An investor is considering two investment options. Option A offers a 4% annual compound interest rate, and Option B offers a 3.75% annual compound interest rate with interest compounded quarterly.

If the investor plans to invest $15,000 for 5 years, which option gains the most interest? 
Question Blank 1 of 2
Option A is the better choice

How much more interest does the better option gain? ( round to the nearest cent ) $

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

\[
A_{\text{A}} = 15000 \left(1 + \frac{0.04}{1}\right)^{1 \times 5}
= 15000 \times (1.04)^5
\]

### **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!

An investor is considering two investment options. Option A offers a 4% annual compound interest rate, and Option B offers a 3.75% annual compound interest rate with interest compounded quarterly. If the investor plans to invest $15,000 for 5 years, which option gains the most interest?

Discover how to compare two investment options: one with a 4% annual compound interest rate and another with 3.75% compounded quarterly. Learn how much more interest the better option gains over 5 years, using the compound interest formula. Ideal for high school students and investors alike.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation