Percentages appear everywhere — sale prices, tax rates, test scores, interest rates, tips, statistics, nutrition labels, and polling data. Most people can handle simple percentages but get confused when the question changes form. Here are all the variations, clearly explained.
The Basic Formula
Percentage = (Part ÷ Whole) × 100. If 18 out of 24 students passed a test, the pass rate is (18 ÷ 24) × 100 = 75%. Simple enough — but the three variables can be shuffled in ways that trip people up.
Finding X% of a Number
"What is 15% of $80?" — multiply: 80 × 0.15 = $12. The shortcut: move the decimal two places left to turn the percent into a decimal, then multiply. 15% = 0.15. For quick mental math, find 10% first (move decimal one place: $8), then adjust (15% = 10% + 5% = $8 + $4 = $12).
What Percent Is X of Y?
"30 is what percent of 120?" — divide and multiply: (30 ÷ 120) × 100 = 25%. This is the most common form in real life: what grade did I get, what percentage of the budget did we spend, what's my return on this investment.
Percentage Increase and Decrease
% Change = ((New − Old) ÷ Old) × 100. A salary going from $50,000 to $54,000 is a (4,000 ÷ 50,000) × 100 = 8% increase. Going from $54,000 back to $50,000 is a (−4,000 ÷ 54,000) × 100 = −7.4% decrease — not 8%. A percentage increase and its reversal are never the same number.
💡 A common mistake: "Prices increased 50%, then dropped 50%" does NOT return to the original price. $100 × 1.5 = $150, then $150 × 0.5 = $75. You're down 25%.
Sales Tax and Discounts
To apply a 8.5% sales tax to $45: multiply by 1.085 = $48.83. To find the pre-tax price from a post-tax total: divide by 1.085. For a 30% discount: multiply by 0.70 (keeping 70% of the price). To find the original price before a 30% discount was applied: divide the sale price by 0.70.
Percentage Points vs Percentages
This distinction matters in news and finance. If interest rates go from 3% to 4%, they've increased by 1 percentage point but by 33% (1 ÷ 3 × 100). When a politician says "crime dropped 10 percentage points," that's very different from "crime dropped 10%."