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 ​$1000 to double if it is invested at 5​% interest compounded monthly.
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Part 1
The money will double in value in approximately 
  
enter your response here years. 
​(Do not round until the final answer. Then round to the nearest tenth as​ needed.)

To solve the compound interest problem using the formula 

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

we need to determine how long it takes for an investment of $ P = 1000 $ dollars to double, meaning $ A = 2000 $ dollars. The interest rate $ r $ is 5% or 0.05, and since the interest is compounded monthly, $ n = 12 $.

1. Substitute the values into the formula:

\[
2000 = 1000 \left(1 + \frac{0.05}{12}\right)^{12t}
\]

2. Simplify the equation:

\[
2 = \left(1 + \frac{0.05}{12}\right)^{12t}
\]

Calculating $ 1 + \frac{0.05}{12} $:

\[
1 + \frac{0.05}{12} = 1 + 0.00416667 \approx 1.00416667
\]

Now, the equation becomes:

\[
2 = (1.00416667)^{12t}
\]

3. Take the natural logarithm of both sides:

\[
\ln(2) = \ln((1.00416667)^{12t})
\]

4. Use the power rule of logarithms:

\[
\ln(2) = 12t \cdot \ln(1.00416667)
\]

5. Solve for $ t $:

\[
t = \frac{\ln(2)}{12 \cdot \ln(1.00416667)}
\]

Calculating the values:

- $ \ln(2) \approx 0.693147 $
- $ \ln(1.00416667) \approx 0.004166 $

Thus,

\[
t \approx \frac{0.693147}{12 \cdot 0.004166} \approx \frac{0.693147}{0.0500} \approx 13.86294
\]

Finally, rounding to the nearest tenth:

\[
t \approx 13.9 \text{ years}
\]

The money will double in value in approximately **13.9 years**. 

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

1. What is the effect of increasing the interest rate on the time to double the investment?
2. How would the time change if the interest were compounded annually instead of monthly?
3. What would be the total amount after 15 years with the same investment and interest rate?
4. How does the formula change if we consider continuous compounding?
5. Can you explain the significance of each variable in the compound interest formula?

**Tip:** Always keep track of the units (e.g., years, months) when solving time-related problems in finance.

Find how long it takes for $1000 to double if it is invested at 5% interest compounded monthly.

Discover how long it takes for a $1000 investment to double at a 5% interest rate compounded monthly, using the compound interest formula. The solution reveals that the money will double in approximately 13.9 years, with detailed steps and calculations.

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