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If an amount grows by a fixed percent each period, \(P(t) = P_0(1 + r)^t\), where \(P_0\) is the starting amount, \(r\) is the rate as a decimal, and \(t\) is the number of periods. For continuous growth, use \(P(t) = P_0e^{rt}\). For example, $500 growing 8% a year for 10 years becomes \(500(1.08)^{10} \approx \$1{,}079.46\). Use a negative rate for decay.
Two Formulas, Two Situations
Growth once per period
\[P(t) = P_0(1 + r)^t\]
- Interest paid once a year
- A population counted every year
- A price that rises 3% each year
Continuous growth
\[P(t) = P_0\,e^{rt}\]
- Bacteria that divide all the time
- Radioactive decay
- Interest compounded continuously
For small rates the two give similar answers. At 8% for 10 years, the first gives $1,079.46 and the second gives $1,112.77, because continuous growth “earns interest on interest” at every instant.
Growth Example: Saving $500
You put $500 in an account that grows 8% per year.
- Growth factor: \(b = 1 + 0.08 = 1.08\)
- After 10 years: \(P(10) = 500(1.08)^{10} \approx 1{,}079.46\)
- Doubling time: \(\frac{\ln 2}{\ln 1.08} \approx 9.0\) years
Decay Example: Losing 12% a Year
A car worth $1,000 (or 1,000 of anything) loses 12% of its value each year. Now \(r = -0.12\), so the factor is \(b = 0.88\), which is less than 1.
\[P(5) = 1000(0.88)^5 \approx 527.73.\]
Its half-life, the time to lose half its value, is \(\frac{\ln 2}{-\ln 0.88} \approx 5.4\) years.
Doubling Time and the Rule of 72
A quick mental trick: divide 72 by the percent rate to estimate the doubling time.
| Rate per period | Rule of 72 estimate | Exact doubling time |
|---|---|---|
| 2% | 36 | 35.0 |
| 6% | 12 | 11.9 |
| 8% | 9 | 9.0 |
| 12% | 6 | 6.1 |
The exact formula is \(t_{\text{double}} = \frac{\ln 2}{\ln(1 + r)}\), or \(\frac{\ln 2}{r}\) for continuous growth.
Finding the Growth Rate From Two Values
Sometimes you know the start and the end, but not the rate. If a town grows from 200 to 300 people in 5 years:
\[200(1 + r)^5 = 300 \;\Rightarrow\; (1 + r)^5 = 1.5 \;\Rightarrow\; r = 1.5^{1/5} – 1 \approx 0.0845.\]
So the town grows about 8.45% per year. Plug that rate into the calculator to predict future years.
Why Calculus Students Care
Continuous growth comes from the differential equation \(\frac{dP}{dt} = rP\): the growth speed is proportional to the amount. Its solution is exactly \(P = P_0e^{rt}\). You can solve it yourself with the differential equation solver (type y' = 0.08y). The derivative of \(e^{rt}\) is \(re^{rt}\), and its antiderivative is \(\frac{1}{r}e^{rt} + C\).
Learn the Math Behind the Exponential Growth Calculator
Exponential growth and decay come from the differential equation y′ = ky, whose solution uses the antiderivative of e^(kt).
Frequently Asked Questions
How do I enter a decay rate?
Type the rate as a negative percent. For 12% decay per year, type -12. The calculator then shows the half-life instead of the doubling time.
What is the difference between exponential and linear growth?
Linear growth adds the same amount each period (like +$40 a year). Exponential growth multiplies by the same factor each period (like ×1.08 a year), so the amount added keeps getting bigger.
Can the time be a fraction?
Yes. The formulas work for any real \(t\), like 2.5 years.
Is compound interest the same as exponential growth?
Yes. Compound interest once per year is \(P_0(1 + r)^t\), and continuous compounding is \(P_0e^{rt}\). Monthly compounding uses \(P_0(1 + r/12)^{12t}\), which falls between the two.