Your credit card has a balance of

​$4100

and an annual interest rate of

18

​%.

You decide to pay off the balance over

five

years. If there are no further purchases charged to the​ card, you must pay

​$104.11

each​ month, and you will pay a total interest of

​$2146.60

.

Assume you decide to pay off the balance over one year rather than

five

.

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.

Let's analyze the problem step by step using the given formula for monthly payments on a loan or credit card balance. The formula for the monthly payment is:

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

where:
- $ P $ is the principal balance ($4100 in this case),
- $ r $ is the annual interest rate (18%, or 0.18 as a decimal),
- $ n $ is the number of compounding periods per year (12 for monthly payments),
- $ t $ is the number of years (5 years and then 1 year for comparison).

We'll calculate the monthly payment ($PMT$) for both scenarios (5 years and 1 year) and then find out how much more you must pay each month and how much less you will pay in total interest.

### Step 1: Calculate the monthly payment for 1 year

Let's use the formula to find the payment over 1 year, then calculate the total amount paid and the interest paid for both 1 year and 5 years. Finally, we'll find the differences.The monthly payment if you pay off the balance over 1 year will be approximately **$375.89**. 

This means you will have to pay **$271.77** more each month compared to the 5-year plan.

Additionally, you will pay **$1,736.13** less in total interest by choosing the 1-year repayment plan over the 5-year plan.

Do you want more details or have any questions?

### Here are 5 related questions:
1. How does changing the interest rate affect the monthly payment amount?
2. What happens to the total interest paid if the repayment period is extended further?
3. How can you calculate the total interest saved by making higher payments?
4. What would the monthly payment be for a 3-year repayment period?
5. How does compounding frequency (e.g., monthly vs. yearly) affect the total interest?

### Tip
Always aim to reduce your debt faster if possible, as this significantly lowers the total interest paid over time.

Your credit card has a balance of $4100 and an annual interest rate of 18%. You decide to pay off the balance over five years. If there are no further purchases charged to the card, you must pay $104.11 each month, and you will pay a total interest of $2146.60. Assume you decide to pay off the balance over one year rather than five. How much more must you pay each month and how much less will you pay in total interest? Use the formula PMT = (P * (r / n)) / (1 - (1 + (r / n))^(-nt)) to determine the regular payment amount.

Explore how much more you need to pay monthly and how much interest you can save by paying off a $4100 credit card balance over one year instead of five, using an 18% annual interest rate. Learn about the loan amortization formula and interest calculation.

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