Sam Long anticipates he will need approximately $225,000 in 15 years to cover his 3-year-old daughter’s college bills for a 4-year degree.How much would he have to invest today at an interest rate of 8 percent compounded semiannually?

To calculate the amount Sam Long would have to invest today, we can use the formula for compound interest:

A = P(1 + r/n)^(nt),

where:
A = the future value of the investment
P = the principal amount (the initial investment)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested for

In this case, Sam Long wants to calculate the principal amount (P), so we rearrange the formula and solve for P:

P = A / (1 + r/n)^(nt)

Given:
A = $225,000
r = 8% = 0.08 (as a decimal)
n = 2 (compounded semiannually)
t = 15 years

Substituting the given values into the formula:

P = $225,000 / (1 + 0.08/2)^(2*15)

Now, let's calculate:

P = $225,000 / (1 + 0.04)^(30)
P = $225,000 / (1.04)^30
P = $225,000 / 2.208
P ≈ $101,970.29

Therefore, Sam Long would have to invest approximately $101,970.29 today at an interest rate of 8 percent compounded semiannually to cover his daughter's college bills in 15 years.

To determine the amount Sam Long would need to invest today at an interest rate of 8 percent compounded semiannually, we need to use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years

In this case, the future value (A) is known: $225,000. The interest rate (r) is 8% or 0.08, compounded semiannually (n = 2), and the time (t) is 15 years. The principal amount (P) is what we need to find.

Plugging in these values into the formula, we get:

$225,000 = P(1 + 0.08/2)^(2 * 15)

Simplifying further:

$225,000 = P(1.04)^30

Now, we can isolate P by dividing both sides of the equation by (1.04)^30:

P = $225,000 / (1.04)^30

Calculating this in a calculator, we find:

P ≈ $66,578.82

Therefore, Sam Long would need to invest approximately $66,578.82 today at an interest rate of 8 percent compounded semiannually to have $225,000 in 15 years to cover his daughter's college bills.

P = Po(1+r)^n = $225,000

r = (8%/2)/100% = 0.04 = Semi-annual % rate expressed as a decimal.

n = 2Comp./yr * 15yrs = 30 Compounding
periods.

Po*(1.04)^30 = 225,000
Po = 225,000/1.04^30 = $69,371.70