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To solve this compound interest problem, we can use the given formula:

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

### Given Information:
- **Initial Amount (P)** = $1100
- **Final Amount (A)** = $2200 (since we're finding how long it takes to double)
- **Interest Rate (r)** = 5% = 0.05
- **Compounding Frequency (n)** = 12 (since it’s compounded monthly)

### Step-by-Step Solution:

1. Substitute the known values into the formula:

   \[
   2200 = 1100 \left(1 + \frac{0.05}{12}\right)^{12t}
   \]

2. Divide both sides by 1100 to isolate the exponential term:

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

3. Simplify the term inside the parentheses:

   \[
   2 = \left(1 + \frac{0.05}{12}\right)^{12t} \approx \left(1.004167\right)^{12t}
   \]

4. To solve for $t$, take the natural logarithm of both sides:

   \[
   \ln(2) = 12t \cdot \ln(1.004167)
   \]

5. Isolate $t$:

   \[
   t = \frac{\ln(2)}{12 \cdot \ln(1.004167)}
   \]

6. Calculate the result:

   \[
   t \approx \frac{0.693147}{12 \times 0.004158} \approx 13.9
   \]

So, it will take approximately **13.9 years** for the investment to double.

Would you like a more detailed explanation, or do you have any questions?

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Here are five related questions for further understanding:

1. What is the formula for compound interest when compounded continuously?
2. How does increasing the compounding frequency affect the time needed to double an investment?
3. What would be the time required to double the investment at a 6% interest rate, compounded monthly?
4. How can you adjust the formula if compounding occurs quarterly instead of monthly?
5. What is the effect of a higher initial principal on the time required to double an investment?

**Tip:** When working with logarithmic equations in compound interest, remember that taking the natural log of both sides simplifies the exponent, making it easier to isolate the variable.

Find how long it takes for $1100 to double if it is invested at 5% interest compounded monthly.

Learn how to calculate the time needed to double a $1100 investment with 5% interest compounded monthly. This solution explores compound interest and exponential growth, making it ideal for high school students in Grades 10-12.

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