Percentage change shows what share of the original value a quantity has grown or shrunk by. It's calculated with one formula: (new − old) ÷ old × 100%. This page breaks down every part of the formula, walks through examples with prices, salaries, and discounts, flags the classic traps, and ends with an interactive trainer where you can practice right away.
What is percentage change
Percentage change is a way to express how much a quantity has changed, as a share of its original value. It's how we compare prices, salaries, follower counts, test scores — any pair of "before" and "after" numbers.
Why percentages instead of dollars or units? An absolute change alone says nothing about scale. A $10 price hike is huge for a $50 item (a fifth of the price!) and barely noticeable for a $1,000 item (just one percent). Percentages put changes on a common scale so you can compare fairly: +20% is +20% whether it's a candy bar or an apartment.
The percentage change formula
The calculation always happens in two steps:
- Find the absolute change — subtract the old value from the new one: \(\Delta = \text{new} - \text{old}\).
- Divide the change by the old value and multiply by 100% — this gives you the percentage change.
The most important part of the formula is the base: you always divide by the old (original) value, because you're comparing against what used to be. Dividing by the new value instead is the single most common mistake in this topic.
Increase or decrease: what the sign of the result tells you
The formula itself tells you the direction of change. If the new value is bigger than the old one, the difference is positive — the quantity increased. If it's smaller, the difference is negative — the quantity decreased. When answering "by how much did it decrease?", the minus sign is usually dropped: you say "decreased by 5%", not "changed by −5%".
Don't confuse "by how much percent" with "how many times" — these are different calculations. "By how much percent" uses the percentage change formula, while "how many times" is simply the new value divided by the old one.
How to increase or decrease a price by 10 percent: the reverse problem
Often the problem is reversed: you know the percentage and need to find the new (or old) value. This is where a change factor comes in handy:
- to increase by \(p\%\) — multiply by \(1 + \frac{p}{100}\). Raising a $1,200 price by 25%: \(1200 \times 1.25 = 1500\);
- to decrease by \(p\%\) — multiply by \(1 - \frac{p}{100}\). A 25% discount on $1,200: \(1200 \times 0.75 = 900\);
- to recover the old value — divide the new value by the same factor: if a 25% price hike brought an item to $1,500, it used to cost \(1500 \div 1.25 = 1200\).
The factor saves you extra steps: no need to work out "how much that is in dollars" and add it separately — one multiplication gives you the answer directly.
Percent vs. percentage points: what's the difference
When a quantity that is itself expressed in percent changes (an interest rate, a turnout rate, a market share), there are two different ways to describe the change:
- percentage points (pp) — a simple difference: a rate was 10%, now it's 12% → it rose by 2 percentage points;
- percent — a relative change using the usual formula: \((12 - 10) \div 10 \times 100\% = 20\%\) → the rate rose by 20 percent.
Both statements are correct and describe the same event, but the numbers differ — which is why news and ads sometimes pick whichever makes a change look bigger or smaller. Always check whether the source means points or percent.
The trap: +50% then −50% is not zero
Percentages of different bases can't be added or subtracted like ordinary numbers. If a quantity grows by \(p\%\) and then drops by the same \(p\%\), it will not return to its starting point: the decrease is calculated from the new, larger base. The combined factor \(\left(1 + \frac{p}{100}\right)\left(1 - \frac{p}{100}\right) = 1 - \left(\frac{p}{100}\right)^2\) is always less than one, so the result always ends up below the start.
Worked examples: calculating percentage change step by step
Example 1. Price went up: from $500 to $600
Problem. A book cost $500 and now costs $600. By what percent did the price increase?
Step 1. Find the change in dollars: \(600 - 500 = 100\).
Step 2. Divide by the old value — the base: \(100 \div 500 = 0.2\).
Step 3. Convert the fraction to a percentage: \(0.2 \times 100\% = 20\%\).
Answer: the price increased by 20%.
Example 2. Bill went down: from $440 to $418
Problem. A monthly phone bill was $440 and became $418. By what percent did it decrease?
Step 1. Change: \(418 - 440 = -22\) — the bill dropped by $22.
Step 2. Divide by the old value: \(-22 \div 440 = -0.05\).
Step 3. In percent: \(-0.05 \times 100\% = -5\%\).
Answer: the bill decreased by 5%. The minus sign signals a decrease; when answering "by how much did it drop", it's usually dropped.
