https://file.aiquickdraw.com/mathbot/file/1727740729395ewc3qzuf.jpg

The problem in the image is as follows:

A portfolio manager puts $1200 into an investment account that earns 4.5% interest compounded monthly. The question is to find how much is in the account after 3 years, and you are asked to use at least 4 non-zero decimal places in intermediate calculations and round the final answer to two decimal places.

### To solve this, we will use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $A$ is the final amount (what we are solving for),
- $P = 1200$ is the principal (initial investment),
- $r = 0.045$ is the annual interest rate (4.5% as a decimal),
- $n = 12$ is the number of times the interest is compounded per year (monthly compounding),
- $t = 3$ is the time the money is invested for, in years.

### Step-by-step calculation:
1. First, calculate the monthly interest rate: 
   \[
   \frac{r}{n} = \frac{0.045}{12} = 0.00375
   \]
2. Then, calculate the exponent:
   \[
   nt = 12 \times 3 = 36
   \]
3. Now, apply the compound interest formula:
   \[
   A = 1200 \left(1 + 0.00375\right)^{36}
   \]
   \[
   A = 1200 \times \left(1.00375\right)^{36}
   \]
4. Use a calculator to find:
   \[
   1.00375^{36} \approx 1.144849
   \]
5. Finally, calculate the final amount $A$:
   \[
   A = 1200 \times 1.144849 \approx 1373.8188
   \]
   Rounding to two decimal places:
   \[
   A \approx 1373.82
   \]

### Final Answer:
After 3 years, the account will have approximately **$1373.82**.

---

Do you have any questions or need further details?

Here are 5 related questions for deeper understanding:
1. What is the impact of different compounding frequencies on the final amount?
2. How would the final amount change if the interest rate was 5% instead of 4.5%?
3. What happens to the investment if compounded annually instead of monthly?
4. How does increasing the investment period to 5 years affect the outcome?
5. How would the formula change for continuous compounding?

**Tip:** Remember, the more frequently interest is compounded, the larger the final amount will be due to the effect of "interest on interest."

A portfolio manager puts $1200 into an investment account that earns 4.5% interest compounded monthly. How much is in the account after 3 years?

Learn how to calculate the future value of $1200 invested at 4.5% interest compounded monthly for 3 years using the compound interest formula. This example highlights key financial concepts such as interest rates and exponential growth.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation