Percent Change: Formula, Calculation, and Examples

What is percent change?

Percent change is a relative measure that expresses the absolute change in a value as a fraction of its starting value, scaled to a percentage.

The percent change formula compares the difference to the original amount.

The standard formula is:

$\text{Percent Change} = \dfrac{\text{Amount of Change}}{\text{Original Value}} \times 100$

This ratio shows the size of the change relative to the initial size of the object, population, or price. A change of $5$ units is highly significant if the starting amount was $10$, but almost unnoticeable if the starting amount was $10{,}000$.

Find the amount of change

The first step in calculating a percentage difference from original amounts is finding the exact size of the change. This is the absolute difference between the two values.

Subtract the original value and the new value. The order of subtraction does not matter as long as you take the positive difference.

If the original value is $40$ and the new value is $50$, the amount of change is $10$. If the original value is $60$ and the new value is $50$, the amount of change is also $10$.

Use the original value as the base

The most important rule of calculating relative change is that the original value is always the denominator.

The original value represents $100\%$ of the starting conditions. If you divide by the new value instead, the resulting percentage will describe the size of the change relative to the end state, which is mathematically incorrect for calculating growth or reduction over time.

Calculate percent increase and decrease

Once you divide the amount of change by the original value, you will have a decimal. Multiply this decimal by $100$ to convert it into a percentage.

When a value grows, it is a percent increase, and when a value shrinks, it is a percent decrease. It is essential to state the direction of the change in your final answer so the meaning of the percentage is clear.

For example, a change from $80$ to $100$ is a $25\%$ increase, while a change from $100$ to $80$ is a $20\%$ decrease. The amounts are the same, but the percentages differ because the starting values differ.

Interpret repeated changes

When multiple percentage changes happen in sequence, they are applied to the new accumulated value each time, not the original starting value.

Because the base value changes after the first step, you cannot simply add or subtract the percentages to find the total change.

If a price increases by $10\%$ and then decreases by $10\%$, the overall change is not zero. The decrease acts on the new, larger amount, pulling the final value lower than the original starting point.

Worked examples

Applying the method to percentage word problems helps to solidify the core steps of identifying the starting value and tracking the difference.

Example 1: Calculating a population increase

Question: A town's population grows from $12{,}000$ to $15{,}000$. What is the percent change?

Method:

1. Find the amount of change by subtracting the original value from the new value.

$15{,}000 - 12{,}000 = 3{,}000$

2. Divide the change by the original value ($12{,}000$).

$\dfrac{3{,}000}{12{,}000} = 0.25$

3. Multiply by $100$ to convert to a percentage.

$0.25 \times 100 = 25\%$

Answer: The percent change is a $25\%$ increase.

Check: Finding $25\%$ of $12{,}000$ yields $3{,}000$. Adding this to $12{,}000$ confirms the new population is $15{,}000$.

Example 2: Calculating a price decrease with decimals

Question: A jacket's price drops from $80$ dollars to $55$ dollars. What is the percent change?

Method:

1. Find the amount of change.

$80 - 55 = 25$

2. Divide the change by the original price ($80$).

$\dfrac{25}{80} = 0.3125$

3. Multiply by $100$ and state the direction.

$0.3125 \times 100 = 31.25\%$

Answer: The percent change is a $31.25\%$ decrease.

Check: Subtracting $31.25\%$ of $80$ from the original $80$ gives exactly $55$.

Example 3: Interpreting consecutive changes

Question: A stock's value rises by $20\%$ in year one, then falls by $20\%$ in year two. What is the overall percent change from the original value?

Method:

1. Assume an original value of $100$ dollars to make calculation simple.
   
2. Calculate the year one increase.

$20\%$ of $100$ dollars is $20$ dollars.

New value = $120$ dollars.

3. Calculate the year two decrease based on the new value.

$20\%$ of $120$ dollars is $24$ dollars.

Final value = $120 - 24 = 96$ dollars.

4. Find the overall percent change from the original $100$ dollars.

Change = $100 - 96 = 4$ dollars.

$\dfrac{4}{100} \times 100 = 4\%$

Answer: The overall percent change is a $4\%$ decrease.

Check: The repeated multipliers are $1.20 \times 0.80 = 0.96$. A multiplier of $0.96$ represents a $4\%$ decrease from the starting value of $1$.

Common mistakes

The most common error is dividing the amount of change by the new value instead of the original value. Always establish which value came first in time or which value is the established baseline before substituting numbers into the formula.

Another frequent mistake is ignoring the direction of the change. Writing "$15\%$" as an answer is incomplete if the context requires knowing whether the quantity went up or down. Always label your final answer as an increase or a decrease.

Related to this is percent error, which measures the inaccuracy of an estimate rather than a chronological change over time. Percent error uses the exact accepted value as the denominator, whereas percent change uses the original starting value.

Frequently asked questions

Can percent change be over $100\%$?

Yes. A percent increase is over $100\%$ if the new value is more than double the original value. For example, an increase from $50$ to $150$ is a change of $100$, which is a $200\%$ increase. A percent decrease, however, cannot exceed $100\%$ unless negative values are allowed in the specific context.

Is percent change the same as a multiplier?

They are closely related but not identical. A percent change tells you how much was added or subtracted. A multiplier tells you what to multiply the original by to find the new value directly. A $20\%$ increase corresponds to a multiplier of $1.20$. Finding the original value before a change occurred is a process known as reverse percentages.