Mistake #1: Reversing Percentage Changes
If a stock drops 50% from $100 to $50, it needs to gain 100% (not 50%) to return to $100.
Why: Percentage calculations use the new base. $50 Γ 50% = $25 (only gets to $75). $50 Γ 100% = $50 (gets back to $100).
Mistake #2: Adding Percentages Incorrectly
A 20% increase followed by 20% decrease does NOT return to original value.
Example: $100 + 20% = $120, then $120 - 20% = $96 (not $100!)
Why: The second 20% is calculated on $120, not $100.
Mistake #3: Confusing Percentage Points vs Percentage
Interest rate changing from 5% to 7% is NOT a 2% increaseβit's a 40% increase.
Percentage point change: 7% - 5% = 2 percentage points
Percentage change: ((7 - 5) Γ· 5) Γ 100 = 40% increase
Frequently Asked Questions
What's the difference between percentage change and percentage difference?
Percentage Change: Measures change from an initial value to a new value. Has direction (increase or decrease) and uses the original value as the base. Formula: ((New - Old) Γ· Old) Γ 100. Example: Price increased from $100 to $120 = 20% change. Percentage Difference: Measures relative difference between two values without regard to which is "original." Uses average of both values as base. Formula: (|V1 - V2| Γ· ((V1 + V2) Γ· 2)) Γ 100. Example: Compare $100 and $120 = 18.18% difference. When to use which: Use percentage change when you have a clear "before" and "after" (sales growth, weight loss). Use percentage difference when comparing two independent values (comparing prices between stores, test scores between groups).
Can percentages exceed 100%?
Yes! Percentages can be any positive or negative number. Above 100%: Common in growth calculations. If investment doubles, that's a 100% increase. If it triples, 200% increase. A 150% increase means the value became 2.5Γ larger. Negative percentages: Indicate decreases. A -50% change means value fell by half. Can go below -100% in percentage change calculations (if new value is negative in some contexts). Examples: Sales increasing 300% means they quadrupled. Stock price up 500% means it's now 6Γ original value. Revenue declining by 80% means only 20% remains.
How do I calculate percentage of percentage (compound percentages)?
You multiply the multipliers. Example: Price increases by 10%, then increases by another 15%. Many people incorrectly add (10% + 15% = 25%). Correct method: $100 Γ 1.10 = $110 (after first increase), then $110 Γ 1.15 = $126.50 (after second). Total change: 26.5%, not 25%. Formula: (1 + %1) Γ (1 + %2) - 1 = total percentage change. For increases/decreases: $100 increased by 20%, then decreased by 15% = $100 Γ 1.20 Γ 0.85 = $102 (net 2% increase). Key insight: Order matters for understanding but not for final resultβ20% increase then 15% decrease = same final value as 15% decrease then 20% increase (both end at $102).
What's the easiest way to calculate tips in my head?
10% method (easiest): Move decimal point one place left. Bill is $47.50, 10% = $4.75. For 20% tip, double it: $9.50. For 15% tip, add half of 10%: $4.75 + $2.38 = $7.13. Round and adjust method: Round bill to nearest $10. $47 becomes $50. 20% of $50 = $10. Subtract a bit since you rounded up: ~$9.50. 15% shortcut (precise): 10% + half of that. Bill $60, 10% = $6, half of $6 = $3, total 15% tip = $9. 20% shortcut: Divide by 5. $75 bill Γ· 5 = $15 tip. Pro tip: Calculate 20% (easy: divide by 5), then adjust down if you want 18% or 15%. $50 bill: 20% = $10, so 15% β $7.50, 18% β $9.
How do I convert fractions to percentages?
Simple method: Divide numerator by denominator, then multiply by 100. Example: 3/4 = 3 Γ· 4 = 0.75 = 75%. Common fractions memorized: 1/4 = 25%, 1/2 = 50%, 3/4 = 75%, 1/3 β 33.33%, 2/3 β 66.67%, 1/5 = 20%, 1/10 = 10%, 1/8 = 12.5%. Shortcut for certain fractions: Multiply numerator and denominator to reach base of 100. Example: 2/5 = (2Γ20)/(5Γ20) = 40/100 = 40%. Or: 7/20 = (7Γ5)/(20Γ5) = 35/100 = 35%. For ratios: "3 out of 5" = 3/5 = 0.6 = 60%.
What does percentage point mean and how is it different from percentage?
Percentage points: Absolute difference between two percentages. Percentage: Relative change. Example: Interest rate goes from 5% to 7%. Percentage point change: 7% - 5% = 2 percentage points (simple subtraction). Percentage change: ((7-5) Γ· 5) Γ 100 = 40% increase (relative change). Why it matters: News often confuses these. "Unemployment rose from 5% to 7%" is a 2 percentage point increase, but a 40% increase in unemployment rate. Election: Candidate A has 40% support, Candidate B has 50%. B is ahead by 10 percentage points, or 25% more support than A. Rule: Use percentage points for absolute differences between percentages. Use percentage for relative changes.
How do sales commissions and discounts work with percentages?
Commission calculation: Sales Amount Γ Commission Rate. Example: Sold $10,000 worth of products at 8% commission = $10,000 Γ 0.08 = $800 earned. Multiple discounts (stacking): Apply sequentially, not additively. Item is $100 with 20% off, then additional 10% off. WRONG: 20% + 10% = 30% = $70 final price. RIGHT: $100 Γ 0.80 = $80 (after first discount), then $80 Γ 0.90 = $72 (after second discount) = 28% total discount. Markup to cover commission: If you want $100 profit after paying 15% commission, you need to charge more. Calculation: $100 Γ· (1 - 0.15) = $100 Γ· 0.85 = $117.65. Discount limits: Multiple 50% off coupons don't equal 100% off (free). First 50% off makes it $50, second 50% off that makes it $25 (75% total discount).
How do percentage increases and decreases relate?
They're not symmetric! If value increases by X%, you need to decrease by MORE than X% to get back to original. Examples: $100 increases 50% to $150. To return to $100, needs to decrease by 33.33% (not 50%). $100 doubles (100% increase) to $200. To return to $100, needs to decrease 50% (not 100%). Formula for return percentage: If increased by X%, decrease needed = X Γ· (1 + X). If increased 25% (1.25), decrease needed = 0.25 Γ· 1.25 = 20%. Why: Percentage calculations use the current value as base. After increase, you're calculating percentages on a larger number. Practical application: Investment drops 50% from $10,000 to $5,000. To recover losses, needs 100% gain (double from $5,000 to $10,000). This is why "preserving capital" is critical in investingβrecovering from losses requires exponentially larger gains.