https://file.aiquickdraw.com/mathbot/file/1726628425708nb5uhuq6.jpg

The question in your photo asks to compare two investment options for $60,000 over 5 years, one offering **4% compounded semiannually** and the other offering **3.25% compounded continuously**.

Let's break down each option and solve it step by step.

### 1. **Investment A: 4% compounded semiannually**
The formula for compound interest is:

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

Where:
- $A$ = the amount of money accumulated after $t$ years, including interest.
- $P$ = the principal amount (initial investment), which is $60,000.
- $r$ = annual interest rate (4% = 0.04).
- $n$ = number of times the interest is compounded per year (semiannually means $n = 2$).
- $t$ = the number of years (5 years).

Using this formula for Investment A:

\[
A = 60,000 \left(1 + \frac{0.04}{2}\right)^{2 \times 5}
\]

### 2. **Investment B: 3.25% compounded continuously**
The formula for continuously compounded interest is:

\[
A = Pe^{rt}
\]

Where:
- $A$ = the final amount.
- $P$ = the principal amount (initial investment), $60,000.
- $r$ = annual interest rate (3.25% = 0.0325).
- $t$ = the number of years (5 years).
- $e$ = the base of the natural logarithm (approximately 2.718).

Using this formula for Investment B:

\[
A = 60,000 \cdot e^{0.0325 \times 5}
\]

Let me calculate both options.After 5 years:

- **Investment A** (4% compounded semiannually) will grow to approximately **$73,139.67**.
- **Investment B** (3.25% compounded continuously) will grow to approximately **$70,586.90**.

### Comparison:
- Investment A will earn more interest by approximately **$2,552.77** over Investment B.

Would you like more details or clarifications?

Here are five related questions to explore further:
1. How does the compounding frequency affect the total amount in general?
2. What is the formula for quarterly compounded interest?
3. How would the results change if the time period were 10 years instead of 5?
4. What is the benefit of continuously compounding interest versus annual compounding?
5. How can this information be used to make better financial decisions?

**Tip:** The more frequent the compounding, the higher the total interest earned, especially over longer periods.

An employee wants to invest $60,000 in a pension plan. One investment offers 4% compounded semiannually. Another offers 3.25% compounded continuously. (a) Which investment will earn more interest in 5 years? (b) How much more will the better plan earn?

Compare two investment options: 4% compounded semiannually and 3.25% compounded continuously over a 5-year period. Learn which plan earns more interest and by how much. This detailed solution breaks down compound interest and continuously compounded formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation