4. You have been saving money for the past few years and have saved $5000 in cash. The goal has been to save up enough money to buy a house. You decided that it would be better to invest the money in a bank until you are ready to make your big purchase. After speaking with a bank representative, you are given two different investment options.

*Option 1-* pays 6.5% per year, compounded quarterly for 10 years.

*Option 2*- pays 6.5% per year, compounded continuously for 10 years.

A) If you decide to invest your money in ***option A***, how much money will you have in your account after 10 years? Round to the nearest cent. Show all your work.

To calculate the amount of money you will have in the account after 10 years with **Option 1**, which offers 6.5% interest per year, compounded quarterly, we'll use the **compound interest formula**:

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

Where:
- $A$ is the amount of money in the account after interest.
- $P$ is the initial principal (in this case, $5000).
- $r$ is the annual interest rate (6.5% or 0.065 as a decimal).
- $n$ is the number of times the interest is compounded per year (quarterly means $n = 4$).
- $t$ is the number of years the money is invested (in this case, 10 years).

### Step 1: Substitute the known values into the formula:
\[
A = 5000 \left(1 + \frac{0.065}{4}\right)^{4 \times 10}
\]

### Step 2: Simplify the expression inside the parentheses:
\[
A = 5000 \left(1 + \frac{0.065}{4}\right)^{40}
= 5000 \left(1 + 0.01625\right)^{40}
= 5000 \left(1.01625\right)^{40}
\]

### Step 3: Calculate $(1.01625)^{40}$:
\[
1.01625^{40} \approx 1.87347
\]

### Step 4: Multiply by the principal:
\[
A = 5000 \times 1.87347 \approx 9367.35
\]

So, the amount of money in your account after 10 years with Option 1 is approximately **$9367.35**.

---

Would you like further details or have any questions? 

Here are 5 related questions to deepen your understanding:
1. What would be the outcome if the interest rate was 5% instead of 6.5%?
2. How does the number of compounding periods affect the total amount?
3. What would happen if you invested for 15 years instead of 10?
4. How does quarterly compounding compare to monthly compounding at the same rate?
5. How does the formula change if the interest rate is compounded annually?

**Tip:** Always double-check the number of compounding periods per year when using the compound interest formula, as it significantly affects the final amount.

You have been saving money for the past few years and have saved $5000 in cash. The goal has been to save up enough money to buy a house. You decided that it would be better to invest the money in a bank until you are ready to make your big purchase. After speaking with a bank representative, you are given two different investment options. Option 1 pays 6.5% per year, compounded quarterly for 10 years. Option 2 pays 6.5% per year, compounded continuously for 10 years. A) If you decide to invest your money in option A, how much money will you have in your account after 10 years? Round to the nearest cent. Show all your work.

Learn how to calculate the future value of $5000 invested for 10 years at a 6.5% annual interest rate, compounded quarterly. This problem explores key financial math concepts such as compound interest and exponential growth, ideal for high school students in Grades 9-12.

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