Jim wants to deposit money in an account to save for a new stereo system in two years. He wants to have $4,000 available at the time. The following rates are available to him:

6.2% simple interest
6.1% compounded annually
5.58% compounded semiannually
5.75% compounded quarterly

Which account(s) should he choose if he wants to invest the smallest amount of money now?

consider the amount of $1 at the end of 2 years

at 6.2% simple ---- 1 + (1)(.062)(2) = $1.124
at 6.1 comp. annually ---- (1.061)^2 = $1.1257
at 5.58% com. semi --- (1.0279)^4 = $1.11636
at 5.75% comp. quarterly -- (1.014375)^8 = 1.12096

so what do you think?

To determine which account(s) Jim should choose in order to invest the smallest amount of money now, we need to calculate the future value of $4,000 under each given interest rate.

1. Simple Interest:
The formula for simple interest is: I = P * r * t
Where:
I = Interest earned
P = Principal amount (initial investment)
r = Interest rate
t = Time period (in years)

To find the principal amount needed, we rearrange the formula:
P = I / (r * t)

In this case, Jim wants to have $4,000 in 2 years, so:
t = 2 years

Using a 6.2% simple interest rate:
I = P * (0.062) * 2
4,000 = P * (0.062) * 2
4,000 = P * 0.124
P = 4,000 / 0.124
P ≈ $32,258.06

Therefore, Jim would need to deposit approximately $32,258.06 to have $4,000 available in 2 years with a 6.2% simple interest rate.

2. Compounded Annually:
The formula for compound interest is: A = P * (1 + r/n)^(n*t)
Where:
A = Future value (including principal and accumulated interest)
P = Principal amount (initial investment)
r = Annual interest rate
n = Number of times interest is compounded per year
t = Time period (in years)

We need to find the principal amount needed based on a future value of $4,000 in 2 years.

Using a 6.1% compounded annually:
4,000 = P * (1 + 0.061/1)^(1*2)
4,000 = P * (1.061)^2
P = 4,000 / (1.061)^2
P ≈ $3,760.98

Therefore, Jim would need to deposit approximately $3,760.98 to have $4,000 available in 2 years with a 6.1% interest rate compounded annually.

3. Compounded Semiannually:
Using a 5.58% compounded semiannually:
4,000 = P * (1 + 0.0558/2)^(2*2)
4,000 = P * (1.0279)^4
P = 4,000 / (1.0279)^4
P ≈ $3,666.09

Therefore, Jim would need to deposit approximately $3,666.09 to have $4,000 available in 2 years with a 5.58% interest rate compounded semiannually.

4. Compounded Quarterly:
Using a 5.75% compounded quarterly:
4,000 = P * (1 + 0.0575/4)^(4*2)
4,000 = P * (1.014375)^8
P = 4,000 / (1.014375)^8
P ≈ $3,618.40

Therefore, Jim would need to deposit approximately $3,618.40 to have $4,000 available in 2 years with a 5.75% interest rate compounded quarterly.

In conclusion, if Jim wants to invest the smallest amount of money now, he should choose the account with a 5.75% interest rate compounded quarterly, as he would need to deposit approximately $3,618.40.