To find the interest rate that will allow Caleb to grow his investment to $300 after 18 years with quarterly compounding, we will use the compound interest formula:

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

Where:

• $A$ is the amount of money accumulated after interest (in this case, $300),
• $P$ is the principal amount (in this case, $120),
• $r$ is the annual interest rate (which we need to find),
• $n$ is the number of times the interest is compounded per year (quarterly means $n = 4$),
• $t$ is the number of years (in this case, 18 years).

Step 1: Plug in the known values

$300 = 120 \left(1 + \frac{r}{4}\right)^{4 \times 18}$

Simplify the exponent:

$300 = 120 \left(1 + \frac{r}{4}\right)^{72}$

Step 2: Solve for $r$

First, divide both sides by 120 to isolate the exponential term:

$\frac{300}{120} = \left(1 + \frac{r}{4}\right)^{72}$

$2.5 = \left(1 + \frac{r}{4}\right)^{72}$

Now, take the 72nd root of both sides:

$\left(1 + \frac{r}{4}\right) = (2.5)^{1/72}$

Use a calculator to find the 72nd root of 2.5:

$(2.5)^{1/72} \approx 1.0115$

Now, subtract 1 from both sides:

$\frac{r}{4} = 1.0115 - 1 = 0.0115$

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

$r = 0.0115 \times 4 = 0.046$

Step 3: Convert to a percentage

Multiply by 100 to get the annual interest rate:

$r = 0.046 \times 100 = 4.6\%$

Final Answer:

The required interest rate is approximately 4.6% per year, compounded quarterly.

Would you like further details or clarification?

Here are some related questions for you:

1. How would the final amount change if the interest were compounded monthly instead of quarterly?
2. How do you calculate the compound interest if the interest is compounded annually?
3. What would happen to the final amount if the interest rate were higher or lower?
4. How can you calculate the time needed to reach a certain amount with a given principal and interest rate?
5. What is the difference between simple and compound interest?

Tip: When solving compound interest problems, remember that the frequency of compounding affects the growth of the investment. More frequent compounding leads to slightly more interest accrued.