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Part 1
Your credit card has a balance of ​$5800 and an annual interest rate of 16​%. You decide to pay off the balance over four years. If there are no further purchases charged to the​ card, you must pay ​$164.40 each​ month, and you will pay a total interest of ​$2091.20. Assume you decide to pay off the balance over one year rather than four. How much more must you pay each month and how much less will you pay in total​ interest?
Use PMT equals StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt EndEndFraction
to determine the regular payment amount.
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Part 1
You will pay ​$

enter your response here more each month.

To solve this problem, we need to calculate the difference in monthly payments and the total interest when paying off the balance over one year rather than four years.

Given:

- Balance $ P = 5800 $
- Annual interest rate $ r = 16\% = 0.16 $
- Number of years for repayment $ t = 4 $ for the first case and $ t = 1 $ for the second case
- Number of payments per year $ n = 12 $ (monthly payments)
- Monthly payment for 4 years $ PMT_{4 \text{ years}} = 164.40 $

We will use the formula for the monthly payment of a loan:

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

### Step 1: Calculate Monthly Payment for 1 Year

Substituting the values into the formula for the 1-year scenario:

- $ P = 5800 $
- $ r = 0.16 $
- $ n = 12 $
- $ t = 1 $

The equation becomes:

\[
PMT_{1 \text{ year}} = \frac{5800 \left(\frac{0.16}{12}\right)}{1 - \left(1 + \frac{0.16}{12}\right)^{-12}}
\]

Let's calculate this.The monthly payment for paying off the balance over 1 year is approximately \$526.24. Therefore, you would need to pay about \$361.84 more each month compared to the 4-year plan.

Next, let's calculate the total interest for each case and find the difference.

### Step 2: Calculate Total Interest for Both Plans

The total amount paid over the loan duration is calculated as:

\[
\text{Total Amount Paid} = PMT \times n \times t
\]

The total interest is then:

\[
\text{Interest} = \text{Total Amount Paid} - P
\]

We'll calculate the total interest for both the 1-year and 4-year plans to determine how much less interest is paid when the balance is paid off in 1 year.The total interest paid over 1 year is approximately \$514.87. When compared to the 4-year plan, you would save about \$1576.33 in interest by paying off the balance in 1 year instead of 4 years.

### Summary:
- **Monthly payment for 1 year**: \$526.24
- **Additional monthly payment**: \$361.84 more each month
- **Total interest for 1 year**: \$514.87
- **Interest savings**: \$1576.33 less in total interest

Would you like further details or have any questions?

### Related Questions:
1. How would the monthly payment change if the interest rate increased to 18%?
2. What if you wanted to pay off the balance in 2 years instead of 4 or 1?
3. How does the length of the repayment period generally affect the total interest?
4. Can you calculate the break-even point when deciding between shorter or longer loan terms?
5. What if the monthly payment includes an additional fixed fee?

### Tip: 
Paying off debt faster reduces the amount of interest paid, but requires higher monthly payments. Balancing between affordability and interest savings is key.

Your credit card has a balance of ​$5800 and an annual interest rate of 16​%. You decide to pay off the balance over four years. If there are no further purchases charged to the​ card, you must pay ​$164.40 each​ month, and you will pay a total interest of ​$2091.20. Assume you decide to pay off the balance over one year rather than four. How much more must you pay each month and how much less will you pay in total​ interest?

Learn how to calculate the difference in monthly payments and total interest for paying off a $5800 credit card debt at a 16% annual interest rate. Compare a 1-year repayment plan to a 4-year plan, and see how much more you pay each month and how much less interest you pay in total. This solution covers loan amortization, algebra, and compound interest concepts, suitable for high school students in Grades 10-12.

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