Why does the formula use e instead of just a regular number?

The continuous compound interest formula comes from a limit: if you compound more and more often (every second, every millisecond, infinitely often), the amount approaches Pe^{rt}. The number e arises naturally from this mathematical limit, and it is the unique base that makes continuous compounding work.

How is continuous compound interest different from regular compound interest?

Regular compound interest—say, compounded monthly—adds interest a fixed number of times per year. Continuous compounding means interest is added infinitely often, at every instant. Because interest is added so frequently, you earn a tiny bit more with continuous compounding than with monthly or yearly compounding at the same rate.

What does it mean if r is negative?

A negative rate models decay, not growth: the value decreases over time. This can represent money losing value due to inflation, radioactive material decaying, or a game character losing power. The same formula applies; just use r < 0, and A will be less than P.

Do I always need a calculator to solve these problems?

If the problem asks for a numerical answer, yes—you'll need to evaluate e^{\text{(some number)}} or \ln(\text{(some number)}), which requires a calculator or computer. However, if the problem says 'leave your answer in terms of e', you can write an exact answer like 1000e^{0.1} without computing the decimal.

Continuous Compound Interest

Q: Do I always need a calculator to solve these problems?

If the problem asks for a numerical answer, yes—you'll need to evaluate e^{\text{(some number)}} or \ln(\text{(some number)}), which requires a calculator or computer. However, if the problem says 'leave your answer in terms of e', you can write an exact answer like 1000e^{0.1} without computing the decimal.

Calculate how much money grows continuously over time, or solve how long growth takes at a given rate of compound interest.

A = Pe^{rt}

Solve a problem with Continuous Compound Interest

Type the problem. The solver will use Continuous Compound Interest where Continuous Compound Interest is the right tool, and tell you when it is not.

What each symbol means

What Continuous Compound Interest takes

Symbol	Meaning
$A$	The final amount after time $t$ has passed; this is what you solve for when you know the starting amount and want to know the future value.
$P$	The principal or initial amount invested; if confused with $A$ , you will calculate the growth in the wrong direction.
$r$	The annual interest rate as a decimal, not a percentage—so $5\% = 0.05$ , not $5$ ; forgetting this conversion makes answers 100 times too large.
$t$	Time in years; if your problem gives time in months or days, you must convert to years first, or the exponential will be completely wrong.

When to use it

Use this formula when interest is compounded continuously, which financial institutions sometimes offer for savings or investment accounts.

Level

Usually taught in: Algebra II

Worked examples

1. Find the amount after 2 years with simple numbers

Problem

An investment of $1000 is placed in an account that earns 5% annual interest compounded continuously. How much money will be in the account after 2 years?

$A = Pe^{rt}$
Write down the continuous compound interest formula.
$A = 1000 \cdot e^{0.05 \cdot 2}$
Substitute $P = 1000$ , $r = 0.05$ (converting 5% to decimal), and $t = 2$ .
$A = 1000 \cdot e^{0.1}$
Multiply the exponent: $0.05 \times 2 = 0.1$ .
$A = 1000 \cdot 1.10517$
Use a calculator to evaluate $e^{0.1} \approx 1.10517$ .
$A = 1105.17$
Multiply to find the final amount in dollars.

Answer: $A = 1105.17 \text{ dollars}$

After 2 years, the original $1000 grows to $1105.17. Notice this is slightly more than what you would get with annual compounding, because continuous compounding adds interest more frequently.

2. Solve for an unknown interest rate

Problem

An investment of $2000 grows to $3000 over 4 years with continuous compounding. What annual interest rate was being earned?

$A = Pe^{rt}$
Start with the continuous compound interest formula.
$3000 = 2000 \cdot e^{4r}$
Substitute the known values: $A = 3000$ , $P = 2000$ , and $t = 4$ .
$\frac{3000}{2000} = e^{4r}$
Divide both sides by 2000.
$1.5 = e^{4r}$
Simplify the fraction.
$\ln(1.5) = 4r$
Take the natural logarithm of both sides to undo the exponential.
$0.40547 = 4r$
Evaluate $\ln(1.5) \approx 0.40547$ with a calculator.
$r = \frac{0.40547}{4}$
Divide both sides by 4.
$r \approx 0.10137$
Calculate the rate as a decimal.

Answer: $r \approx 0.10137 \text{ or about } 10.14\%$

Working backward from the final amount requires logarithms because the unknown is in the exponent. The account was earning approximately 10.14% annual interest. This example shows why natural logarithm ( $\ln$ ) is essential when solving for $r$ or $t$ .

3. Word problem: game character's power growth

Problem

In a fantasy video game, a character's magical power grows continuously while channeling a spell. The growth rate is 12% per minute. If the character starts with 500 magical power points, how many power points will they have after 5 minutes of channeling?

