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. 


To solve this problem, we will use the formula for compound interest, which is:

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

Where:
- $ A $ is the final amount (total balance).
- $ P $ is the initial principal (in this case, $1,000,000).
- $ r $ is the annual interest rate (expressed as a decimal).
- $ n $ is the number of times the interest is compounded per year.
- $ t $ is the number of years.

### Part A: Plan 1 - 6% interest, compounded monthly
For Plan 1:
- $ P = 1,000,000 $
- $ r = 6\% = 0.06 $
- $ n = 12 $ (since interest is compounded monthly)
- $ t = 5 $ years

We can now calculate the amount $ A $ using the formula:

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

Now we can plug in the numbers to find the result.

### Part B: Plan 2 - 12% interest, compounded monthly
For Plan 2:
- $ P = 1,000,000 $
- $ r = 12\% = 0.12 $
- $ n = 12 $
- $ t = 5 $ years

We will use the same formula:

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

I'll now calculate both values.### Part A: Plan 1 - 6% interest, compounded monthly
The final balance after 5 years in Plan 1 is approximately:

\[
A_1 = 1,348,850.15
\]

### Part B: Plan 2 - 12% interest, compounded monthly
The final balance after 5 years in Plan 2 is approximately:

\[
A_2 = 1,816,696.70
\]

### Part C: Which plan gives a greater balance, and by how much?

Plan 2 gives a greater balance. To find how much more you earn with Plan 2:

\[
\text{Difference} = A_2 - A_1 = 1,816,696.70 - 1,348,850.15 = 467,846.55
\]

So, by choosing Plan 2, you would earn **$467,846.55** more over the 5 years.

Would you like more details or have any further questions?

---

Here are 5 related questions for deeper understanding:
1. How would the final balance change if the interest is compounded quarterly instead of monthly?
2. What is the effect of increasing the time period to 10 years for both plans?
3. How would the results change if Plan 2 compounded annually instead of monthly?
4. What happens to the final amount if the principal was doubled to $2,000,000?
5. How does simple interest compare to compound interest for both plans?

**Tip:** The more frequently interest is compounded, the more you earn over time. Monthly compounding typically yields more than annual compounding.

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? B) How much money will you have at the end of the 5 years in plan 2? C) Which plan will give you a greater balance? How much more do you earn by choosing the better plan?

Explore the comparison between two savings plans with interest rates of 6% and 12%, compounded monthly over 5 years. Learn how to calculate compound interest step-by-step and determine which plan yields a greater balance, including detailed explanations and formula applications. Suitable for students in Grades 9-12.

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