Percentage Change Calculator

What's the key thing to understand about The Formula?

Percentage change = ((New Value minus Original Value) divided by Original Value) times 100. A positive result indicates an increase; negative indicates a decrease. Example: a price rises from £80 to £96: change = ((96-80)/80) × 100 = 20% increase. A salary falls from £42,000 to £38,000: change = ((38,000-42,000)/42,000) × 100 = -9.5% decrease. The formula always uses the original value as the denominator — this is why a 50% fall followed by a 50% rise does not return to the original: a 50% fall from 100 gives 50, and a 50% rise from 50 only gives 75, not back to 100.

What do I need to know about Common Applications?

Price inflation, salary increases, investment returns, population growth, test score changes, statistical comparisons. Any time you want to know 'how much has this changed relatively?'

What's the difference between Percentage Points and Percentage Change?

If a tax rate rises from 20% to 25%, that's a 5 percentage point increase but a 25% change. These are different and frequently confused — always specify which you mean in formal contexts.

What do I need to know about Percentage Change in Context?

Percentage change requires careful interpretation. A 50% fall followed by a 50% rise does not return to the original value: 100 falls 50% to 50, then rises 50% to 75 — a 25% net loss. The arithmetic mean of percentage changes is misleading for cumulative performance; the geometric mean (CAGR) is correct for multi-period return calculation. Share prices: a stock that drops 50% requires a 100% gain to return to its starting price — asymmetric recovery that catches many investors by surprise when estimating how much a loss needs to be recovered before breaking even.

What should I know about Why Percentage Changes Don't Reverse Themselves?

A common and costly misconception is that an increase and an equal-sized decrease cancel out. They don't, because each percentage is calculated from a different base value. If a stock price rises 50% from $100, it reaches $150 (× 1.50). If it then falls 50%, it doesn't return to $100 — it falls to $75 (× 0.50 of $150), a net loss of 25% from the starting point. The same asymmetry works in reverse: if a $100 item is marked down 50% to $50, it then needs a 100% increase — not 50% — to get back to $100, because the 50% increase would only be calculated on the smaller $50 base. This is why retailers' 'price slashed then restored' claims, and headlines about markets 'recovering' after a crash, deserve a second look: recovering to the original value always requires a larger percentage gain than the percentage that was lost. The rule of thumb: a decrease of p% requires an increase of p/(1−p) × 100% to fully reverse it.

What do I need to know about Percentage Change with Negative Values?

The standard formula, (new − old) / |old| × 100%, needs care when the starting value is negative, such as a business tracking profit that moves from a loss to a gain. Using the absolute value of the old figure in the denominator keeps the sign of the result meaningful: if profit moves from −$10,000 (a loss) to $5,000 (a profit), the change is (5,000 − (−10,000)) / |−10,000| × 100% = 15,000/10,000 × 100% = 150%, correctly showing a large positive improvement. Without taking the absolute value, dividing by a negative number would flip the sign and make an improvement look like a decline. A special case arises when the starting value is exactly zero — percentage change is mathematically undefined here, since you cannot express a change as a percentage of nothing (a company with $0 revenue that earns $10,000 hasn't grown by 'a percentage', it has gone from no revenue to some revenue). In these situations, it's clearer to report the absolute change in dollars rather than force a percentage figure.

What do I need to know about Average Percentage Change Over Multiple Periods (CAGR)?

When a value changes by a different percentage each period — like an investment that grows 10% one year and falls 5% the next — the "average" percentage change isn't the simple arithmetic mean of those percentages; it's the compound annual growth rate (CAGR). CAGR is calculated as [(ending value / starting value)^(1/n)] - 1, where n is the number of periods, and it gives the single steady annual rate that would take the starting value to the ending value over that span. For an investment going from $1,000 to $1,331 over 3 years, CAGR = (1331/1000)^(1/3) - 1 = 1.1 - 1 = 10%, even if the actual year-by-year changes varied. This is why investment returns are reported as CAGR rather than an average of yearly percentage changes, which would overstate performance whenever returns are volatile.

Worked Example: Why Successive Percentage Changes Don't Simply Add Up

A common mistake is assuming a 20% discount followed by a further 15% discount equals a flat 35% off. It doesn't, because the second discount applies to the already-reduced price, not the original. Starting from £100: after 20% off, price = 100 × (1 − 0.20) = £80. After a further 15% off that new price: 80 × (1 − 0.15) = £68. The total reduction from £100 to £68 is £32, which as a percentage of the original price is 32% — one percentage point less than the naive 35% total. The same logic applies to successive increases: a 10% pay rise followed by a 10% pay cut does not return you to your starting salary. £100 → ×1.10 = £110 → ×0.90 = £99, a net 1% loss, because the 10% cut is calculated on the larger, already-increased figure.

Successive Percentage Changes: Reference Table

The pattern to notice: the 'actual' combined effect is always found by multiplying the two decimal factors together (e.g. 1.10 × 0.90 = 0.99 for the +10%/-10% row), never by adding or subtracting the two percentages directly.

Worked Example: Reversing a Percentage Change to Find the Original Value

A common real-world question runs the formula backwards: if a price after a 15% increase is £92, what was the original price? The percentage-change formula is new = original × (1 + rate), so original = new ÷ (1 + rate) = 92 ÷ 1.15 = £80. Checking: £80 × 1.15 = £92, confirming the answer. The same reverse logic applies to a decrease: if a discounted price of £68 reflects a 20% reduction, the original price is 68 ÷ (1 − 0.20) = 68 ÷ 0.80 = £85. A frequent mistake here is subtracting (or adding) the percentage from the final figure directly — taking 15% of £92 (£13.80) and subtracting it from £92 to get £78.20, which is wrong, because the 15% increase was always a percentage of the smaller original price, not of the larger final one.

Index Numbers: Percentage Change Anchored to a Base Year

Economists and statisticians often track change over many years using an index number rather than restating the percentage change every single year against a moving previous value. An index sets a chosen base year to exactly 100, and every other year is expressed as a percentage of that base: if prices were 100 in the base year and have risen 30% by year five, the index reads 130. The advantage is that comparing any two years becomes a single subtraction: a move from an index of 130 to an index of 143 is simply a 13-percentage-point rise on the index, which also happens to equal (143−130)/130 ≈ 10% growth over that particular stretch, calculated the normal percentage-change way, with the base year acting as a fixed reference point rather than everything being chained to the immediately preceding year. UK and US consumer price indices (CPI) are built exactly this way, which is why inflation is often reported both as 'the index level' and as 'percentage change year-on-year' side by side.

Worked Example: Comparing Two Different Growth Rates Fairly

Comparing raw differences instead of percentage change can be misleading when the starting points differ. Shop A's weekly sales rose from 200 units to 240 units (a rise of 40 units); Shop B's rose from 1,000 units to 1,050 units (a rise of 50 units). By raw numbers, Shop B grew more. But as percentage change, Shop A: (240−200)/200 × 100 = 20%; Shop B: (1,050−1,000)/1,000 × 100 = 5%. Shop A actually grew four times faster relative to its own starting size, even though its absolute increase was smaller — exactly the kind of comparison percentage change exists to make fair, since it accounts for scale rather than comparing raw totals from very different baselines.