The Standard Percentage Change Formula
New Value = final / current / ending valueOld Value = original / initial / starting valuePositive result = increaseNegative result = decreaseThe denominator is always the Old (original) value — never the new value.
The percentage change formula compares a new value to an original baseline and expresses the difference as a percentage of that original. The rule that applies in every version of this formula: divide by the Old value, never by the New value. Swapping these is the most common calculation error, and it produces a different — incorrect — result.
When the result is positive, the value went up. When it is negative, the value went down. The sign tells you direction; the number tells you how much.
Variable names across different contexts
The formula variable "Old Value" goes by many names depending on where you encounter it. All of these mean exactly the same thing in the formula:
| Variable in Formula | Also called |
|---|---|
| Old Value | Original value, Initial value, Starting value, Base value, V₁, Previous value |
| New Value | Final value, Current value, Ending value, V₂, Revised value |
This matters for students moving between textbooks, and for analysts applying the same formula across different tools. Whether your spreadsheet labels a column "Previous Revenue" or "V₁," it goes in the denominator.
Percentage Increase and Decrease Formulas
The unified formula above handles both increases and decreases — a negative result simply signals a decrease. Some textbooks split this into two separate formulas so that a decrease always produces a positive number. Both approaches are correct; use whichever your context requires.
The unified formula is mathematically equivalent and handles both directions. The separate versions exist because some teachers and style guides prefer a percentage decrease expressed as a positive number. The magnitude is always the same — only the sign convention differs.
Worked example — percentage increase
A software subscription rises from $120/year to $156/year. What is the percentage change?
Identify the values: Old = $120, New = $156
Subtract: New − Old = 156 − 120 = 36
Divide by Old: 36 ÷ 120 = 0.30
Multiply by 100: 0.30 × 100 = 30%
✓ The subscription increased by 30%.
Worked example — percentage decrease
A flight ticket drops from $450 to $315. What is the percentage change?
Identify the values: Old = $450, New = $315
Subtract: New − Old = 315 − 450 = −135
Divide by Old: −135 ÷ 450 = −0.30
Multiply by 100: −0.30 × 100 = −30%
✓ The ticket price decreased by 30%. The negative sign confirms it is a decrease — this is correct, not an error.
For a detailed walkthrough of step-by-step calculations, real-world scenarios, and further practice, the percentage increase and decrease guide covers these in depth.
How to Use the Percentage Change Formula
Applying the formula follows four steps in the same order every time, regardless of whether you are working with prices, population figures, test scores, or financial results.
Identify the old and new values. The old value is your starting point. The new value is where you ended up — or the current figure you are comparing against. If the data gives you a "before" and "after," the before is always Old.
Subtract: New Value − Old Value. A positive number here means the value went up. A negative number means it went down. This step gives you the absolute change.
Divide the result by the old value. This converts the absolute change into a proportion of the original. If the old value is negative (a loss, a temperature below zero, a negative balance), use its absolute value: |Old Value|.
Multiply by 100. This scales the proportion to a percentage. Positive = percentage increase. Negative = percentage decrease.
Percentage Change Formula in Excel and Google Sheets
When your data is in a spreadsheet, the percentage change formula translates directly into a cell formula. The main thing to handle is the case where your original value might be negative — that is what the ABS() function takes care of.
Standard Excel formula
=(B2-A2)/ABS(A2)
Where A2 holds the Old Value (original) and B2 holds the New Value (current). Then format the cell as Percentage (Home → Number → % button in Excel; Format → Number → Percent in Google Sheets) to display 0.3 as 30%.
When the Old Value is negative, dividing without ABS() flips the sign of the result. Using ABS(A2) makes sure the denominator is always a positive magnitude, so the direction of change is determined solely by whether New is larger or smaller than Old.
Simpler version (when Old Value is always positive)
=(B2-A2)/A2
This version works whenever your original values are guaranteed to be positive — most revenue, pricing, and quantity data. If there is any chance of a zero or negative in the original column, use the ABS() version instead.
Worked example in Excel
| A — Old Value | B — New Value | C — Formula | Result (formatted as %) |
|---|---|---|---|
| 120 | 156 | =(B2-A2)/ABS(A2) | 30% |
| 450 | 315 | =(B3-A3)/ABS(A3) | −30% |
| −50 | −20 | =(B4-A4)/ABS(A4) | 60% |
The third row shows the ABS() function at work: going from a loss of −50 to a loss of −20 is a 60% improvement (the loss shrunk). Without ABS(), the same formula would return −60% — the wrong sign.
The formula syntax is identical in Google Sheets. After entering the formula, go to Format → Number → Percent to display the result as a percentage.
Percentage Change Formula for Negative Numbers
When the original value is a negative number — a reported loss, a temperature below zero, an account with a negative balance — the standard formula can produce a misleading result if you use the raw negative denominator. The fix is straightforward: always divide by the absolute value of the original.
The absolute-value formula
The vertical bars | | mean absolute value — always the positive magnitude of the original number.
Worked example — improving from a loss
A company's quarterly loss improved from −$80,000 to −$32,000. What was the percentage change?
Values: Old = −80,000; New = −32,000
Subtract: New − Old = −32,000 − (−80,000) = 48,000
Divide by |Old|: 48,000 ÷ 80,000 = 0.60
Multiply by 100: 0.60 × 100 = 60%
✓ The company's loss improved by 60% — the loss fell from $80k to $32k, which is a meaningful improvement correctly captured as a positive percentage change.
Edge case — when the old value is zero
When the original value is exactly zero, the percentage change formula cannot be applied — you would be dividing by zero. Report the absolute change instead (for example: "revenue increased from $0 to $50,000"). There is no meaningful percentage change from zero.
Reversing the Percentage Change Formula
Sometimes you know the final value and the percentage change but need to work backwards to find the original. These are the reverse formulas:
A product costs $207 after a 15% price increase. What was the original price?
New = $207, rate = 15%, so the multiplier is (1 + 15/100) = 1.15
Old = 207 ÷ 1.15 = $180
✓ The original price was $180. Check: (207 − 180) / 180 × 100 = 15%. ✓
A jacket is on sale for $51 — 15% off. What was the original price?
New = $51, rate = 15%, so the multiplier is (1 − 15/100) = 0.85
Old = 51 ÷ 0.85 = $60
✓ The original price was $60. Check: (51 − 60) / 60 × 100 = −15%. ✓
To reverse a 20% increase, don't subtract 20% from the new value. That gives 80% of the new value — not the original. A 20% increase followed by a 20% decrease does not return to the starting point. Always divide by (1 + rate/100) or (1 − rate/100).
Rate of Change Over Time (Multi-Period Formula)
The standard percentage change formula measures change between two points. When you want to describe how something grew or shrank across multiple periods — and find the consistent rate per period — you need the rate of change over time formula:
New = final valueOld = starting valuen = number of periodsWhen n is in years, this formula is also called CAGR — Compound Annual Growth Rate.
Worked example — population growth
A city had a population of 500,000 in 2016 and 680,000 in 2026. What was the annual growth rate?
Values: Old = 500,000; New = 680,000; n = 10 years
Ratio: 680,000 ÷ 500,000 = 1.36
Apply the exponent: 1.36^(1/10) = 1.36^0.1 ≈ 1.0311
Subtract 1 and multiply: (1.0311 − 1) × 100 = 3.11% per year
✓ The city's population grew at an average annual rate of 3.11% over ten years.
Business application — CAGR
When n represents years, this formula gives the Compound Annual Growth Rate (CAGR) — the standard measure used in financial reporting, investor presentations, and business cases. CAGR smooths out year-to-year fluctuations and gives a single, comparable growth rate.
Revenue grew from $2M to $3.5M over 5 years. What is the CAGR?
Ratio: 3.5 ÷ 2 = 1.75; n = 5
1.75^(1/5) = 1.75^0.2 ≈ 1.1184
(1.1184 − 1) × 100 = 11.84% per year
✓ CAGR = 11.84%/year. Total revenue grew 75% over 5 years, but the equivalent annual rate is 11.84%.
Percentage Change vs. Percentage Difference vs. Percentage Points
Three measures with similar names but different meanings — using the wrong one is a frequent source of confusion in reports and presentations. Take an interest rate that moves from 4% to 5%:
Directional comparison
Uses the original value as the base. Tells you the relative change from a starting point. Order matters: Old → New.
Symmetric comparison
Uses the average as the base. No defined "old" or "new." Appropriate when comparing two independent values.
