1. You win the lottery and get $1,000,000. You decide that you want to invest all of the money in a  savings account for 5 years. A bank representative offers you two different savings plans. 
Plan 1-The bank gives you a 6% interest rate and compounds the interest each month. Plan 2- The bank gives you a 12% interest rate and compounds the interest each month. 
A) How much money will you have at the end of the 5 years in plan 1? Round to the nearest cent.  Show all your work.  


To solve this problem, we'll use the compound interest formula:

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

Where:
- $A$ is the amount of money accumulated after $t$ years, including interest.
- $P$ is the principal amount (the initial amount of money).
- $r$ is the annual interest rate (decimal form).
- $n$ is the number of times interest is compounded per year.
- $t$ is the time the money is invested for in years.

### Plan 1: 6% interest rate compounded monthly

Given:
- $P = 1,000,000$ (the initial amount)
- $r = 0.06$ (6% annual interest rate)
- $n = 12$ (compounded monthly)
- $t = 5$ years

#### Step-by-step Calculation:

1. Substitute the given values into the formula:

\[
A = 1,000,000 \left(1 + \frac{0.06}{12}\right)^{12 \times 5}
\]

2. Simplify inside the parentheses:

\[
A = 1,000,000 \left(1 + 0.005\right)^{60}
\]

\[
A = 1,000,000 \left(1.005\right)^{60}
\]

3. Calculate $\left(1.005\right)^{60}$:

\[
A = 1,000,000 \times 1.348850
\]

4. Final amount $A$:

\[
A = 1,348,850.00
\]

### Answer:
At the end of 5 years, you will have approximately **$1,348,850.00** in Plan 1.

---

Would you like a more detailed explanation of any step? 

Here are five related questions to consider:
1. How does compounding frequency affect the final amount in an investment?
2. What is the difference between simple interest and compound interest?
3. How would the amount change if the interest were compounded daily instead of monthly?
4. What would happen if the interest rate was lower, say 4%, with monthly compounding?
5. How do banks determine the compounding periods for different accounts?

**Tip:** Compounding more frequently leads to more accumulated interest over time, as interest is being calculated and added to the principal more often.

You win the lottery and get $1,000,000. You decide that you want to invest all of the money in a savings account for 5 years. A bank representative offers you two different savings plans. Plan 1-The bank gives you a 6% interest rate and compounds the interest each month. Plan 2- The bank gives you a 12% interest rate and compounds the interest each month. How much money will you have at the end of the 5 years in plan 1? Round to the nearest cent.

Find out how to calculate the final amount when investing $1,000,000 at a 6% annual interest rate compounded monthly for 5 years. Using the compound interest formula, we walk through step-by-step calculations to determine the final amount, approximately $1,348,850. Perfect for high school students learning about exponential growth and compound interest.

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