https://file.aiquickdraw.com/mathbot/file/1729441058749sohf3yy0.jpg

It looks like the question asks you to find the accumulated value of an investment of $20,000 for 6 years at an interest rate of 1.5%, compounded at different frequencies (semiannually, quarterly, monthly). 

The relevant formula for compound interest is:

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

Where:
- $ A $ is the accumulated amount (future value).
- $ P $ is the principal amount ($20,000).
- $ r $ is the annual interest rate (1.5% or 0.015).
- $ n $ is the number of times the interest is compounded per year.
- $ t $ is the time the money is invested for (6 years).

### a. Semiannually (n = 2)
\[
A = 20000 \left(1 + \frac{0.015}{2}\right)^{2 \times 6}
\]
\[
A = 20000 \left(1 + 0.0075\right)^{12}
\]
\[
A \approx 20000 \times 1.093575
\]
\[
A \approx 21,681.52
\]

This matches the value you provided for part a.

### b. Quarterly (n = 4)
\[
A = 20000 \left(1 + \frac{0.015}{4}\right)^{4 \times 6}
\]
\[
A = 20000 \left(1 + 0.00375\right)^{24}
\]
\[
A \approx 20000 \times 1.093757
\]
\[
A \approx 21,684.46
\]

This matches the value you provided for part b.

### c. Monthly (n = 12)
For part c, let's calculate the accumulated value when compounded monthly:

\[
A = 20000 \left(1 + \frac{0.015}{12}\right)^{12 \times 6}
\]
\[
A = 20000 \left(1 + 0.00125\right)^{72}
\]
\[
A \approx 20000 \times 1.093887
\]
\[
A \approx 21,686.55
\]

So, for part c, the accumulated value when compounded monthly is approximately **$21,686.55**.

---

Let me know if you'd like more details or have any questions! Here are 5 related questions to expand your understanding:

1. How does changing the compounding frequency affect the accumulated value?
2. What would happen if the interest rate were higher, say 2%, for the same time period?
3. How would you calculate the accumulated value if the interest were compounded continuously?
4. What is the difference between simple interest and compound interest?
5. If the investment was for 10 years instead of 6, how would the result change?

**Tip**: Increasing the frequency of compounding slightly increases the accumulated value, but the effect diminishes as the compounding frequency becomes higher.

Find the accumulated value of an investment of $20,000 for 6 years at an interest rate of 1.5%, compounded semiannually, quarterly, and monthly.

Find the accumulated value of a $20,000 investment for 6 years at an interest rate of 1.5%, compounded semiannually, quarterly, and monthly. The solution uses the compound interest formula and demonstrates how different compounding frequencies affect the future value of the investment.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation