Calculation
Mathematical Formula For Calculating Percentage Rate Growth
$$\text{Growth Rate} = \left( \left( \frac{\text{Final Value}}{\text{Initial Value}} \right)^{\frac{1}{\text{Time Period}}} – 1 \right) \times 100$$
Example Broken Down With Steps
$$
\text{Given: Initial Value = 100, Final Value = 200, Years = 3}
$$
$$
\text{Step 1: Divide final value by initial value: } \frac{200}{100} = 2
$$
$$
\text{Step 2: Take the cube root: } \sqrt[3]{2} = 1.2599
$$
$$
\text{Step 3: Subtract 1: } 1.2599 – 1 = 0.2599
$$
$$
\text{Step 4: Multiply by 100: } 0.2599 \times 100 = 25.99
$$
$$
\text{Result: Annual Growth Rate is 25.99%.}
$$
Further Explained
Example: A value grew from 100 to 200 over 3 years.
Steps:
- Divide the final value by the initial value:
$$ \frac{200}{100} = 2 $$ - Take the cube root (since it spans 3 years):
$$ \sqrt[3]{2} = 1.2599 $$ - Subtract 1:
$$ 1.2599 – 1 = 0.2599 $$ - Multiply by 100:
$$ 0.2599 \times 100 = 25.99 $$ - Conclusion: The annual growth rate is 25.99%.
The Rule of 72: Estimate Doubling Time Instantly
The Rule of 72 is a shortcut for estimating how long it takes for a value to double at a constant percentage growth rate. Divide 72 by the growth rate to get the approximate number of periods.
Doubling time ≈ 72 / Growth rate (%)
Example Broken Down With Steps
An investment grows at 8% per year. How long until it doubles?
Step 1: Divide 72 by the rate: 72 / 8 = 9
Result: The investment will approximately double in 9 years.
Reference Table
Annual growth rateApproximate years to double2%36 years3%24 years4%18 years6%12 years8%9 years10%7.2 years12%6 years
The Rule Works in Reverse Too
If you know how long something took to double, you can estimate the growth rate:
Growth rate ≈ 72 / Doubling time
If a city’s population doubled in 24 years:
72 / 24 = 3% annual growth rate
Why 72?
The number 72 is used because it is a close approximation that is easy to divide mentally. It works well for rates between 2% and 15%. For very high rates, the estimate becomes less accurate.
The exact formula for doubling time uses logarithms: t = ln(2) / ln(1 + r), where r is the rate as a decimal. The Rule of 72 is a practical approximation that requires no logarithms.
Use the Percentage Rate Growth Calculator above to compute exact growth rates for any values.