The Basic Formula for Percentage Change

Percentage change measures how much something has grown or shrunk as a portion of its starting value. The formula is straightforward: subtract the old value from the new value, divide by the old value, then multiply by 100.

Written as an equation, it looks like this:

(New Value − Old Value) ÷ Old Value × 100 = Percentage Change

The result tells you the direction and size of the change. A positive number means growth. A negative number means decline. The larger the absolute number, the bigger the shift.

Key Takeaways

  • Percentage change always divides the difference by the starting amount, not the ending amount, which is why the order matters.
  • A 50% increase followed by a 50% decrease does not return you to the original value because the second calculation uses a larger base.
  • Percentage change works the same way whether you are tracking prices, populations, test scores, or any other measurable quantity.
  • Negative percentage change means the value went down; positive means it went up.
  • The formula fails only when the old value is zero, because you cannot divide by zero.

Working Through a Real Example

Suppose a stock price was $40 last month and is $50 today. The new value is $50. The old value is $40. The difference is $10.

Divide $10 by $40 to get 0.25. Multiply by 100 to get 25. The stock rose 25%.

Now suppose the stock falls from $50 back to $40. The new value is $40. The old value is $50. The difference is still $10, but now you divide by $50, not $40. That gives you 0.20, or 20%. The stock fell 20%, not 25%, even though it moved the same dollar amount. This is why the starting value always matters.

Why the Order of Values Matters

Many people mix up which number goes where. The rule is straightforward: the old value (or starting value) always goes in the denominator — the bottom of the division. The new value minus the old value always goes in the numerator — the top.

If you reverse them, you get the wrong answer. A $10 change on a base of $40 is not the same as a $10 change on a base of $50, even though the dollar amount is identical. Percentage change measures the shift relative to where you started, so the starting point is what you divide by.

This is also why a 50% gain followed by a 50% loss does not bring you back to the original value. If you start with $100 and gain 50%, you have $150. If you then lose 50% of $150, you lose $75 and end up with $75. The second loss is calculated on the larger amount, so it erases more value than the first gain created.

Handling Negative Numbers and Decreases

When a value goes down, the percentage change is negative. If a price drops from $80 to $60, the difference is −$20. Divide by $80 to get −0.25, or −25%. The negative sign tells you the direction of the change.

The same formula works whether the old value, the new value, or both are negative. What matters is the direction of movement from one to the other. If you owe $100 in debt and now owe $75, the debt decreased by 25%. If you owe $100 and now owe $150, the debt increased by 50%.

One exception: if the old value is zero, the formula breaks down because you cannot divide by zero. In that case, percentage change is undefined. You can describe the change in absolute terms (you went from zero to ten), but not as a percentage.

Common Mistakes to Avoid

The most frequent error is dividing by the new value instead of the old value. This gives you a different number that does not represent the true percentage change. Always ask yourself: "What was the starting point?" That is your denominator.

Another mistake is forgetting to multiply by 100. If you stop after dividing, you get a decimal (0.25) instead of a percentage (25%). The multiplication by 100 converts the decimal into the percentage form people expect to see.

A third error is confusing percentage change with percentage points. If a test score goes from 70% to 80%, that is a 10 percentage point increase. But the percentage change is different: (80 − 70) ÷ 70 × 100 = 14.3%. Percentage points measure the absolute difference between two percentages. Percentage change measures how much the first percentage grew relative to itself.

Using Percentage Change in Daily Situations

Percentage change appears everywhere. Retail stores use it to show discounts: if a shirt was $40 and now costs $30, the discount is (30 − 40) ÷ 40 × 100 = −25%. Salary increases use it: if you earned $50,000 and now earn $55,000, your raise is (55,000 − 50,000) ÷ 50,000 × 100 = 10%. Population growth, inflation rates, and test score improvements all use the same calculation.

The formula is identical regardless of what you are measuring. The only thing that changes is what the numbers represent. Whether you are tracking dollars, people, or percentage points, the math stays the same.

Frequently Asked Questions

What if the old value is negative?

The formula still works. If a temperature dropped from −10 degrees to −5 degrees, the change is (−5 − (−10)) ÷ (−10) × 100 = 5 ÷ (−10) × 100 = −50%. The negative sign in the denominator flips the sign of the result, which correctly shows that the temperature rose (became less cold) even though both values are negative.

Can percentage change be more than 100%?

Yes. If something doubles, that is a 100% increase. If it triples, that is a 200% increase. If a company's revenue went from $1 million to $5 million, the change is (5 − 1) ÷ 1 × 100 = 400%. There is no upper limit to percentage change.

Is there a difference between percentage change and percent change?

No. The terms are interchangeable. Both refer to the same calculation and the same result. You may also hear it called "rate of change" or "relative change," though those terms sometimes have slightly different meanings in specialized contexts.

How do I calculate percentage change over multiple years?

Use the same formula, but make sure your old value is the earliest year and your new value is the latest year. If a population was 100,000 in 2010 and 150,000 in 2020, the percentage change is (150,000 − 100,000) ÷ 100,000 × 100 = 50%. This tells you the total change over the decade, not the change per year.