To solve this problem, we need to break it down into steps based on the information provided. This is a common credit card debt problem where we use the formula for amortization, particularly for paying off a loan or a debt with fixed monthly payments.

Given Information:

• Initial debt (principal), $P = 7900$ dollars
• Annual interest rate, $r = 16.7\%$
• Monthly payment, $M = 196$ dollars
• The interest is compounded monthly, so the monthly interest rate $r_m = \frac{16.7\%}{12} = \frac{0.167}{12} \approx 0.01392$ (monthly rate)

We will now go through the three parts of the question:

(a) Calculate the time to pay off the debt with a $196 monthly payment.

The time to pay off the debt can be found using the following formula:

$n = \frac{\log\left( \frac{M}{M - P \cdot r_m} \right)}{\log(1 + r_m)}$

Where:

• $n$ is the number of months it will take to pay off the debt.
• $M$ is the monthly payment.
• $P$ is the principal (the original amount of the loan).
• $r_m$ is the monthly interest rate.

(b) Calculate the total interest paid.

The total interest paid can be calculated as:

$\text{Total interest} = (n \times M) - P$

Where $n$ is the number of months found in part (a).

(c) Find the amount saved if twice the monthly payment is made.

If the monthly payment is doubled to $2M$, the time to pay off the debt will decrease. The new time to pay off the debt can be calculated using the same formula for $n$ but replacing $M$ with $2M$:

$n_{\text{double}} = \frac{\log\left( \frac{2M}{2M - P \cdot r_m} \right)}{\log(1 + r_m)}$

The amount saved over the lifetime of the debt will be the difference in total interest paid when making the original payment vs. the doubled payment:

$\text{Amount saved} = \left( n \times M \right) - \left( n_{\text{double}} \times 2M \right)$

Let me calculate these values for you.Here are the results:

(a) Time to pay off the debt:

It will take approximately 59.55 months (or about 5 years) to pay off the debt with a $196 monthly payment.

(b) Total interest paid:

The total interest paid over the lifetime of the debt will be $3,772.73.

(c) Amount saved by doubling the payment:

If you double the monthly payment to $392, you will save $2,337.04 over the lifetime of the debt.

Would you like more details or have any questions?

Here are 5 related questions you might be interested in:

1. How does increasing the monthly payment affect the time to pay off the debt?
2. How can you calculate the effective interest rate on a loan?
3. What would happen if the interest rate was lower or higher?
4. How does compound interest work in loan repayment?
5. What strategies can help pay off credit card debt faster?

Tip: Paying even a small additional amount each month can significantly reduce the total interest paid on a loan.