Suppose a credit card was used to make a ​$7800 purchase at 17.4​% interest with a monthly payment of ​$187. 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 problem, we’ll use the amortization formula for credit card debt, where payments are made monthly and a fixed interest rate is applied. Here are the steps for each part of the question.

### Given Information
- **Principal (P)**: $7,800
- **Annual Interest Rate (APR)**: 17.4%
- **Monthly Payment**: $187

The monthly interest rate is calculated as:
\[
\text{Monthly Interest Rate} = \frac{17.4\%}{12} = \frac{0.174}{12} \approx 0.0145 \text{ or } 1.45\% \text{ per month}
\]

### Part (a): Calculate the Time to Pay Off the Debt

To calculate the time $ n $ (in months) to pay off the debt, we use the formula:
\[
n = \frac{\ln \left( \frac{M}{M - P \cdot r} \right)}{\ln(1 + r)}
\]
where:
- $ M $ is the monthly payment ($187),
- $ P $ is the principal ($7,800),
- $ r $ is the monthly interest rate (0.0145).

Let's substitute these values into the formula.

### Part (b): Calculate the Total Interest Paid

After calculating $ n $, we can determine the total interest paid:
\[
\text{Total Interest} = (M \cdot n) - P
\]
where $ M \cdot n $ is the total amount paid over all months, and $ P $ is the original loan amount.

### Part (c): Calculate Savings with Double Monthly Payments

If we double the monthly payment to $ 2 \times 187 = 374 $ and recalculate the time to pay off the debt, we’ll be able to determine the amount saved. 

To find the savings:
1. Calculate the new repayment time $ n' $ with a payment of $374.
2. Find the new total interest paid with the higher payment.
3. The savings are then the difference in total interest between the original and doubled payment plans.

Let me calculate these parts step-by-step.### Solution Summary:

1. **Part (a): Time to Pay Off Debt with $187 Monthly Payment**
   - It will take approximately **64.49 months** (about 5 years and 5 months) to pay off the debt with the given monthly payment of $187.

2. **Part (b): Total Interest Paid**
   - The total interest paid over this time will be **$4,259.70**.

3. **Part (c): Savings with Double Monthly Payment**
   - If the monthly payment is doubled to $374, it will take **25.02 months** (about 2 years and 1 month) to pay off the debt.
   - The total interest paid with the doubled payment will be **$1,555.76**.
   - **Savings** in interest by paying double would be **$2,703.94**.

Would you like more detailed calculations, or have any specific questions?

### Related Questions:
1. How would a lower interest rate affect the time to pay off the debt?
2. What if the payment amount was increased by only $50 each month?
3. How does credit card interest typically compound, and why is it costly?
4. What other factors could help reduce interest payments on debt?
5. How can one calculate monthly payments for other loan types, like mortgages?

### Tip:
Paying more than the minimum on high-interest debt significantly reduces both the interest paid and the payoff time.

Suppose a credit card was used to make a $7800 purchase at 17.4% interest with a monthly payment of $187. 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.

Calculate the time to pay off a $7,800 credit card debt with a 17.4% interest rate and a monthly payment of $187. Learn how to determine the total interest paid and potential savings when doubling the monthly payment, using amortization and interest formulas. Ideal for Grades 11-12 math students.

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