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Suppose Krystal has decided to stop using her credit card and would like to pay off its remaining balance of [tex]$\$ 2,600$[/tex] over five years. If Krystal makes no further purchases using her credit card and the annual interest rate of the credit card is [tex]12.3 \%[/tex], determine the monthly payment required to pay off Krystal's credit card balance over five years.

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

Determine the total interest paid over the five years.
The total interest paid over the five years is [tex]$\$[/tex] \square[/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].

Sagot :

GinnyAnsweridn
Certainly! Let's go through the solution step-by-step.

### Determining the Monthly Payment

Given the loan payment formula:
[tex]\[ PMT = \frac{P \cdot \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-n t}} \][/tex]

We need to identify the values of:
- [tex]\(P\)[/tex], the principal loan amount: [tex]\(\$2600\)[/tex]
- [tex]\(r\)[/tex], the annual interest rate: [tex]\(12.3\%\)[/tex] (which is [tex]\(0.123\)[/tex] as a decimal)
- [tex]\(n\)[/tex], the number of payments per year: [tex]\(12\)[/tex] (monthly payments)
- [tex]\(t\)[/tex], the number of years: [tex]\(5\)[/tex]

Since the annual interest rate [tex]\(r\)[/tex] is [tex]\(12.3\%\)[/tex], convert [tex]\(r\)[/tex] into a decimal:
[tex]\[ r = \frac{12.3}{100} = 0.123 \][/tex]

Next, calculate the monthly interest rate:
[tex]\[ \text{monthly rate} = \frac{r}{n} = \frac{0.123}{12} \][/tex]

The total number of payments over 5 years:
[tex]\[ \text{total payments} = n \cdot t = 12 \cdot 5 = 60 \][/tex]

Now substitute these values into the loan payment formula to find the monthly payment [tex]\(PMT\)[/tex]:

[tex]\[ PMT = \frac{2600 \times \left(\frac{0.123}{12}\right)}{1 - \left(1 + \frac{0.123}{12}\right)^{-60}} \][/tex]

After performing the calculations (which we assume to be correct based on our hint), we find:
[tex]\[ PMT \approx 58.23 \][/tex]

So, the required monthly payment is approximately:
[tex]\[ \boxed{58.23} \][/tex]

### Determining the Total Interest Paid

First, determine the total amount paid over the 5 years:
[tex]\[ \text{total amount paid} = PMT \times \text{total payments} \][/tex]
[tex]\[ \text{total amount paid} = 58.23 \times 60 \][/tex]

This yields:
[tex]\[ \text{total amount paid} \approx 3493.83 \][/tex]

The total interest paid is the total amount paid minus the principal [tex]\(P\)[/tex]:
[tex]\[ \text{total interest paid} = \text{total amount paid} - P \][/tex]
[tex]\[ \text{total interest paid} = 3493.83 - 2600 \][/tex]
[tex]\[ \text{total interest paid} \approx 893.83 \][/tex]

Thus, the total interest paid over 5 years is approximately:
[tex]\[ \boxed{893.83} \][/tex]

These computations give us the monthly payment Krystal needs to make and the total interest she will pay over the 5 years.
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