Question: Jason has $3,589.35 in credit card debt. The annual interest rate on the unpaid balance is 19.6% compounded monthly. If Jason wants to pay off his credit card debt in 2 years, what is his monthly payment, assuming he makes no additional purchases on this card? (Suggestion: Use the formula for the present value of an ordinary annuity. Round your answer to the nearest cent.)
1. **State the problem:**
Jason owes $3,589.35 on a credit card with an annual interest rate of 19.6% compounded monthly. He wants to pay off the debt in 2 years with equal monthly payments. We need to find the monthly payment amount.
2. **Formula used:**
The present value $P$ of an ordinary annuity (monthly payments) is given by:
$$P = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$
where:
- $PMT$ is the monthly payment,
- $r$ is the monthly interest rate (annual rate divided by 12),
- $n$ is the total number of payments (months).
3. **Identify values:**
- $P = 3589.35$
- Annual interest rate = 19.6%, so monthly rate $r = \frac{19.6}{100 \times 12} = 0.0163333$
- Number of months $n = 2 \times 12 = 24$
4. **Rearrange formula to solve for $PMT$:**
$$PMT = P \times \frac{r}{1 - (1 + r)^{-n}}$$
5. **Calculate denominator:**
$$1 - (1 + 0.0163333)^{-24} = 1 - (1.0163333)^{-24}$$
Calculate $(1.0163333)^{-24}$:
$$ (1.0163333)^{24} \approx e^{24 \times \ln(1.0163333)} \approx e^{24 \times 0.0162} = e^{0.3888} \approx 1.475$$
So,
$$ (1.0163333)^{-24} = \frac{1}{1.475} \approx 0.678$$
Therefore,
$$1 - 0.678 = 0.322$$
6. **Calculate $PMT$:**
$$PMT = 3589.35 \times \frac{0.0163333}{0.322} = 3589.35 \times 0.0507 = 181.94$$
7. **Final answer:**
Jason's monthly payment to pay off the debt in 2 years is approximately **$181.94**.
**Note:** The value $8719.26$ mentioned in the prompt does not relate to this calculation and is ignored here.