# How to Calculate Growth Rate or Percent Change

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Percent change is a common method of describing differences due to change over time, such as population growth. There are three methods you can use to calculate percent change, depending on the situation: the straight-line approach, the midpoint formula or the continuous compounding formula.

## Straight-Line Percent Change

The straight-line approach is better for changes that don't need to be compared to other positive and negative results.

1. Write the straight-line percent change formula, so you have a foundation from which to add your data. In the formula, "V0" represents the initial value, while "V1" represents the value after a change. The triangle simply represents change.

2. Substitute your data for the variables. If you had a breeding population that grew from 100 to 150 animals, then your initial value would be 100 and your subsequent value after change would be 150.

3. Subtract the initial value from the subsequent value to calculate the absolute change. In the example, subtracting 100 from 150 gives you a population change of 50 animals.

4. Divide the absolute change by the initial value to calculate the rate of change. In the example, 50 divided by 100 calculates a 0.5 rate of change.

5. Multiply the rate of change by 100 to convert it to a percent change. In the example, 0.50 times 100 converts the rate of change to 50 percent. However, if the numbers were reversed such that the population decreased from 150 to 100, the percent change would be -33.3 percent. So a 50 percent increase, followed by a 33.3 percent decrease returns the population to the original size; this incongruity illustrates the "end-point problem" when using the straight-line method to compare values that may rise or fall.

## The Midpoint Method

If comparisons are required, the midpoint formula is often a better choice, because it gives uniform results regardless of the direction of change and avoids the "end-point problem" found with the straight-line method.

1. Write the midpoint percent change formula in which "V0" represents the initial value and "V1" is the later value. The triangle means "change." The only difference between this formula and the straight-line formula is that the denominator is the average of the starting and ending values rather than simply the starting value.

2. Insert the values in place of the variables. Using the straight-line method's population example, the initial and subsequent values are 100 and 150, respectively.

3. Subtract the initial value from the subsequent value to calculate the absolute change. In the example, subtracting 100 from 150 leaves a difference of 50.

4. Add the initial and subsequent values in the denominator and divide by 2 to calculate the average value. In the example, adding 150 plus 100 and dividing by 2 produces an average value of 125.

5. Divide the absolute change by the average value to compute the midpoint rate of change. In the example, dividing 50 by 125 produces a rate of change of 0.4.

6. Multiply the rate of change by 100 to convert it to a percentage. In the example, 0.4 times 100 calculates a midpoint percent change of 40 percent. Unlike the straight-line method, if you reversed the values such that the population decreased from 150 to 100, you get a percent change of -40 percent, which only differs by the sign.

## Average Annual Continuous Growth Rate

The continuous compounding formula is useful for average annual growth rates that steadily change. It is popular because it relates the final value to the initial value, rather than just providing the initial and final values separately – it gives the final value in context. For example, saying a population grew by 15 animals isn’t as meaningful as saying it showed a 650 percent increase from the initial breeding pair.

1. Write down the average annual continuous growth rate formula, where "N0" represents the initial population size (or other generic value), "Nt" represents the subsequent size, "t" represents the future time in years and "k" is the annual growth rate.

2. Substitute the actual values for the variables. Continuing with the example, if the population grew over the course of 3.62 years, substitute 3.62 for the future time and use the same 100 initial and 150 subsequent values.

3. Divide the future value by the initial value to calculate the overall growth factor in the numerator. In the example, 150 divided by 100 results in a 1.5 growth factor.

4. Take the natural log of the growth factor to calculate the overall growth rate. In the example, enter 1.5 into a scientific calculator and press "ln" to get 0.41.

5. Divide the result by the time in years to calculate the average annual growth rate. In the example, 0.41 divided by 3.62 produces an average annual growth rate of 0.11 in a continuously growing population.

6. Multiply the growth rate by 100 to convert to a percentage. In the example, multiplying 0.11 times 100 gives you an average annual growth rate of 11 percent.

#### Tips

• Some financial investments, such as savings accounts or bonds, compound periodically instead of continuously.