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. 

C) Which plan will give you a greater balance? How much more do you earn by choosing the better plan? Explain. 

______________


To determine which plan gives a greater balance, we need to use the formula for compound interest:

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

Where:
- $A$ is the final amount (balance after interest),
- $P$ is the initial principal (in this case, $1,000,000),
- $r$ is the annual interest rate (6% for Plan 1, 12% for Plan 2),
- $n$ is the number of times the interest is compounded per year (monthly compounding means $n = 12$),
- $t$ is the number of years (5 years in this case).

### **Plan 1: 6% Interest Compounded Monthly**
- $P = 1,000,000$
- $r = 0.06$
- $n = 12$
- $t = 5$

Substitute into the formula:

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

### **Plan 2: 12% Interest Compounded Monthly**
- $P = 1,000,000$
- $r = 0.12$
- $n = 12$
- $t = 5$

Substitute into the formula:

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

Now, let's calculate both balances.### Results:
- **Plan 1 (6% interest compounded monthly)**: \$1,348,850.15
- **Plan 2 (12% interest compounded monthly)**: \$1,816,696.70

### Conclusion:
Plan 2 gives a greater balance. By choosing Plan 2, you would earn **\$467,846.55** more compared to Plan 1 after 5 years.

Would you like more details or have any questions? Here are some follow-up questions that expand on this topic:

1. How would the balance change if the compounding were done annually instead of monthly?
2. What is the impact of increasing the investment period to 10 years on both plans?
3. How would Plan 1 perform with quarterly compounding instead of monthly?
4. What happens if the initial investment is increased to \$2,000,000?
5. Can you calculate the effective annual rate (EAR) for both plans?

**Tip:** When comparing savings plans, always check the compounding frequency, as it can significantly impact your final return.

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.

C) Which plan will give you a greater balance? How much more do you earn by choosing the better plan? Explain.

Explore a detailed comparison of two savings plans: Plan 1 with 6% interest compounded monthly and Plan 2 with 12% interest compounded monthly. Learn how to calculate compound interest and determine which plan gives a higher balance over 5 years. Suitable for students in Grades 10-12.

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