You see "20% off" on a price tag — can you calculate the final cost in your head? Or maybe you need to reverse a sales tax to find the pre-tax price. Percentages appear constantly in everyday life, yet they're surprisingly easy to get wrong. This guide walks through the most common real-world percentage scenarios so you know exactly which formula to use.
1. The Basics: "%" Just Means Divide by 100
Percent literally means "per hundred." 15% = 15 ÷ 100 = 0.15. This conversion is the foundation of every percentage calculation.
Three core questions and their formulas:
- What percent is A of B? →
A ÷ B × 100 - What is X% of B? →
B × X ÷ 100 - A is X% of what number B? →
A ÷ (X ÷ 100)
"Part ÷ Whole = Percentage" and "Whole × Percentage = Part." Keep these two directions in mind and you can handle almost every everyday percentage problem.
2. Discounts: What Does "20% Off" Actually Mean?
There are two ways to express a discount, and mixing them up is a common source of confusion:
| Common phrasing | You pay | You save |
|---|---|---|
| 10% off | 90% of original price | 10% |
| 20% off | 80% of original price | 20% |
| 30% off | 70% of original price | 30% |
| 50% off (half price) | 50% of original price | 50% |
Example: an item originally priced at $120 is "20% off":
- Sale price =
120 × (1 − 0.20) = 120 × 0.80 = $96 - Amount saved =
120 − 96 = $24
Reverse: you saved $30 on a $150 item — what's the discount percentage?
- Discount % =
30 ÷ 150 × 100 = 20%
3. Tax: Adding and Removing Tax Are Not Symmetric
This is one of the most frequent calculation mistakes. Using a 10% sales tax as an example:
Pre-tax → Tax-inclusive (adding tax):
- Tax-inclusive = Pre-tax × (1 + 10%) = Pre-tax × 1.10
- Example: Pre-tax $100 → Tax-inclusive
100 × 1.10 = $110
Tax-inclusive → Pre-tax (removing tax):
- Pre-tax = Tax-inclusive ÷ 1.10
- Example: Tax-inclusive $110 → Pre-tax
110 ÷ 1.10 = $100
To remove a 10% tax, some people calculate
110 × 0.90 = $99 — this is wrong. The tax was calculated on the pre-tax amount, not the tax-inclusive amount. Always divide by (1 + tax rate) to reverse a tax.
4. Growth Rate and Percentage Change
The formula for percentage change is:
% Change = (New Value − Old Value) ÷ Old Value × 100%
Common examples:
- Revenue grew from $80K to $100K → Growth =
(100 − 80) ÷ 80 × 100% = 25% - Weight dropped from 75kg to 70kg → Change =
(70 − 75) ÷ 75 × 100% ≈ −6.7%(negative = decrease) - Price increased from $200 to $230 → Increase =
(230 − 200) ÷ 200 × 100% = 15%
Key rule: the denominator is always the old (baseline) value, not the new one.
5. The Compounding Trap: You Can't Just Add Percentages
An item goes up 20%, then goes on sale for 20% off. You might think it's back to the original price — but it's not.
- Original price $100 → Up 20% →
100 × 1.2 = $120 - Then 20% off →
120 × 0.8 = $96 - Final price is $96 — 4% below the original price
The reason: the two changes use different base values. The 20% increase is applied to $100; the 20% decrease is applied to $120. Percentage changes multiply, they don't simply add or cancel.
A savings account with 3% annual interest doesn't earn "6% over two years" — it earns
1.03 × 1.03 = 1.0609, or about 6.09%. The difference seems small, but it compounds significantly over long time horizons.
6. Proportions: What Percentage Is This of the Total?
Formula: Proportion = Item Amount ÷ Total Amount × 100%
| Expense | Amount | Share |
|---|---|---|
| Rent | $1,500 | 1,500 ÷ 4,500 ≈ 33.3% |
| Food | $1,200 | 1,200 ÷ 4,500 ≈ 26.7% |
| Transport | $800 | 800 ÷ 4,500 ≈ 17.8% |
| Entertainment | $1,000 | 1,000 ÷ 4,500 ≈ 22.2% |
| Total | $4,500 | 100% |
7. Goal Achievement Rate
Achievement Rate = Actual ÷ Target × 100%
- Monthly sales target: 100 units, actual: 73 → Achievement rate =
73% - Target: 50 features, completed: 38 → Completion rate =
76%
Achievement rates can exceed 100%, meaning the target was surpassed.
8. Summary
Percentage calculations look simple but hide several traps:
- Discounts: "X% off" means you pay (100 − X)% of the price, not X%
- Tax reversal: Divide by (1 + tax rate) to remove tax — don't multiply by (1 − tax rate)
- Growth rate: Always use the old (baseline) value as the denominator; the result can be negative
- Compounding trap: Sequential percentage changes multiply — they don't simply add or cancel
For anything more complex than mental arithmetic, a dedicated calculator can save time and eliminate errors.