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Suppose Emily would like to purchase a car for [tex]$\$26,540$[/tex], including all taxes and fees. Emily would like to make a down payment of [tex]$\[tex]$2,600$[/tex][/tex] and then finance the remaining cost using a three-year loan at 7.3\%.

Determine the monthly payment required to repay the loan and the total interest paid over the life of the loan. Round solutions to the nearest cent, if necessary.

The monthly payment is [tex]$\$ \square$[/tex].

The total interest paid is [tex]$\[tex]$ \square$[/tex][/tex].

Hint: Related Formula
The loan payment formula for fixed installment loans is given by the expression
[tex]
PMT = \frac{P \left( \frac{r}{n} \right)}{\left[ 1 - \left(1 + \frac{r}{n}\right)^{-nt} \right]}
[/tex]
where PMT is the periodic payment required to repay a loan of [tex]$P$[/tex] dollars, paid [tex]$n$[/tex] times per year over [tex][tex]$t$[/tex][/tex] years, at an annual interest rate of [tex]$r\%$[/tex].



Answer :

GinnyAnswer
To determine the monthly payment required to repay the loan and the total interest paid, let’s follow the given loan payment formula step-by-step:

[tex]\[ \text{PMT} = \frac{P \left( \frac{r}{n} \right)}{1 - \left(1 + \frac{r}{n} \right)^{-nt}} \][/tex]

### Step 1: Identify the necessary values:
- Total cost of the car [tex]\( C = \$26,540 \)[/tex]
- Down payment [tex]\( D = \$2,600 \)[/tex]
- Annual interest rate [tex]\( r = 7.3\% = 0.073 \)[/tex]
- Number of payments per year [tex]\( n = 12 \)[/tex]
- Loan term in years [tex]\( t = 3 \)[/tex]

### Step 2: Calculate the principal amount to be financed
[tex]\[ P = C - D = 26,540 - 2,600 = 23,940 \][/tex]

### Step 3: Plug the values into the loan payment formula:
[tex]\[ P = 23,940 \][/tex]
[tex]\[ r = 0.073 \][/tex]
[tex]\[ n = 12 \][/tex]
[tex]\[ t = 3 \][/tex]

Now, compute the monthly interest rate:
[tex]\[ \frac{r}{n} = \frac{0.073}{12} \approx 0.0060833333 \][/tex]

### Step 4: Plug these values into the given formula:
[tex]\[ \text{PMT} = \frac{23,940 \times 0.0060833333}{1 - (1 + 0.0060833333)^{-36}} \][/tex]

Solving the denominator first:
[tex]\[ 1 + \frac{r}{n} = 1 + 0.0060833333 = 1.0060833333 \][/tex]
[tex]\[ (1.0060833333)^{-36} \approx 0.8066305683 \][/tex] (raising to the power of -36)

Then:
[tex]\[ 1 - 0.8066305683 = 0.1933694317 \][/tex]

Now, solving the entire fraction:
[tex]\[ \text{PMT} = \frac{23,940 \times 0.0060833333}{0.1933694317} \approx 742.49 \][/tex]

So, the monthly payment (PMT) is:
[tex]\[ \text{PMT} = \$742.49 \][/tex]

### Step 5: Calculate the total payment and the total interest paid
[tex]\[ \text{Total Payment} = \text{PMT} \times n \times t \][/tex]
[tex]\[ \text{Total Payment} = 742.49 \times 12 \times 3 = 26,729.64 \][/tex]

[tex]\[ \text{Total Interest Paid} = \text{Total Payment} - P \][/tex]
[tex]\[ \text{Total Interest Paid} = 26,729.64 - 23,940 = 2,789.49 \][/tex]

Thus:
- The monthly payment is [tex]\(\$742.49\)[/tex].
- The total interest paid over the life of the loan is [tex]\(\$2,789.49\)[/tex].

To summarize:
- The monthly payment is [tex]\(\$742.49\)[/tex].
- The total interest paid is [tex]\(\$2,789.49\)[/tex].

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