Why isn't a 50% rise cancelled by a 50% fall?
Because each percentage is taken from a different base. $100 up 50% is $150; $150 down 50% is $75, not $100. To fully reverse a +50% move you need a −33.3% move.
Calculate percentages, percentage change, increase, and decrease. Solve 'what is X% of Y' and 'X is what percent of Y' instantly.
X% of Y = (X ÷ 100) × Y. X is what % of Y = (X ÷ Y) × 100. % change = ((New − Old) ÷ Old) × 100.
25% of 200 = (25 ÷ 100) × 200 = 50. Going from 100 to 150 is a ((150 − 100) ÷ 100) × 100 = +50% change; going back from 150 to 100 is only −33.3%, because the base changed.
Because each percentage is taken from a different base. $100 up 50% is $150; $150 down 50% is $75, not $100. To fully reverse a +50% move you need a −33.3% move.
If a rate moves from 4% to 6%, that is a rise of 2 percentage points but a 50% percentage change. Financial reporting uses percentage points to avoid this ambiguity.
To add 20%: multiply by 1.20. To remove 20% that is already included: divide by 1.20, not multiply by 0.80. $120 including 20% tax has a net price of $100, not $96.
Sale price = Original × (1 − discount ÷ 100). A $80 item at 35% off costs 80 × 0.65 = $52, saving $28.
Use ((Year 2 − Year 1) ÷ Year 1) × 100. Revenue moving from $420,000 to $505,000 is ((505,000 − 420,000) ÷ 420,000) × 100 = +20.2%.
If 15% of a number is 45, the original is 45 ÷ 0.15 = 300. Divide the part by the decimal form of the percentage.