Example 3. How to raise a price by 10 percent
Problem. An item costs $800. The seller raises the price by 10%. What is the new price?
Solving with the factor. Raising by 10% means multiplying by \(1 + \frac{10}{100} = 1.1\):
\(800 \times 1.1 = 880\).
Checking with the percentage change formula: \((880 - 800) \div 800 \times 100\% = 10\%\) — it checks out.
And lowering the price by 10% means multiplying by \(1 - 0.1 = 0.9\): \(800 \times 0.9 = 720\).
Example 4. The reverse problem: what was the price before the discount
Problem. After a 20% discount, a jacket costs $1,440. What did it cost before the discount?
Step 1. A 20% discount means \(100\% - 20\% = 80\%\) of the old price is left — a factor of \(0.8\).
Step 2. Divide the new price by the factor: \(1440 \div 0.8 = 1800\).
Check: \(1800 \times 0.8 = 1440\) — correct.
Answer: before the discount, the jacket cost \(1,800. A common mistake is adding 20% to the new price (\)1440 \times 1.2 = 1728$): that's wrong, because the discount was calculated from the old price, not the new one.
Common mistakes when calculating percentage change
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Dividing the change by the new value instead of the old one: (600 − 500) ÷ 600 = 16.7%.
The base is always the old value, because you're comparing against what it used to be: (600 − 500) ÷ 500 × 100% = 20%. Before calculating, clearly identify which number is the original one.
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Adding or subtracting percentages of different bases: "+50% then −50% — back to the start", "a 20% discount plus another 30% — 50% total".
Each next percentage is calculated from the new base. Multiply the factors instead: 1.5 × 0.5 = 0.75 (a net −25%); 0.8 × 0.7 = 0.56 (a combined 44% discount, not 50%).
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Treating "decrease by 20%" as "multiply by 0.2".
Multiplying by 0.2 leaves 20% of the value (an 80% decrease!). Decreasing BY 20% means multiplying by 1 − 0.2 = 0.8.
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Confusing percent with percentage points: a rate rising from 10% to 12% is called "a 2% increase".
By 2 percentage points is the simple difference. In percent, the rate rose by (12 − 10) ÷ 10 × 100% = 20%. Use the word "points" for the difference between two percentages.
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Confusing "how many times" with "by how much percent": tripling is called "a 300% increase".
Tripling is +200%: out of the new value's 300%, 100% is the original amount, so only 200% was added. The formula won't lie: (300 − 100) ÷ 100 × 100% = 200%.
Questions and answers
How do I calculate how many percent one number is greater than another?
Same formula: divide the difference by the number you're comparing TO, then multiply by 100%. "By what percent is 80 greater than 64?" — (80 − 64) ÷ 64 × 100% = 25%. Note that "by what percent is 64 less than 80" gives a different answer — (80 − 64) ÷ 80 × 100% = 20% — because the base is different.
How do I calculate percentage change with a calculator or in Excel?
On a calculator: divide the new value by the old one, subtract 1, and multiply by 100. For example, 600 ÷ 500 = 1.2 → 1.2 − 1 = 0.2 → 20%. In Excel or Google Sheets: if the old value is in cell A1 and the new one in B1, use the formula =(B1-A1)/A1 and format the cell as a percentage.
What if the old value is zero?
Percentage change is undefined — you can't divide by zero. If a channel had 0 subscribers and now has 50, it's not correct to talk about "growth by some percent" — that's growth from zero. In such cases, state the absolute change instead (+50 subscribers).
How is percentage change different from percentage difference?
Percentage change is directional: there's an "old" and a "new" value, and you divide by the old one. Percentage difference compares two values on equal footing, with no order: the absolute difference is divided by their average. In school problems and everyday life (prices, salaries, discounts), it's almost always percentage change you need.
If a quantity increased by 100%, how many times did it grow?
2 times. A 100% increase means an equal amount was added to the original quantity. Likewise, a 200% increase means 3 times, and a 50% increase means 1.5 times. In reverse: "grew 4 times" means "by 300%".
How do I calculate a price change in percent over several months?
There are two options. The overall change over the period is the usual formula between the first and last value. But if you know the percent for each month separately, you can't add them — you need to multiply the factors instead: a 10% rise followed by a 20% rise gives 1.1 × 1.2 = 1.32, i.e. +32%, not +30%.
