Use the formula AequalsUpper P left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript nt
 to solve the compound interest problem.
Find how long it takes for ​$2000 to double if it is invested at 2​% interest compounded monthly.

To solve for the time it takes for an investment to double using the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- $A$ is the amount of money accumulated after n years, including interest.
- $P$ is the principal amount (the initial amount of money).
- $r$ is the annual interest rate (decimal).
- $n$ is the number of times that interest is compounded per year.
- $t$ is the time in years.

### Given:
- $P = 2000$
- $A = 4000$ (since we want to double the investment)
- $r = 0.02$ (2% as a decimal)
- $n = 12$ (monthly compounding)

### Steps:

1. Substitute the known values into the formula:

\[
4000 = 2000 \left(1 + \frac{0.02}{12}\right)^{12t}
\]

2. Simplify:

\[
4000 = 2000 \left(1 + \frac{0.02}{12}\right)^{12t}
\]

Divide both sides by 2000:

\[
2 = \left(1 + \frac{0.02}{12}\right)^{12t}
\]

3. Calculate $1 + \frac{0.02}{12}$:

\[
1 + \frac{0.02}{12} = 1 + 0.00166667 \approx 1.00166667
\]

4. Now we have:

\[
2 = (1.00166667)^{12t}
\]

5. Take the logarithm of both sides:

\[
\log(2) = \log((1.00166667)^{12t})
\]

Using the property of logarithms:

\[
\log(2) = 12t \cdot \log(1.00166667)
\]

6. Solve for $t$:

\[
t = \frac{\log(2)}{12 \cdot \log(1.00166667)}
\]

7. Calculate:

- $\log(2) \approx 0.3010$
- $\log(1.00166667) \approx 0.000724$

Now plug in these values:

\[
t = \frac{0.3010}{12 \cdot 0.000724} \approx \frac{0.3010}{0.008688} \approx 34.6
\]

### Conclusion:
It takes approximately **34.6 months** (or about **2.88 years**) for $2000 to double at a 2% interest rate compounded monthly.

Would you like more details or have any questions? Here are some related questions you might consider:

1. How does the compounding frequency affect the total amount earned?
2. What would the doubling time be if the interest rate was increased to 4%?
3. How can you calculate the total amount after a certain number of years with compound interest?
4. What is the difference between simple interest and compound interest?
5. How does inflation impact the real value of your investment over time?

**Tip:** Consider using a financial calculator or software to easily compute compound interest scenarios.

Find how long it takes for $2000 to double if it is invested at 2% interest compounded monthly.

Discover how to calculate the time it takes for an investment of $2000 to double at a 2% interest rate compounded monthly. This detailed solution covers the compound interest formula and logarithmic calculations, suitable for Grades 10-12 students.

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