Percent Change Formula

Percent change measures how much a quantity has increased or decreased relative to its original value, calculated as new - original/original x 100\%.

The Formula

Percent Change=New−OriginalOriginal×100%\text{Percent Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%

When to use: If a price goes from $50 to $60, the change is $10. Compared to the original $50, that's 1050=20%\frac{10}{50} = 20\% increase.

Quick Example

Price: $80→$60  ⟹  Change=60−8080×100%=−25% (25% decrease)\text{Price: } \$80 \to \$60 \implies \text{Change} = \frac{60-80}{80} \times 100\% = -25\% \text{ (25\% decrease)}

Notation

Δ%=New−OldOld×100%\Delta\% = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%; positive means increase, negative means decrease

What This Formula Means

Percent change measures how much a quantity has increased or decreased relative to its original value, calculated as new−originaloriginal×100%\frac{\text{new} - \text{original}}{\text{original}} \times 100\%.

If a price goes from $50 to $60, the change is $10. Compared to the original $50, that's 1050=20%\frac{10}{50} = 20\% increase.

Formal View

Δ%=xnew−xoldxold×100%\Delta\% = \frac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}} \times 100\% where xold≠0x_{\text{old}} \neq 0

Worked Examples

Example 1

easy
A jacket originally costs $60\$60 and is on sale for $45\$45. What is the percent decrease?

Answer

25% decrease25\% \text{ decrease}

First step

1
Find the change: 60−45=1560 - 45 = 15.

Full solution

  1. 2
    Divide by the original value: 1560=0.25\frac{15}{60} = 0.25.
  2. 3
    Convert to a percentage: 0.25×100=25%0.25 \times 100 = 25\%.
Percent change is calculated as changeoriginal×100%\frac{\text{change}}{\text{original}} \times 100\%. A decrease means the new value is less than the original.

Example 2

medium
A town's population grew from 12,00012{,}000 to 15,00015{,}000. What is the percent increase?

Example 3

hard
A stock price rises by 20%20\% one year and falls by 20%20\% the next year. If it started at $100\$100, what is the final price and the overall percent change?

Common Mistakes

  • Dividing by the new value instead of the original - the denominator is always the starting amount.
  • Forgetting the sign - new less than original is a decrease (negative), not a positive change.
  • Computing percent change of the difference alone - use new−oldold\frac{\text{new}-\text{old}}{\text{old}}, not just the difference times 100.

Why This Formula Matters

Percent change is how prices, populations, and scores are compared fairly regardless of size — a \$10 rise means more on a \$50 item than a \$500 one. The classic trap is dividing by the new value or the change itself instead of the original. Recognizing it by "Is a change being compared to the original starting value?" — rather than by familiar numbers — is what lets a student tell it apart from percent of a number and percentages and ratio in a mixed problem set.

Frequently Asked Questions

What is the Percent Change formula?

Percent change measures how much a quantity has increased or decreased relative to its original value, calculated as new−originaloriginal×100%\frac{\text{new} - \text{original}}{\text{original}} \times 100\%.

How do you use the Percent Change formula?

If a price goes from $50 to $60, the change is $10. Compared to the original $50, that's 1050=20%\frac{10}{50} = 20\% increase.

What do the symbols mean in the Percent Change formula?

Δ%=New−OldOld×100%\Delta\% = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%; positive means increase, negative means decrease

Why is the Percent Change formula important in Math?

Percent change is how prices, populations, and scores are compared fairly regardless of size — a \$10 rise means more on a \$50 item than a \$500 one. The classic trap is dividing by the new value or the change itself instead of the original. Recognizing it by "Is a change being compared to the original starting value?" — rather than by familiar numbers — is what lets a student tell it apart from percent of a number and percentages and ratio in a mixed problem set.

What do students get wrong about Percent Change?

The procedure for percent change is the easy part; the trap is dividing by the new value instead of the original. Asking "Is a change being compared to the original starting value?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Percent Change formula?

Before studying the Percent Change formula, you should understand: percentages, subtraction, division.

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