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 vs. 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.