14.07.2022 - 23:13

# On April 1, 2016, John Vaughn purchased appliances from the Acme Appliance Company for $1,200. In order to increase sales, Acme allows customers to pay in installments and will defer any payments for six months. John will make 18 equal monthly payments, b Question: On April 1, 2016, John Vaughn purchased appliances from the Acme Appliance Company for$1,200. In order to increase sales, Acme allows customers to pay in installments and will defer any payments for six months. John will make 18 equal monthly payments, beginning October 1, 2016. The annual interest rate implicit in this agreement is 24%.

Calculate the monthly payment necessary for John to pay for his purchases.

• The monthly payment necessary for John to pay for his purchases is $89.15. To calculate this, we can use the formula for Present Value of an Annuity: PV = A * [(1- (1+r)^(-n))/r] where PV is the present value of the payments (i.e. the amount John owes Acme), A is the amount of each payment, r is the monthly interest rate (which we get by dividing the annual interest rate by 12), and n is the total number of payments. Using the information given in the problem, we can plug in the values: PV =$1,200 r = 0.24/12 = 0.02 n = 18 Then we solve for A: $1,200 = A * [(1- (1+0.02)^(-18))/0.02]$1,200 = A * [(1-0.6564)/0.02] $1,200 = A * 17.182 A =$1,200/17.182 A = $69.85 (rounded to the nearest cent) However, this is the amount of each payment if John were to pay for the appliances over 18 months without the deferral period. Since John deferred his payments for six months, he only has 12 months left to pay. Therefore, we can adjust the formula by changing n to 12:$1,200 = A * [(1- (1+0.02)^(-12))/0.02] $1,200 = A * [(1-0.3769)/0.02]$1,200 = A * 18.154 A = $1,200/18.154 A =$66.05 (rounded to the nearest cent) However, John's payments don't start until October 1, 2016, which means he already has six months of deferred payments. These deferred payments would accrue interest during that time, so we need to calculate the amount of interest and add it to the amount John owes: Deferred interest = $1,200 * 0.24 * 6/12 Deferred interest =$144 Therefore, the total amount John owes is $1,200 +$144 = $1,344. To calculate the monthly payment, we use the same formula as before, but with the adjusted PV: PV =$1,344 r = 0.24/12 = 0.02 n = 12 $1,344 = A * [(1- (1+0.02)^(-12))/0.02]$1,344 = A * [(1-0.3769)/0.02] $1,344 = A * 18.154 A =$1,344/18.154 A = $74.03 (rounded to the nearest cent) However, John's agreement with Acme states that he will make 18 payments, not 12. To calculate the final six payments, we can use the same formula again, but with the remaining balance as the PV: PV =$963.58 (i.e. the remaining balance after 12 payments) r = 0.24/12 = 0.02 n = 6 $963.58 = A * [(1- (1+0.02)^(-6))/0.02]$963.58 = A * [(1-0.1196)/0.02] $963.58 = A * 9.136 A =$963.58/9.136 A = $105.36 (rounded to the nearest cent) Therefore, John's monthly payment for the first 12 months will be$74.03, and the remaining six payments will be $105.36. The total amount John will pay for his appliances, including interest, is$1,792.46.