Final answer:
To find the size of the periodic payment, use the formula for the present value of an annuity. To find the outstanding principal, use the formula for the future value of an annuity.
Step-by-step explanation:
To determine the size of the periodic payment, we can use the formula for the present value of an annuity:
PV = R * (1 - (1 + i/n)^(-n*t)) / (i/n)
In this case, the present value (PV) is $31,000, the interest rate (i) is 12%, the number of payments per year (n) is 2 (since interest is compounded semi-annually), and the total number of years (t) is 12. Solving for the periodic payment (R), we find:
R = PV * (i/n) / (1 - (1 + i/n)^(-n*t))
Substituting the given values, we have:
R = $31,000 * (0.12/2) / (1 - (1 + 0.12/2)^(-2*12))
Calculating this expression gives us the size of the periodic payment.
To find the outstanding principal remaining after a certain number of years, we can use the formula for the future value of an annuity:
FV = R * ((1 + i/n)^(n*t) - 1) / (i/n)
In this case, we can substitute the given values into the formula to find the outstanding principal.