The Three Core Percentage Problems
Almost every percentage situation in real life is one of three types. Type 1: "What is X% of Y?" — multiply: 20% of $85 = 0.20 × 85 = $17. Type 2: "X is what percent of Y?" — divide and multiply: 18 out of 40 = (18 ÷ 40) × 100 = 45%. Type 3: "X is Y% of what number?" — divide: $15 is 20% of what? = 15 ÷ 0.20 = $75. Recognizing which type you are facing instantly tells you the right calculation.
Calculating Discounts and Sale Prices
For a 30% discount on a $120 item, calculate the discount: 0.30 × $120 = $36, making the sale price $84. The faster method: the item costs 70% of its original price, so $120 × 0.70 = $84. When discounts stack, they do not simply add up. A "20% off, then an additional 10% off" is not 30% off. You get 20% first ($100 → $80), then 10% off the reduced price ($80 × 0.90 = $72). The combined discount is 28%, not 30%.
Percentage Change: Increases and Decreases
Percentage change formula: ((New – Old) ÷ Old) × 100. If a stock goes from $50 to $65: ((65 – 50) ÷ 50) × 100 = +30%. If it drops from $65 back to $50: ((50 – 65) ÷ 65) × 100 = –23.1%. Note the asymmetry: a 30% gain followed by a 23.1% loss brings you back to even. This is why a 50% loss requires a 100% gain to break even — something many investors overlook.
Quick Mental Math Tricks
The commutative property of percentages: X% of Y = Y% of X. So 7% of 50 = 50% of 7 = 3.5. This often transforms a hard problem into a trivial one. To calculate 15%: find 10% (move decimal left), then add half of that. To estimate sales tax of 8.25%, round to 8% for quick calculations. For restaurant math, if the bill ends in a round number, 20% is just moving the decimal and doubling.
When comparing percentage increases across different base values, remember that the same percentage means very different absolute amounts. A 10% raise on a $30,000 salary is $3,000. A 10% raise on a $100,000 salary is $10,000. Always consider both the percentage and the absolute number to get the full picture.