To find the interest rate for an investment with compounded monthly interest, we use the compound interest formula:

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

Where:

• $A$ is the amount after time $t$ (final balance),
• $P$ is the principal (initial investment),
• $r$ is the annual interest rate (which we need to find),
• $n$ is the number of times interest is compounded per year,
• $t$ is the number of years.

Given:

• $A = 90,481$ (final balance),
• $P = 34,000$ (initial investment),
• $t = 18$ years,
• $n = 12$ (compounded monthly).

Substitute the given values into the formula:

$90,481 = 34,000 \left(1 + \frac{r}{12}\right)^{12 \times 18}$

$90,481 = 34,000 \left(1 + \frac{r}{12}\right)^{216}$

Now, solve for $r$:

1. Divide both sides by 34,000:

$\frac{90,481}{34,000} = \left(1 + \frac{r}{12}\right)^{216}$

$2.6612 = \left(1 + \frac{r}{12}\right)^{216}$

2. Take the 216th root of both sides:

$\left(2.6612\right)^{\frac{1}{216}} = 1 + \frac{r}{12}$

$1.004502 \approx 1 + \frac{r}{12}$

3. Subtract 1 from both sides:

$0.004502 \approx \frac{r}{12}$

4. Multiply both sides by 12 to solve for $r$:

$r \approx 0.054024$

5. Convert $r$ to a percentage by multiplying by 100:

$r \approx 5.40\%$

Thus, the interest rate is approximately 5.40%.

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Would you like more details on any of these steps or have any other questions?

Here are five related questions:

1. What would the final balance be if the interest rate was 6% instead?
2. How does the number of compounding periods per year affect the final balance?
3. How would the calculation change if interest were compounded daily?
4. What if the investment term was extended by 5 more years—how would the final balance change?
5. How does this formula relate to continuous compounding?

Tip: When solving compound interest problems, always double-check the number of compounding periods per year, as this significantly affects the final outcome.