is obviously 2. Same number.
- For discounts, think “what fraction do I pay”. 25% off means you pay ¾. A $59.99 jacket at 25% off is about $45 before you reach for a phone.
Percentage questions, answered
How do I calculate percentage change between two numbers?
What rise do I need to recover a 50% fall?
How do I work out a percentage of a number?
How do I take a percentage off a price?
How do I work out what percent one number is of another?
How is percentage change calculated?
Why is a 50% increase not cancelled by a 50% decrease?
What is a percentage point?
Does the calculator store what I type?
Where a percentage usually goes next
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