Find the present value of $14000 due in 9 years at the given rate of interest.

(a) 2%/year compounded semiannually?

(b) 5%/year compounded semiannually?

a) PV = 1400(1.01)^-18

= ....

b) PV = 1400(1.025)^-18
= ....

To find the present value of an amount due in the future at a given rate of interest, we use the formula for present value:

Present Value = Future Value / (1 + r/n)^(n*t)

where:
- Future Value is the amount due in the future,
- r is the annual interest rate (as a decimal),
- n is the number of compounding periods per year, and
- t is the number of years.

(a) Using a rate of 2%/year compounded semiannually:
- Future Value = $14,000
- r = 2% = 0.02 (as a decimal)
- n = 2 (compounded semiannually)
- t = 9 years

To calculate the present value, substitute the given values into the formula:

Present Value = $14,000 / (1 + 0.02/2)^(2*9)

Simplify the formula:

Present Value = $14,000 / (1 + 0.01)^(18)

Calculate the present value:

Present Value ≈ $14,000 / (1.01)^18 ≈ $10,734.64

Therefore, the present value of $14,000 due in 9 years at a rate of 2%/year compounded semiannually is approximately $10,734.64.

(b) Using a rate of 5%/year compounded semiannually:
- Future Value = $14,000
- r = 5% = 0.05 (as a decimal)
- n = 2 (compounded semiannually)
- t = 9 years

Substitute the given values into the formula:

Present Value = $14,000 / (1 + 0.05/2)^(2*9)

Simplify the formula:

Present Value ≈ $14,000 / (1 + 0.025)^(18)

Calculate the present value:

Present Value ≈ $14,000 / (1.025)^18 ≈ $8,323.25

Therefore, the present value of $14,000 due in 9 years at a rate of 5%/year compounded semiannually is approximately $8,323.25.