Answer:
a. It will take approximately 40 months
b. The total amount Paul has to pay is approximately $8,000
c. The amount Paul pays each month is approximately $316.59
d. The amount Paul has to pay is approximately $7,598.16
e. Lower monthly payment
f. Lower total payment amount (amount to be paid)
Step-by-step explanation:
a. The loan amount, PV = $7,000
The annual interest rate, r = 8%
Option 1; The amount of equal monthly payment to repay the loan = $200
Option 2; The number of equal monthly payment to repay the loan = 24
The formula for the present value of an annuity is given as follows;
[tex]PMT = \dfrac{\dfrac{r}{12} \times PV}{1 - \left (1 + \dfrac{r}{12} \right)^{-n}}[/tex]
Where;
PMT = The monthly payment
n = The number of months
Where PMT = $200, we have;
[tex]200 = \dfrac{\dfrac{0.08}{12} \times 7,000}{1 - \left (1 + \dfrac{0.08}{12} \right)^{-n}}[/tex]
[tex]1 - \dfrac{\dfrac{0.08}{12} \times 7,000}{200} = \left (1 + \dfrac{0.08}{12} \right)^{-n}[/tex]
[tex]1 - \dfrac{7}{30} = \dfrac{23}{30} = \left ( \dfrac{151}{150} \right)^{-n}[/tex]
[tex]-n = \dfrac{ ln \left(\dfrac{23}{30} \right)}{ln\left(\dfrac{151}{150} \right) } \approx -39.988[/tex]
∴ The number of months it will take for Paul to repay the loan, n ≈ 39.988 ≈ 40 months
b. The total amount Paul has to pay, A = PMT × n
Therefore, b plugging in the values of PMT, and 'n', we get;
A ≈ $200 × 40 = $8,000
c. Using option 2, we have;
n = 24
Therefore;
[tex]PMT = \dfrac{\dfrac{0.08}{12} \times 7,000}{1 - \left (1 + \dfrac{0.08}{12} \right)^{-24}} \approx 316.59[/tex]
The monthly payment, PMT ≈ $316.59
d. The total amount Paul has to pay using option 2, A ≈ 316.59 × 24 = 7,598.16
The total amount Paul has to pay using option 2, A ≈ $7,598.16
e. A reason why Paul might choose option 1 is that option 1 offers lower monthly payment
f. A reason why Paul may choose option 2 is that option 2 offers a lower total amount that he has to pay.
A) It will take 48 months for Paul to repay the loan.
B) Paul will have to pay $ 9,523.42.
C) Paul should pay $ 340.20 per month.
D) Paul will have to pay $ 8164.80.
E) Paul might choose option 1 because his monthly payments would be lower and, therefore, more accessible.
F) Paul might choose option 2 because his total payout would be much less than option 1.
Since Paul wants to buy a car, and he needs to take out a loan for $ 7000, and the car salesman offers him a loan with an interest rate of 8%, compounded annually, and Paul considers two options to repay the loan (Option 1: Pay $ 200 each month, until the loan is fully repaid, and Option 2: Make 24 equal monthly payments), for:
The following calculations must be performed:
A) 7000 x 1.08 ^ 4 = 9,523.42
200 x 12 x 4 = 9600
Therefore, it will take 48 months for Paul to repay the loan.
B) Paul will have to pay $ 9,523.42.
C) 7000 x 1.08 ^ 2 = 8,164.80
8,164.80 / 24 = 340.20
Therefore, Paul should pay $ 340.20 per month.
D) Paul will have to pay $ 8164.80.
E) Paul might choose option 1 because his monthly payments would be lower and, therefore, more accessible.
F) Paul might choose option 2 because his total payout would be much less than option 1.
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