Suppose a credit card was used to make a ​$12 comma 600 purchase at 14.7​% interest with a monthly payment of ​$289. Complete the following.
​(a) Calculate the time it will take to pay off the debt making only the given payment each month.
​(b) Calculate the total interest paid.
​(c) Find the amount of money saved over the lifetime of the debt if twice the amount of the given monthly payment is paid each month.

To solve this, we will use the formulas related to paying off loans, specifically those involving a fixed monthly payment on a balance with interest.

**Given Information:**
- Principal (initial balance): $ P = 12,600 \, \text{dollars} $
- Interest rate: $ r = 14.7\% $ annually or $ \frac{14.7}{12} = 1.225\% $ monthly
- Monthly payment: $ M = 289 \, \text{dollars} $

We'll use the formula for the time it takes to pay off the debt with a fixed monthly payment:

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

Where:
- $ n $ is the number of months
- $ r $ is the monthly interest rate as a decimal
- $ P $ is the principal (initial balance)
- $ M $ is the monthly payment

### (a) Calculate the time it will take to pay off the debt making only the given payment each month.

Let’s compute this now.The time it will take to pay off the debt making only the given payment of $289 each month is approximately **63 months**.

### (b) Calculate the total interest paid.

To calculate the total interest, we can use the following steps:
1. First, find the total amount paid over the 63 months, which is $ 63 \times 289 $.
2. Then, subtract the original loan amount $ P = 12,600 $ from this total.

Let’s compute the total interest now.The total interest paid over the course of the loan is approximately **$5,528.35**.

### (c) Find the amount of money saved over the lifetime of the debt if twice the amount of the given monthly payment is paid each month.

If the monthly payment is doubled to $ 2 \times 289 = 578 \, \text{dollars} $, the loan will be paid off much faster, reducing the total interest paid. We will now recalculate the time to pay off the debt with this new payment and compute the new total interest paid. Let's calculate it.If twice the monthly payment (\$578) is made each month:

- The time to pay off the debt would be approximately **26 months**.
- The total interest paid would be approximately **$2,147.97**.
- The amount saved in interest over the lifetime of the debt would be approximately **$3,380.38**.

Would you like more details on any of these calculations or further clarifications?

Here are 5 related questions for further exploration:
1. How does changing the interest rate affect the total interest paid?
2. What happens if a payment lower than the required amount is made each month?
3. How would making one extra payment per year affect the time to pay off the loan?
4. Can compounding frequency (monthly vs. yearly) impact the total cost of the loan?
5. What would be the effect of an initial lump sum payment on the overall interest paid?

**Tip**: Paying even a small extra amount toward your

Suppose a credit card was used to make a $12,600 purchase at 14.7​% interest with a monthly payment of ​$289. Complete the following. (a) Calculate the time it will take to pay off the debt making only the given payment each month. (b) Calculate the total interest paid. (c) Find the amount of money saved over the lifetime of the debt if twice the amount of the given monthly payment is paid each month.

Learn how to calculate the time required to pay off a $12,600 credit card debt at 14.7% interest with a monthly payment of $289. This solution covers key loan amortization concepts, interest calculations, and provides insights on how doubling monthly payments can save you money. Suitable for advanced high school or college students.

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