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Katelyn plans to apply for a [tex]$\$[/tex]10,000[tex]$ loan at an interest rate of $[/tex]5.6\%[tex]$ for 5 years. Use the monthly payment formula to complete the statement.

\[
M = \frac{P \cdot r \cdot (1 + r)^{n}}{(1 + r)^{n} - 1}
\]

Where:
- $[/tex]M[tex]$ = monthly payment
- $[/tex]P[tex]$ = principal
- $[/tex]r[tex]$ = monthly interest rate
- $[/tex]n[tex]$ = number of payments (months)

Rounded to the nearest cent, Katelyn's monthly payment for the loan is \$[/tex] [tex]$\square$[/tex].


Sagot :

To find Katelyn's monthly payment for her loan, we will use the monthly payment formula for a fixed-rate mortgage or installment loan:

[tex]\[ M = \frac{P \cdot \left(\frac{1}{1}\right) \left(1 + \frac{t}{2}\right)^{12}}{\left(1 + \frac{1}{2}\right)^{2 \pi} - 1} \][/tex]

Given the values:
- Principal amount [tex]\( P = 10{,}000 \)[/tex] dollars
- Annual interest rate [tex]\( r = 5.6\% \)[/tex]
- Loan term [tex]\( t = 5 \)[/tex] years

First, convert the interest rate to a monthly rate in decimal form:
[tex]\[ r \text{ (annual)} = 0.056 \][/tex]
[tex]\[ \text{monthly interest rate} = \frac{r}{12} = \frac{0.056}{12} \approx 0.00467 \][/tex]

Next, calculate the total number of monthly payments:
[tex]\[ n = t \times 12 = 5 \times 12 = 60 \][/tex]

Now, use the monthly payment formula:
[tex]\[ M = P \cdot \frac{r \cdot (1 + r)^n}{(1 + r)^n - 1} \][/tex]

Substitute the given values into the formula:
[tex]\[ M = \frac{10{,}000 \times 0.00467 \times (1 + 0.00467)^{60}}{(1 + 0.00467)^{60} - 1} \][/tex]

Upon calculation, the monthly payment is found and rounded to the nearest cent:
[tex]\[ M \approx 191.47 \][/tex]

Therefore, Katelyn's monthly payment for the loan, rounded to the nearest cent, is \$ 191.47.