A debt of $8,800 is to be amortized with 8 equal semiannual payments of $1,417.12. If the annual interest rate is 12% compounded semiannually, find the unpaid balance immediately after the 5th payment. Round your answer to the nearest cent.
A)$227.28
B)$3,787.98
C)$140,155.21
D)$6,968.44
E)$9,638.81

Question

A debt of $8,800 is to be amortized with 8 equal semiannual payments of $1,417.12. If the annual interest rate is 12% compounded semiannually, find the unpaid balance immediately after the 5th payment. Round your answer to the nearest cent. ​
A)$227.28
B)$3,787.98
C)$140,155.21
D)$6,968.44
E)$9,638.81

Quizplus · Accepted Answer

The unpaid balance after the 5th payment can be calculated using the formula for the remaining balance of an amortizing loan, which involves calculating the present value of the remaining payments. Given an annual interest rate of 12% compounded semiannually, the semiannual interest rate is 6%. After 5 payments, there are 3 payments left. The formula for the present value of an annuity can be used here:$$ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) $$Where:- $PV$ is the present value or the unpaid balance we're solving for,- $P$ is the payment amount ($1,417.12),- $r$ is the semiannual interest rate (6% or 0.06),- $n$ is the number of remaining payments (3).Plugging in the numbers:$$ PV = 1417.12 \times \left( \frac{1 - (1 + 0.06)^{-3}}{0.06} \right) $$$$ PV = 1417.12 \times \left( \frac{1 - (1.06)^{-3}}{0.06} \right) $$$$ PV = 1417.12 \times \left( \frac{1 - 0.839619}{0.06} \right) $$$$ PV = 1417.12 \times \left( \frac{0.160381}{0.06} \right) $$$$ PV = 1417.12 \times 26.73017 $$$$ PV \approx 3787.98 $$Therefore, the unpaid balance immediately after the 5th payment is approximately $3,787.98.

A Debt of $8,800 Is to Be Amortized with 8