Absolute arithmetic difference
Simply subtracts one percentage from another. No division. Reports the gap in the same units as the original percentages.
In the interest rate example: "The rate rose by 25%" (percentage change), "The two rates differ by 22.2%" (percentage difference), and "The rate rose by 1 percentage point" (percentage points) are all numerically correct — but they describe different things. Mixing them is a frequent error in journalism and business reporting. For more on how statistical context shapes interpretation, see the statistical interpretation guide.
Common Mistakes When Using the Percentage Change Formula
Wrong: (156 − 120) ÷ 156 × 100 = 23.1%. Right: (156 − 120) ÷ 120 × 100 = 30%. The denominator is always the original. The two formulas measure different things — the second one is not the percentage change.
Without |Old|: (−20 − (−50)) ÷ −50 × 100 = −60%. With |Old|: (−20 − (−50)) ÷ 50 × 100 = +60%. The first result gives the wrong sign — a genuine improvement in a loss registers as a decrease instead of an increase.
A 15% increase followed by a 10% decrease is not a net 5% change. The two percentages apply to different bases. The correct calculation is (1 + 0.15) × (1 − 0.10) − 1 = 1.035 − 1 = 3.5% net increase.
An interest rate rising from 4% to 5% is a 25% percentage change (relative) and a 1 percentage point increase (absolute). Reporting a 25% increase in an interest rate will alarm readers who expect a 1-unit shift in the rate itself.
Practice Problems
Work through each problem before revealing the solution.
A company's revenue was $4.2M in Q1 and $5.46M in Q2. What is the percentage change?
Revenue grew by 30% from Q1 to Q2.
A region's temperature dropped from −8°C to −14°C. What is the percentage change?
(−14 − (−8)) ÷ |−8| × 100 = −6 ÷ 8 × 100 = −75%.
The temperature fell by 75% relative to its original magnitude.
A product's price is $85 after a 25% reduction. What was the original price?
Old = 85 ÷ (1 − 25/100) = 85 ÷ 0.75 = $113.33.
Check: (85 − 113.33) ÷ 113.33 × 100 ≈ −25%. ✓
An investment grew from $10,000 to $14,641 over 4 years. What was the annual growth rate (CAGR)?
= (1.4641^0.25 − 1) × 100
= (1.10 − 1) × 100 = 10% per year.
Check: $10,000 × 1.10⁴ = $10,000 × 1.4641 = $14,641. ✓
Frequently Asked Questions
The percentage change formula is: Percentage change = ((New Value − Old Value) ÷ Old Value) × 100. A positive result means the value increased; a negative result means it decreased. Always use the original (old) value as the denominator — never the new value.
Use =(B2-A2)/ABS(A2) where A2 is the old value and B2 is the new value, then format the cell as a percentage. If the old value is always positive, the simpler =(B2-A2)/A2 works. The ABS() function handles cases where the original value might be negative.
Percent change is directional — it uses the original value as the denominator and measures change from a defined starting point. Percent difference is symmetric — it uses the average of both values, so order doesn't matter. Use percent change for before/after comparisons; use percent difference when comparing two independent values with no clear "original."
Yes. A negative percentage change means the value decreased. For example, if a price falls from $100 to $75, the percentage change is (75 − 100) ÷ 100 × 100 = −25%. The negative sign indicates a decrease — this is the correct result, not a calculation error.
Use the absolute value of the original in the denominator: ((New − Old) ÷ |Old|) × 100. For example, if a loss changes from −$50,000 to −$20,000, the change is (30,000 ÷ 50,000) × 100 = +60%. If the original is exactly zero, the formula is undefined — report the absolute change instead.
For a percentage increase: Old = New ÷ (1 + rate/100). For a percentage decrease: Old = New ÷ (1 − rate/100). Example: a price of $138 after a 15% increase means the original was 138 ÷ 1.15 = $120.
Use the rate of change over time formula: ((New ÷ Old)^(1/n) − 1) × 100, where n is the number of periods. When n is in years, this is also called CAGR (Compound Annual Growth Rate). Example: revenue growing from $2M to $3.5M over 5 years = ((3.5/2)^(1/5) − 1) × 100 ≈ 11.84% per year.
Formula reference checked against standard statistics and mathematics usage. For the broader mathematical treatment of relative change, see the Wikipedia article on relative change. | Published: November 13, 2026 by Minsa A