Question: A landscape architect accrued $7,862.57 in credit card debt. If the architect makes a monthly payment of $400 (and makes no additional charges on the account), how many months will it take to repay the debt if the annual interest rate on the credit card is 18.6%? (Round your answer up to the nearest month.)
1. **State the problem:**
We need to find the number of months it will take to repay a credit card debt of $7,862.57 with monthly payments of $400 and an annual interest rate of 18.6%.
2. **Identify the formula:**
This is a loan amortization problem where the debt decreases with monthly payments and accrues monthly interest.
The monthly interest rate $i$ is given by:
$$i = \frac{18.6}{100 \times 12} = 0.0155$$
The formula for the number of months $n$ to pay off a loan with principal $P$, monthly payment $M$, and monthly interest rate $i$ is:
$$n = \frac{\log\left(\frac{M}{M - iP}\right)}{\log(1 + i)}$$
3. **Calculate monthly interest rate:**
$$i = \frac{18.6}{100 \times 12} = 0.0155$$
4. **Plug in values:**
$$P = 7862.57, \quad M = 400$$
Calculate the numerator:
$$\frac{M}{M - iP} = \frac{400}{400 - 0.0155 \times 7862.57} = \frac{400}{400 - 121.80} = \frac{400}{278.20} \approx 1.4377$$
Calculate the logarithms:
$$\log(1.4377) \approx 0.1573$$
$$\log(1 + 0.0155) = \log(1.0155) \approx 0.0067$$
5. **Calculate $n$:**
$$n = \frac{0.1573}{0.0067} \approx 23.48$$
6. **Interpretation:**
It will take approximately 23.48 months to repay the debt.
Since the problem asks to round up to the nearest month:
$$n = 24 \text{ months}$$
7. **Conclusion:**
The architect will need to make payments for 24 months to fully repay the debt.
**Note:** The answer 9 months is incorrect based on the calculation with interest.