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.

B) How much money will you have at the end of the 5 years in plan 2? Round to the nearest cent.  Show all your work.

C) Which plan will give you a greater balance? How much more do you earn by choosing the better plan? Explain. 
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To solve these problems, we will use the **compound interest formula**, which is:

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

Where:
- $A$ is the amount of money after interest (the future value).
- $P$ is the principal amount (the initial investment).
- $r$ is the annual interest rate (in decimal form).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the number of years.

Given:
- $P = 1,000,000$
- $t = 5$ years
- Interest is compounded monthly, so $n = 12$.

### Part A: Plan 1 (6% interest rate compounded monthly)

For Plan 1:
- $r = 6\% = 0.06$
- $n = 12$
- $t = 5$

The formula for Plan 1 is:

\[
A_1 = 1,000,000 \left(1 + \frac{0.06}{12}\right)^{12 \times 5}
\]

Let's compute this.

### Part B: Plan 2 (12% interest rate compounded monthly)

For Plan 2:
- $r = 12\% = 0.12$
- $n = 12$
- $t = 5$

The formula for Plan 2 is:

\[
A_2 = 1,000,000 \left(1 + \frac{0.12}{12}\right)^{12 \times 5}
\]

We will compute this as well.

### Part C: Compare the balances and find which plan gives a greater return

Finally, we will subtract the balances from each other to find out how much more you would earn by choosing the better plan.

Let's compute these values now.### Part A: Plan 1 (6% interest rate compounded monthly)

Using the formula, the total amount at the end of 5 years is:

\[
A_1 = 1,000,000 \left(1 + \frac{0.06}{12}\right)^{12 \times 5} = 1,348,850.15
\]

So, you will have **$1,348,850.15** at the end of 5 years in Plan 1.

---

### Part B: Plan 2 (12% interest rate compounded monthly)

Using the same formula for Plan 2, the total amount at the end of 5 years is:

\[
A_2 = 1,000,000 \left(1 + \frac{0.12}{12}\right)^{12 \times 5} = 1,816,696.70
\]

So, you will have **$1,816,696.70** at the end of 5 years in Plan 2.

---

### Part C: Comparison

Plan 2 gives you a higher balance. The difference between the two plans is:

\[
1,816,696.70 - 1,348,850.15 = 467,846.55
\]

Thus, by choosing Plan 2, you will earn **$467,846.55** more than Plan 1 over the 5 years.

Would you like more details or have any questions? Here are some related questions to explore:

1. What happens if the interest is compounded quarterly instead of monthly?
2. How would the balance change if the investment duration was 10 years instead of 5?
3. What is the formula for continuous compounding, and how would it apply to these plans?
4. How does changing the interest rate affect the compound interest over shorter periods, like 1 or 2 years?
5. Can you calculate the effective annual rate (EAR) for both plans to compare them?

**Tip:** Compounding more frequently (like monthly instead of annually) increases the final amount due to interest being added 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. 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. B) How much money will you have at the end of the 5 years in plan 2? Round to the nearest cent. Show all your work. C) Which plan will give you a greater balance? How much more do you earn by choosing the better plan? Explain.

Explore a detailed solution to a compound interest problem involving a $1,000,000 investment over 5 years. Compare two plans: a 6% and a 12% interest rate, both compounded monthly. Learn how to use the compound interest formula and find out which plan yields a higher balance and by how much. Suitable for high school students studying financial mathematics.

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