Question

Paul wants to buy a car. He needs to take out a loan for $7000. The car salesman offers him a loan with an interest rate of 8%, compounded annually. Paul considers two options to repay the loan.
Option 1: Pay $200 each month, until the loan is fully repaid
Option 2: Make 24 equal monthly payments.
a. Use option 1 to calculate
the number of months it will take for Paul to repay the loan.
b. Use option 1 to calculate the total amount that Paul has to pay.
c. Use option 2 to calculate the amount Paul pays each month.
d. Use option 2 to calculate the total amount that Paul has to pay.
e. Give a reason why Paul might choose option 1.
f. Give a reason why Paul might choose option 2.

Answer 1

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;

$PMT = \dfrac{\dfrac{r}{12} \times PV}{1 - \left (1 + \dfrac{r}{12} \right)^{-n}}$

Where;

PMT = The monthly payment

n = The number of months

Where PMT = $200, we have;

$200 = \dfrac{\dfrac{0.08}{12} \times 7,000}{1 - \left (1 + \dfrac{0.08}{12} \right)^{-n}}$

$1 - \dfrac{\dfrac{0.08}{12} \times 7,000}{200} = \left (1 + \dfrac{0.08}{12} \right)^{-n}$

$1 - \dfrac{7}{30} = \dfrac{23}{30} = \left ( \dfrac{151}{150} \right)^{-n}$

$-n = \dfrac{ ln \left(\dfrac{23}{30} \right)}{ln\left(\dfrac{151}{150} \right) } \approx -39.988$

∴ 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;

$PMT = \dfrac{\dfrac{0.08}{12} \times 7,000}{1 - \left (1 + \dfrac{0.08}{12} \right)^{-24}} \approx 316.59$

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.