To solve this problem, we will calculate the total amount of interest earned by each option and determine which option gains more.

Step 1: Formula for compound interest

The formula for compound interest is:

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

• $A$ = the future value of the investment/loan, including interest.
• $P$ = the principal investment amount (initial deposit).
• $r$ = annual interest rate (decimal).
• $n$ = the number of times the interest is compounded per year.
• $t$ = the time the money is invested for, in years.

Step 2: Calculating Interest for Option A

For Option A, the interest is compounded annually:

• Principal ($P$) = $15,000
• Interest rate ($r$) = 4% = 0.04
• Compounding frequency ($n$) = 1 (since it's compounded annually)
• Time ($t$) = 5 years

= 15000 \times (1.04)^5$$ ### **Step 3: Calculating Interest for Option B** For Option B, the interest is compounded quarterly: - Principal ($$P$$) = $15,000 - Interest rate ($$r$$) = 3.75% = 0.0375 - Compounding frequency ($$n$$) = 4 (since it's compounded quarterly) - Time ($$t$$) = 5 years $$A_{\text{B}} = 15000 \left(1 + \frac{0.0375}{4}\right)^{4 \times 5} = 15000 \times \left(1 + 0.009375\right)^{20}$$ Now, let me calculate both final amounts.### Results: - **Interest earned in Option A**: $3,249.79 - **Interest earned in Option B**: $3,077.66 - **Difference**: Option A earns **$172.14** more interest than Option B (rounded to the nearest cent). So, Option A is indeed the better choice, and it earns **$172.14** more in interest. Would you like further details or clarification? Here are some additional questions to explore: 1. What would happen if the investment period was 10 years instead of 5? 2. How would the results change if Option B had monthly compounding instead of quarterly? 3. How would the interest difference change if the principal amount were higher? 4. What if Option A had a higher interest rate, say 4.5%? 5. How do the results differ for smaller or larger initial investments? **Tip:** When interest is compounded more frequently (like quarterly or monthly), the amount tends to grow faster, even if the interest rate is slightly lower!