$A = Pe^{rt}$
Use the continuous compound interest formula; here the 'interest' is magical power growth.
$A = 500 \cdot e^{0.12 \cdot 5}$
Substitute $P = 500$ (starting power), $r = 0.12$ (converting 12% to decimal), and $t = 5$ (minutes).
$A = 500 \cdot e^{0.6}$
Multiply the exponent: $0.12 \times 5 = 0.6$ .
$A = 500 \cdot 1.8221$
Use a calculator to evaluate $e^{0.6} \approx 1.8221$ .
$A = 911.06$
Multiply to get the final power level.

Answer: $A \approx 911.06 \text{ power points}$

The character's power nearly doubles in 5 minutes because continuous growth at 12% per minute is extremely fast. This problem shows that the continuous compound interest formula applies anywhere exponential growth happens, not just in banking.

Common mistakes

Where Continuous Compound Interest usually goes wrong

The mistake	Why it is wrong	The fix
Using the full percentage value instead of the decimal: $A = 1000 \cdot e^{5 \cdot 2}$ for a 5% rate.	Percentages must be converted to decimals: $5\% = 0.05$ , not $5$ ; using 5 instead makes the exponent 100 times larger, causing a drastically inflated answer.	Always convert percentages to decimals before substituting: $5\% = 0.05$ , $12\% = 0.12$ , and so on.
Treating $e^{rt}$ as $e^r \cdot t$ and computing $(2.718...)^{0.05} \times 2$ instead of $(2.718...)^{0.1}$ .	The exponent $rt$ is a single number; you must multiply $r$ and $t$ first, then raise $e$ to that power. Multiplying $e^r$ by $t$ gives a linear answer instead of exponential growth.	Always compute the exponent $rt$ as one step, then raise $e$ to that result.
Using $A = P(1 + r/n)^{nt}$ when the problem says 'compounded continuously'.	Continuous compounding is mathematically different from discrete compounding; it uses $e$ and applies when interest is added infinitely often. Using the wrong formula gives a slightly smaller (and incorrect) answer.	Check the problem statement: if it says 'compounded continuously', use $A = Pe^{rt}$ . If it says 'compounded quarterly' or 'compounded monthly', use $A = P(1 + r/n)^{nt}$ instead.

Tips and when to use something else

Convert interest rates from percentages to decimals before plugging into the formula— $5\% = 0.05$ , not $5$ .
When solving for time or rate, you will need the natural logarithm ( $\ln$ ) to undo the exponential. Without it, you'll be stuck with $e$ in your equation.
Use this formula only when the problem explicitly says 'compounded continuously'. For discrete compounding intervals (yearly, quarterly, monthly), use $A = P(1 + r/n)^{nt}$ instead.
Most scientific calculators and graphing calculators have an $e^x$ button. If you need to evaluate $e^{0.1}$ , press the $e^x$ button, enter 0.1, and hit equals.

Related formulas

Euler's NumberEuler's Number e is a fundamental constant approximately equal to 2.71828 that models continuous exponential growth and decay.Natural LogarithmNatural logarithm finds the power to which e must be raised to get a number, and it's essential for solving equations involving exponential functions.Exponential GrowthUse Exponential Growth when a quantity grows by multiplying by a constant factor each time period, like bacteria populations or money earning compound interest.Compound InterestCompound Interest calculates how money grows when interest earns interest, used for investments, savings, and loans over time.

Frequently asked questions

Why does the formula use $e$ instead of just a regular number?: The continuous compound interest formula comes from a limit: if you compound more and more often (every second, every millisecond, infinitely often), the amount approaches $Pe^{rt}$ . The number $e$ arises naturally from this mathematical limit, and it is the unique base that makes continuous compounding work.
How is continuous compound interest different from regular compound interest?: Regular compound interest—say, compounded monthly—adds interest a fixed number of times per year. Continuous compounding means interest is added infinitely often, at every instant. Because interest is added so frequently, you earn a tiny bit more with continuous compounding than with monthly or yearly compounding at the same rate.
What does it mean if $r$ is negative?: A negative rate models decay, not growth: the value decreases over time. This can represent money losing value due to inflation, radioactive material decaying, or a game character losing power. The same formula applies; just use $r < 0$ , and $A$ will be less than $P$ .
Do I always need a calculator to solve these problems?: If the problem asks for a numerical answer, yes—you'll need to evaluate $e^{\text{(some number)}}$ or $\ln(\text{(some number)})$ , which requires a calculator or computer. However, if the problem says 'leave your answer in terms of $e$ ', you can write an exact answer like $1000e^{0.1}$ without computing the decimal.

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Reviewed 2026-09-18