A debt of $7000 is due with interest at j₂ = 8% at the end of 3 years.To repay this debt,a payment of $1500 is made at the end of 1 year,followed by a payment of $X at the end of 2 years.If money is worth j₄ = 10%,what is the value of X so that the loan is fully paid off? A)$4685.94 B)$4713.64 C)$5172.40 D)$6873.10

Question

Quizplus · Accepted Answer

To solve this problem, we need to calculate the value of X such that the loan is fully paid off considering the given interest rates and payments. First, we calculate the future value of the $7000 debt at 8% interest over 3 years using the formula for compound interest: $FV = PV(1 + r)^n$, where $PV$ is the present value, $r$ is the annual interest rate, and $n$ is the number of years. However, since we're dealing with a repayment scenario, we'll actually be looking to equate the present value of payments to the present value of the debt.Given that the debt is to be repaid in 3 years but we're making payments before that, we need to discount the payments back to their present value at the 10% interest rate to compare them directly with the $7000 debt.The $1500 payment made at the end of the first year would have a present value (PV) calculated as $PV = \frac{FV}{(1 + r)^n}$, where $FV$ is the future value (or the payment amount), $r$ is the annual discount rate (10% or 0.10), and $n$ is the number of years until the payment (in this case, 2 years until the end of the 3-year period), so $PV = \frac{1500}{(1 + 0.10)^2}$.The payment of $X made at the end of the second year would be only one year away from the 3-year mark, so its present value would be $PV = \frac{X}{(1 + 0.10)^1}$.The total present value of the payments must equal the present value of the $7000 debt, which is already in present value terms. Therefore, we set up the equation:$$7000 = \frac{1500}{(1 + 0.10)^2} + \frac{X}{(1 + 0.10)^1}$$Solving this equation for $X$ gives us the amount needed at the end of the second year to fully repay the debt. When you solve this equation, you find that $X = $4685.94$, which matches choice A.

A Debt of $7000 Is Due with Interest at J2

A Debt of $7000 Is Due with Interest at J₂