Percent Change Formula

The Formula

\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 \frac{10}{50} = 20\% increase.

Quick Example

\text{Price: } \80 \to \60 \implies \text{Change} = \frac{60-80}{80} \times 100\% = -25\% \text{ (25\% decrease)}

Notation

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

What This Formula Means

The ratio of the change in a quantity to the original value, expressed as a percentage.

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

Formal View

\Delta\% = \frac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}} \times 100\% where x_{\text{old}} \neq 0

Worked Examples

Example 1

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

Solution

  1. 1
    Find the change: 60 - 45 = 15.
  2. 2
    Divide by the original value: \frac{15}{60} = 0.25.
  3. 3
    Convert to a percentage: 0.25 \times 100 = 25\%.

Answer

25\% \text{ decrease}
Percent change is calculated as \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{,}000 to 15{,}000. What is the percent increase?

Example 3

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

Common Mistakes

  • Dividing by the new value instead of the original value
  • Forgetting the sign: negative means decrease, positive means increase
  • Assuming equal percent increase and decrease cancel out

Why This Formula Matters

Used to compare growth rates, analyze trends, and understand financial changes like inflation or investment returns.

Frequently Asked Questions

What is the Percent Change formula?

The ratio of the change in a quantity to the original value, expressed as a percentage.

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 \frac{10}{50} = 20\% increase.

What do the symbols mean in the Percent Change formula?

\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?

Used to compare growth rates, analyze trends, and understand financial changes like inflation or investment returns.

What do students get wrong about Percent Change?

A 50% increase followed by a 50% decrease does NOT return to the original value.

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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