Percentage Calculations: Every Formula, Every Variation, and the Stacked Discount Trap
If a store takes 50% off a $100 item and then takes an additional 20% off, the final price is $40 — not $30. Percentages don't add; they compound. The second 20% is off the already-discounted price, not the original. This is one of six common percentage errors that cost people money in everyday financial decisions. The math is straightforward once you understand the four fundamental question types.
Key Takeaways
- Four problem types, four formulas — identify which type before calculating; most errors come from using Type 4 when Type 1 is needed
- Percentages don't add — successive discounts compound: 50% off then 20% off = 60% total off, not 70%
- Percentage change always divides by the original — (new − old) / old × 100; using the new value as denominator is a common mistake
- Percentage points ≠ percentages — going from 10% to 15% is a 5 percentage point increase but a 50% relative increase
- Markup ≠ margin — 25% markup on cost ≠ 25% margin on revenue; these are different calculations with different denominators
The Stacked Discount Problem
When multiple percentage discounts apply to the same item, they multiply — they don't add. The formula for two stacked discounts d₁ and d₂:
Final price = Original × (1 − d₁) × (1 − d₂)
A $100 item with 50% off, then 20% off: $100 × 0.50 × 0.80 = $40. Total discount = 60%, not 70%. The combined effective discount = 1 − (1 − d₁)(1 − d₂) = 1 − 0.4 = 0.60.
Common Percentage Errors and How to Avoid Them
Error: Adding stacked discounts
❌ "40% off + 20% off = 60% off" → Wrong: 40% then 20% = 52% total
✓ (1−0.4)×(1−0.2) = 0.48 → 52% off
Error: Wrong denominator in % change
❌ Price went from $80→$100: (100−80)/100 × 100 = 20% (wrong denominator)
✓ (100−80)/80 × 100 = 25% increase
Error: Percentage points vs percent
"Interest rates rose from 2% to 4%" — this is a 2 percentage point increase AND a 100% relative increase. Both are correct; context determines which to state.
Markup vs Margin: The Business Percentage Trap
Markup and margin both describe profit as a percentage — but they use different denominators, producing very different numbers for the same situation. Markup divides by cost; margin divides by revenue.
| Concept | Formula | Example: Cost $60, Price $100 |
|---|---|---|
| Markup | (Price − Cost) / Cost × 100 | (100−60)/60 × 100 = 66.7% markup |
| Gross margin | (Price − Cost) / Price × 100 | (100−60)/100 × 100 = 40% margin |
| Discount | (Original − Sale) / Original × 100 | Denominator = original price |
| % increase | (New − Old) / Old × 100 | Denominator = starting value |
| % decrease | (Old − New) / Old × 100 | Denominator = starting value (same) |
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