A woman takes out a loan to be paid back over 4 years at j₁₂ = 6% with monthly payments of 281.82.After 14 payments,the outstanding balance on the loan is $8791.53.She then misses the next 4 payments (from time 15 to 18).She begins making regular monthly payments at time 19.If she still wants to pay back her loan in full by the end of 4 years,what is the size of her new monthly payment?

Question

Quizplus · Accepted Answer

To solve the problem, we first need to find the original loan amount. We can use the formula for present value of an annuity to do this:

PV = PMT x ((1 - (1 + r)^-n) / r)

Where:
PMT = monthly payment = $281.82
r = monthly interest rate = 6% / 12 = 0.5%
n = number of payments = 4 x 12 = 48

Plugging in the values, we get:

PV = $281.82 x ((1 - (1 + 0.005)^-48) / 0.005)
PV = $12,000

So the original loan amount was $12,000.

After 14 payments, the outstanding balance on the loan is $8,791.53. We can use the formula for remaining balance of an annuity to find out how much of the loan is left to be paid:

RB = PV x ((1 + r)^n - (1 + r)^p) / ((1 + r)^n - 1)

Where:
RB = remaining balance
PV = present value = $12,000
r = monthly interest rate = 0.5%
n = total number of payments = 48
p = number of payments made = 14

Plugging in the values, we get:

RB = $12,000 x ((1 + 0.005)^48 - (1 + 0.005)^14) / ((1 + 0.005)^48 - 1)
RB = $8,791.53

So after 14 payments, there is $8,791.53 left to be paid.

The woman then misses the next 4 payments, so the number of payments left to be made is 48 - 14 - 4 = 30. We can use the same formula as before to find out how much the monthly payment needs to be in order to pay off the remaining balance in 30 months:

PMT = RB x r / (1 - (1 + r)^-n)

Where:
PMT = monthly payment
RB = remaining balance = $8,791.53
r = monthly interest rate = 0.5%
n = number of payments = 30

Plugging in the values, we get:

PMT = $8,791.53 x 0.005 / (1 - (1 + 0.005)^-30)
PMT = $322.68

So the new monthly payment needs to be $322.68, which is closest to option C, $322.68.

A Woman Takes Out a Loan to Be Paid Back