How Percentages Actually Work
Percentages seem simple until they don't. Here's how to think about them clearly — including the mistakes most people make with percentage increases and decreases.
“Percent” means per hundred — from Latin per centum. A percentage is just a ratio expressed as a fraction of 100.
50% = 50/100 = 0.5. That’s the whole foundation. But percentages cause surprising amounts of confusion when applied to real situations, because there are a few non-obvious properties that catch people off guard.
The Basic Conversions
| Percentage | Decimal | Fraction |
|---|---|---|
| 50% | 0.5 | 1/2 |
| 25% | 0.25 | 1/4 |
| 10% | 0.1 | 1/10 |
| 1% | 0.01 | 1/100 |
| 0.5% | 0.005 | 1/200 |
To convert a percentage to a decimal, divide by 100 (or move the decimal point two places left). To go the other way, multiply by 100.
Finding a Percentage of a Number
“What is 15% of 80?”
Translate to math: 15/100 × 80 = 0.15 × 80 = 12.
A faster mental math shortcut: 10% of any number is just that number with a decimal shift. 10% of 80 is 8. 5% is half of that: 4. So 15% = 8 + 4 = 12.
Percentage Increase and Decrease
If a price rises from $40 to $50, what’s the percentage increase?
Formula: (new − old) / old × 100
(50 − 40) / 40 × 100 = 10/40 × 100 = 25%
A 25% increase. The key: you always divide by the original value.
For a decrease from $50 to $40:
(40 − 50) / 50 × 100 = −10/50 × 100 = −20%
A 20% decrease. Notice that a 25% increase and a 20% decrease are not inverses — they change the base.
The Asymmetry Trap
This is where most people get confused.
If something increases by 50% and then decreases by 50%, do you end up where you started?
No. Start with 100. Increase by 50%: 100 × 1.5 = 150. Decrease by 50%: 150 × 0.5 = 75. You end up at 75, not 100.
The decrease was applied to the higher value, so 50% of 150 is 75, not 50.
In general: percentage changes are multiplicative, not additive. A 20% increase followed by a 20% decrease equals 1.2 × 0.8 = 0.96 — a net 4% loss.
Applying a Percentage Increase
To increase a number by p%: multiply by (1 + p/100).
- 20% increase on $80:
80 × 1.20 = $96 - 15% increase on $200:
200 × 1.15 = $230
To decrease by p%: multiply by (1 − p/100).
- 30% discount on $150:
150 × 0.70 = $105 - 8% tax reduction on $1000:
1000 × 0.92 = $920
Reversing a Percentage Change
If an item costs $120 after a 20% increase, what was the original price?
You might think: subtract 20% from 120. 120 × 0.8 = $96. But that’s wrong.
The correct approach: 120 / 1.20 = $100.
You divide by the multiplier used to apply the increase. This trips people up constantly with sale prices: “30% off means I can find the original by adding 30%” — but that’s not right. If you paid $70 after a 30% discount, the original was $70 / 0.70 = $100, not $70 × 1.30 = $91.
Percentage Points vs Percentages
These are different things that sound the same.
If a tax rate increases from 20% to 25%, it increased by 5 percentage points — but by 25% as a relative change.
(25 − 20) / 20 × 100 = 25% relative increase, but only 5 percentage point increase.
Politicians and journalists sometimes conflate these intentionally or accidentally. “Interest rates rose by 1%” is ambiguous — did they rise from 3% to 3.03% (1% relative change) or from 3% to 4% (1 percentage point absolute change)? The latter is 33x larger.
Tips and Tax
A common real-world use: calculating a tip or sales tax.
15% tip on $44:
- 10% of 44 = 4.40
- 5% = 2.20
- Total tip: 6.60
20% tip on $44:
- 10% of 44 = 4.40
- Double it: 8.80
Adding 8.5% tax to $60:
60 × 1.085 = $65.10- Or: 8% = $4.80, 0.5% = $0.30 → tax = $5.10 → total = $65.10
Compound vs Simple Interest
Simple interest applies a percentage to the original principal only:
$1000 at 5% for 3 years = $1000 × 0.05 × 3 = $150 interest
Compound interest applies the percentage to the running balance:
$1000 at 5% annually: Year 1: $1050 → Year 2: $1102.50 → Year 3: $1157.63
The difference compounds (literally) over time. After 30 years, simple interest gives $2,500; compound gives $4,322. This is why savings accounts and investment returns are quoted as compound, while simple interest is typically used in installment loan calculations.
The Quick Summary
- Percent = per hundred. Convert to decimal by dividing by 100.
- Percentage change =
(new − old) / old × 100. Always divide by the original. - Percentage changes are multiplicative: +20% then −20% ≠ 0.
- To reverse a percentage change, divide by the multiplier (not apply the inverse percentage).
- “Percentage points” and “percent change” are different — context matters.