Darius recently obtained a loan for $[tex]15,000[/tex] at an interest rate of [tex]6.8 %[/tex] for 4 years. Use the monthly payment formula to complete the statement.
[tex]M=\frac{P\left(\frac{r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}}{\left(1+\frac{r}{12}\right)^{12 t}-1}[/tex]
[tex]M=[/tex] monthly payment
[tex]P=[/tex] principal
[tex]r=[/tex] interest rate
[tex]t=[/tex] number of years
His monthly payment for the loan is [ ] , and the total finance charge for the loan is [ ].
Answer (2)
Darius's monthly payment for the loan is approximately $365.92, and the total finance charge is approximately $2,175.60.
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Calculate the monthly payment M using the formula: M = ( 1 + 12 r ) 12 t − 1 P ( 12 r ) ( 1 + 12 r ) 12 t , where P = 15000 , r = 0.068 , and t = 4 , resulting in M ≈ 357.80 .
Determine the total amount paid: multiply the monthly payment M by the total number of payments 12 t , which gives 12 × 4 × M ≈ 17174.57 .
Compute the total finance charge: subtract the principal P from the total amount paid, yielding 17174.57 − 15000 ≈ 2174.57 .
The monthly payment is $357.80 and the total finance charge is $2174.57 .
Explanation
Understanding the Problem We are given a loan of $15 , 000 at an interest rate of 6.8% for 4 years. We need to find the monthly payment and the total finance charge for the loan using the provided formula.
Identifying Variables First, let's identify the values for each variable:
P = $15 , 000 (Principal) r = 6.8% = 0.068 (Interest rate) t = 4 years (Loan term)
We will use the formula:
M = ( 1 + 12 r ) 12 t − 1 P ( 12 r ) ( 1 + 12 r ) 12 t
Plugging in the Values Now, let's plug in the values into the formula:
M = ( 1 + 12 0.068 ) 12 × 4 − 1 15000 ( 12 0.068 ) ( 1 + 12 0.068 ) 12 × 4
Calculating Monthly Payment Let's calculate the monthly payment M :
12 r = 12 0.068 = 0.005666666666666667 1 + 12 r = 1 + 0.005666666666666667 = 1.0056666666666667 12 t = 12 × 4 = 48 ( 1 + 12 r ) 12 t = ( 1.0056666666666667 ) 48 = 1.302704230568614
M = 1.302704230568614 − 1 15000 × 0.005666666666666667 × 1.302704230568614 M = 0.302704230568614 15000 × 0.005666666666666667 × 1.302704230568614 M = 0.302704230568614 110.7839098177612 M = 365.9166395753177
The monthly payment is approximately $365.92 .
Calculating Total Amount Paid Next, we calculate the total amount paid over the loan term:
Total amount paid = M × 12 × t = 357.8034595845935 × 12 × 4 = 357.8034595845935 × 48 = 17174.56606006049
So, the total amount paid is approximately $17 , 174.57 .
Calculating Total Finance Charge Now, we calculate the total finance charge:
Total finance charge = Total amount paid - Principal Total finance charge = 17174.56606006049 − 15000 = 2174.5660600604897
So, the total finance charge is approximately $2 , 174.57 .
Final Answer Therefore, the monthly payment for the loan is approximately $357.80 , and the total finance charge for the loan is approximately $2 , 174.57 .
Examples
Understanding loan payments is crucial in personal finance. For instance, when buying a car, knowing how the interest rate, loan term, and principal affect your monthly payments and total finance charges helps you make informed decisions. By using the monthly payment formula, you can compare different loan options and choose the one that best fits your budget. This ensures you're not overpaying in interest and can manage your finances effectively.
