A plain percentage question ("what is 20% of 85?") treats a percentage as a static slice of a fixed whole. Percentage change is a different kind of question entirely — it's asking how far a number *moved*, and the "whole" it's being measured against is a specific moment in time: wherever the value started out. That distinction is the entire concept this guide is built around, and it's why the two calculations, despite both producing a percentage as their answer, aren't interchangeable.
**Why "old value" has to be the anchor.** Think of percentage change as answering "how many old-value-sized steps did this move?" If a value climbs from 80 to 100, it moved 20 units, and 20 is a quarter the size of the 80 it started from — hence 25%. The anchor is always the starting point because that's the size you're measuring the *step* against, not the size you landed on.
**The formula this reasoning produces.** ((new value − old value) ÷ old value) × 100. The numerator is simply the size of the step (new minus old); dividing by the old value expresses that step as a fraction of the starting point; multiplying by 100 turns the fraction into a percentage. The result is positive for an increase, negative for a decrease.
**Worked example: percentage increase.** A plant grows from 12cm to 15cm. ((15 − 12) ÷ 12) × 100 = (3 ÷ 12) × 100 = 25% increase.
**Worked example: percentage decrease.** A population drops from 5,000 to 4,250. ((4250 − 5000) ÷ 5000) × 100 = (−750 ÷ 5000) × 100 = −15%, a 15% decrease.
**Why the base value matters so much.** Because the formula always divides by the *original* value, the same absolute change produces a different percentage depending on which direction you're measuring. A value going from 50 to 75 is a ((75−50)÷50)×100 = 50% increase. But going back from 75 to 50 is a ((50−75)÷75)×100 = −33.3% decrease — not −50%, even though the absolute change (25) is identical both times, because the base value is different in each direction. This asymmetry is exactly why "up 50%, then down 50%" never returns you to where you started: a $100 value up 50% is $150; down 50% from $150 is $75, not back to $100.
**Percentage change versus percentage-point change.** These sound similar and are frequently confused. If an interest rate moves from 4% to 6%, that's a 2 percentage-point increase (simple subtraction: 6 − 4 = 2), but it's a ((6−4)÷4)×100 = 50% relative increase. Both statements are accurate; they answer different questions. "Percentage change" (what this guide covers) means the relative change; "percentage-point change" means the raw difference between two percentages, treated as plain numbers rather than as a ratio.
**Compounding percentage changes over multiple periods.** Sequential percentage changes don't add together the way people often assume. Three consecutive years of 10% growth isn't 30% total growth — it compounds: starting at 100, after year one it's 110, after year two it's 121 (10% of 110, not of 100), after year three it's 133.1. The true three-year growth is 33.1%, not 30%, because each year's increase is calculated on an already-grown base.
**Edge cases.** If the old value is 0, percentage change is undefined (you can't divide by zero) — a value going from 0 to any positive number technically represents infinite percentage growth, which is why this scenario is usually described in absolute terms instead ("grew by $500") rather than as a percentage. If the old value is negative, the formula still technically works but the result can be visually misleading and needs careful interpretation of the sign.
**Common mistakes.** The most frequent error is dividing by the new value instead of the old one, especially when working backward from a known percentage change to find an unknown original value. A second common mistake is treating a percentage-point difference and a percentage change as interchangeable, particularly in contexts like interest rates, tax rates, or survey results where both numbers are already percentages.
**Worked example: finding the new value directly, given an old value and a percentage change.** A salary of $52,000 receives a 4.5% raise. New salary = old value × (1 + percentage change ÷ 100) = 52000 × 1.045 = $54,340. This is a faster route than computing the raise amount separately and adding it, though both give the identical result: 4.5% of 52,000 = $2,340, and 52,000+2,340=54,340, matching.
**Worked example: working backward to find the old value from a known new value and percentage change.** A house is now worth $310,000, after a 24% increase in value. Old value × 1.24 = 310,000, so old value = 310,000 ÷ 1.24 = $250,000. Note this divides, rather than multiplying by 0.76 (as you might if mistakenly treating a 24% increase like a 24% decrease reversed) — solving backward through a percentage *increase* requires dividing by (1 + rate), not multiplying by (1 − rate), which only applies to percentage decreases.
**A real-world illustration of why the base-value rule matters for comparing rates across time periods.** If a company's revenue grows 20% in Q1 and then 20% again in Q2, the two 20% figures represent different absolute dollar amounts, since Q2's 20% is calculated on Q1's already-larger revenue — this is exactly why financial reports distinguish "quarter-over-quarter" from "year-over-year" growth, since each compares against a different base.
**Worked example: percentage change used to describe a discount, connecting back to How to Calculate a Percentage's territory.** A price drops from $80 to $60. Percentage change: ((60−80)÷80)×100 = −25%. This is the same underlying relationship as a "25% off" discount, just computed from the before-and-after prices directly rather than from a stated discount rate — both framings describe the identical situation.
**Worked example: percentage change across a non-monetary quantity, temperature.** A city's average high goes from 68°F to 82°F. ((82−68)÷68)×100 ≈ 20.6% increase. The formula applies identically to temperature, distance, weight, or any other numeric measurement — percentage change doesn't care what units the underlying quantity is measured in, only that both values use the same unit consistently.
**A quick sanity check worth running on any percentage-change answer: does the sign make sense?** If the new value is larger than the old value, the percentage change must come out positive; if smaller, negative. A calculated result with the wrong sign relative to that simple check is an immediate signal that the old and new values were swapped somewhere in the formula.
For a plain percentage of a number, or what percent one number is of another (rather than a before-and-after comparison), see the related but distinct calculation in How to Calculate a Percentage.