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Math  /  Algebra

QuestionSusie takes out a car loan for $21,000\$ 21,000 for a term of 3 years at 6%6 \% interest compounded monthly. (a) Find her monthly payment. (b) Find the total amount she pays for the car. (c) Find the total amount of interest she pays.

Studdy Solution

STEP 1

1. The principal amount of the car loan is $21,000.
2. The loan term is 3 years.
3. The interest rate is 6% per annum, compounded monthly.
4. We need to find the monthly payment, total amount paid, and total interest paid.

STEP 2

1. Calculate the monthly interest rate.
2. Calculate the number of payments.
3. Use the loan amortization formula to find the monthly payment.
4. Calculate the total amount paid over the term of the loan.
5. Calculate the total interest paid.

STEP 3

Calculate the monthly interest rate.
The annual interest rate is 6%, so the monthly interest rate is:
r=6%12=0.5%=0.005 r = \frac{6\%}{12} = 0.5\% = 0.005

STEP 4

Calculate the number of payments.
The loan term is 3 years, and payments are made monthly, so the number of payments is:
n=3×12=36 n = 3 \times 12 = 36

STEP 5

Use the loan amortization formula to find the monthly payment.
The formula for the monthly payment M M is:
M=Pr(1+r)n(1+r)n−1 M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
where: - P=21000 P = 21000 is the principal, - r=0.005 r = 0.005 is the monthly interest rate, - n=36 n = 36 is the number of payments.
Substitute the values into the formula:
M=210000.005(1+0.005)36(1+0.005)36−1 M = 21000 \frac{0.005(1 + 0.005)^{36}}{(1 + 0.005)^{36} - 1}
Calculate M M :
M≈210000.005×1.196681.19668−1 M \approx 21000 \frac{0.005 \times 1.19668}{1.19668 - 1} M≈210000.00598340.19668 M \approx 21000 \frac{0.0059834}{0.19668} M≈21000×0.03043 M \approx 21000 \times 0.03043 M≈638.51 M \approx 638.51

STEP 6

Calculate the total amount paid over the term of the loan.
The total amount paid is the monthly payment multiplied by the number of payments:
Total Amount Paid=M×n \text{Total Amount Paid} = M \times n Total Amount Paid=638.51×36 \text{Total Amount Paid} = 638.51 \times 36 Total Amount Paid≈22986.36 \text{Total Amount Paid} \approx 22986.36

STEP 7

Calculate the total interest paid.
The total interest paid is the total amount paid minus the principal:
Total Interest Paid=Total Amount Paid−P \text{Total Interest Paid} = \text{Total Amount Paid} - P Total Interest Paid=22986.36−21000 \text{Total Interest Paid} = 22986.36 - 21000 Total Interest Paid≈1986.36 \text{Total Interest Paid} \approx 1986.36
The monthly payment is approximately $638.51 \$638.51 , the total amount paid is approximately $22,986.36 \$22,986.36 , and the total interest paid is approximately $1,986.36 \$1,986.36 .

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