Your credit card has a balance of ​$2600 and an annual interest rate of 14​%. You decide to pay off the balance over three years. If there are no further purchases charged to the​ card, you must pay ​$88.84 each​ month, and you will pay a total interest of ​$598.24. Assume you decide to pay off the balance over one year rather than three. How much more must you pay each month and how much less will you pay in total​ interest?
Use PMT equals StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt EndEndFraction
to determine the regular payment amount.
Question content area bottom
Part 1
You will pay ​$

enter your response here more each month.
​(Round to the nearest cent as​ needed.)

To solve this, we need to determine how much more you would pay each month and the reduction in total interest when you decide to pay off the balance in one year instead of three years. We'll use the PMT formula:

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

Where:
- $ P = 2600 $ (principal or balance)
- $ r = 0.14 $ (annual interest rate)
- $ n = 12 $ (number of payments per year)
- $ t $ is the time in years (1 year and 3 years for comparison)

---

### Step 1: Calculate the monthly payment for 1 year ($ t = 1 $):
\[
PMT = \frac{2600 \cdot \left(\frac{0.14}{12}\right)}{1 - \left(1 + \frac{0.14}{12}\right)^{-12}}
\]

### Step 2: Calculate the monthly payment for 3 years ($ t = 3 $):
\[
PMT = \frac{2600 \cdot \left(\frac{0.14}{12}\right)}{1 - \left(1 + \frac{0.14}{12}\right)^{-36}}
\]

We'll compute these values.### Results:
1. **Monthly Payment for 1 Year:** \$233.45
2. **Monthly Payment for 3 Years:** \$88.86
3. **Additional Monthly Payment:** \$144.58

---

### Step 3: Calculate the total interest savings.
For this, we need to calculate the total payments for both plans and subtract the principal to find the interest paid.### Results:
1. **Total Interest Paid for 1 Year:** \$201.36
2. **Total Interest Paid for 3 Years:** \$599.03
3. **Interest Savings by Paying in 1 Year:** \$397.67

---

### Final Answer:
- You will pay **\$144.58 more each month**.
- You will pay **\$397.67 less in total interest**.

Would you like a detailed explanation of the calculations or have any specific questions?  
Here are 5 related questions you might explore:  

1. How does the interest rate affect the total interest paid over time?  
2. What would the monthly payments be if the balance were higher, such as \$5000?  
3. How is the formula affected if payments are made biweekly instead of monthly?  
4. How can extra payments impact the overall interest savings?  
5. How does changing the term length (e.g., 5 years) influence the monthly payments and interest?  

**Tip:** Paying off debts faster not only saves on interest but also improves your credit score by reducing credit utilization.

Your credit card has a balance of $2600 and an annual interest rate of 14%. You decide to pay off the balance over three years. If there are no further purchases charged to the card, you must pay $88.84 each month, and you will pay a total interest of $598.24. Assume you decide to pay off the balance over one year rather than three. How much more must you pay each month and how much less will you pay in total interest?

Explore how to calculate the difference in monthly payments and total interest when paying off a $2600 credit card balance at 14% interest over one year versus three years. Understand the PMT formula and how faster repayment reduces interest costs, suitable for students in Grades 11-12.

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