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Compute exact percentage proportions, relative percentage increases and decreases, symmetric percentage differences, reverse percentage markdowns, and compounding growth rates.
P% to decimal multiplier r = rac{P}{100} or compute part-to-whole ratio ext{Ratio} = rac{V_{ ext{part}}}{V_{ ext{whole}}}. ext{Result} = V_{ ext{whole}} imes left( rac{P}{100}
ight). Reverse calculation (original price before markup/tax): V_{ ext{orig}} = rac{V_{ ext{final}}}{1 + (P/100)}.Delta% = left( rac{V_{ ext{final}} - V_{ ext{initial}}}{|V_{ ext{initial}}|}
ight) imes 100%. Positive values denote appreciation; negative values denote depreciation. ext{Diff}% = rac{|V_1 - V_2|}{(V_1 + V_2) / 2} imes 100%.Percentage Change measures directional growth or reduction from a specific initial baseline over time: ((Final - Initial) / Initial) * 100%. Percentage Difference compares two static measurements where neither is the 'original', dividing the absolute difference by the average midpoint: (|V1 - V2| / ((V1+V2)/2)) * 100%.
Because the base changes. If an item costs $100, a 20% discount reduces it to $80 ($100 - $20). Increasing $80 by 20% adds only $16 ($80 * 0.20), resulting in $96. To recover from an X% loss, you need a larger percentage gain: 1 / (1 - X) - 1.
To find the original price before an added tax/markup (T%), divide by (1 + T/100). For example, $115 with 15% VAT has an original price of 115 / 1.15 = $100. To find the price before a discount (D%), divide by (1 - D/100). For example, a $60 item with a 25% discount had an original price of 60 / 0.75 = $80.
Percentage points measure the arithmetic difference between two percentages. If an interest rate climbs from 4% to 5%, it increased by 1 percentage point (5 - 4 = 1), but by 25 percent in relative terms ((5 - 4)/4 = 0.25 = 25%). Conflating these leads to massive statistical misinterpretations.