Paul is going to buy a collectible vintage painting from a local art gallery The painting is priced at \( \$ 600 \) in the gallery The gallery owner does accept credit cards but prefers cash. In fact, he offers to give Paul a \( 5 \% \) discount if he can pay in cash Paul doesn't have any cash but can get a cash advance on his credit card. His credit card has an APR of \( 16 \% \) on credit purchases and a \( 32 \% \) APR on Lash advances. Assumirig Paul wents to pay the painting off over 12 monilhs, which of the following is true? a Paul will pay a total of \( \$ 696 \), over 12 months, it he purchases the painting with his credit card b. Paul will pay a total of \( \$ 853.28 \), over 12 months, if he purchases the painting with his credit cand c Paul will pay a total of \( \$ 78240 \), over 12 months, if he purchases the painting with a cash advance lor discountcd purchase price. d Paul wil pay a total of \( \$ 708.90 \), over 12 months, it he purchases the painting wilt a cash advance for discounted purchasi price

To solve this problem, we need to calculate the total amount Paul will pay in each scenario. Let's break it down step by step. ### Step 1: Calculate the discounted price if paid in cash The original price of the painting is \( P = 600 \). The gallery offers a \( 5\% \) discount for cash payments. \[ \text{Discount} = 0.05 \times P = 0.05 \times 600 = 30 \] \[ \text{Discounted Price} = P - \text{Discount} = 600 - 30 = 570 \] ### Step 2: Calculate the total payment if using a credit card without cash advance The APR for credit purchases is \( 16\% \). To find the monthly interest rate, we divide by \( 12 \): \[ \text{Monthly Interest Rate} = \frac{0.16}{12} = \frac{0.16}{12} \approx 0.01333 \] Using the formula for the total payment over 12 months for a loan, we can calculate the total amount paid: \[ \text{Total Payment} = P \times (1 + r)^n \] where \( r \) is the monthly interest rate and \( n \) is the number of months. ### Step 3: Calculate the total payment if using a cash advance The APR for cash advances is \( 32\% \). The monthly interest rate is: \[ \text{Monthly Interest Rate} = \frac{0.32}{12} = \frac{0.32}{12} \approx 0.02667 \] The total payment for the discounted price of \( 570 \) over 12 months is calculated similarly: \[ \text{Total Payment} = \text{Discounted Price} \times (1 + r)^n \] Now, let's calculate the total payments for both scenarios. ### Calculations 1. Total payment using credit card (without cash advance): - \( P = 600 \) - \( r = 0.01333 \) - \( n = 12 \) 2. Total payment using cash advance (with discounted price): - \( P = 570 \) - \( r = 0.02667 \) - \( n = 12 \) Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(570\left(1+0.02667\right)^{12}\) - step1: Add the numbers: \(570\times 1.02667^{12}\) - step2: Convert the expressions: \(570\left(\frac{102667}{100000}\right)^{12}\) - step3: Simplify: \(570\times \frac{102667^{12}}{100000^{12}}\) - step4: Rewrite the expression: \(10\times 57\times \frac{102667^{12}}{100000^{12}}\) - step5: Rewrite the expression: \(10\times 57\times \frac{102667^{12}}{10^{60}}\) - step6: Reduce the numbers: \(57\times \frac{102667^{12}}{10^{59}}\) - step7: Multiply: \(\frac{57\times 102667^{12}}{10^{59}}\) Calculate or simplify the expression \( 600 * (1 + 0.01333)^12 \). Calculate the value by following steps: - step0: Calculate: \(600\left(1+0.01333\right)^{12}\) - step1: Add the numbers: \(600\times 1.01333^{12}\) - step2: Convert the expressions: \(600\left(\frac{101333}{100000}\right)^{12}\) - step3: Simplify: \(600\times \frac{101333^{12}}{100000^{12}}\) - step4: Rewrite the expression: \(200\times 3\times \frac{101333^{12}}{100000^{12}}\) - step5: Rewrite the expression: \(200\times 3\times \frac{101333^{12}}{200^{12}\times 500^{12}}\) - step6: Reduce the numbers: \(3\times \frac{101333^{12}}{200^{11}\times 500^{12}}\) - step7: Multiply the fractions: \(\frac{3\times 101333^{12}}{200^{11}\times 500^{12}}\) ### Step 4: Results of the calculations 1. **Total payment using credit card (without cash advance)**: \[ \text{Total Payment} \approx 703.33 \] 2. **Total payment using cash advance (with discounted price)**: \[ \text{Total Payment} \approx 781.71 \] ### Step 5: Compare with the options given Now, let's compare these results with the options provided: a. Paul will pay a total of \( \$ 696 \) over 12 months if he purchases the painting with his credit card. **(Not true, as he pays approximately \( 703.33 \))** b. Paul will pay a total of \( \$ 853.28 \) over 12 months if he purchases the painting with his credit card. **(Not true, as he pays approximately \( 703.33 \))** c. Paul will pay a total of \( \$ 782.40 \) over 12 months if he purchases the painting with a cash advance for the discounted purchase price. **(Not true, as he pays approximately \( 781.71 \))** d. Paul will pay a total of \( \$ 708.90 \) over 12 months if he purchases the painting with a cash advance for the discounted purchase price. **(Not true, as he pays approximately \( 781.71 \))** ### Conclusion None of the options provided are correct based on the calculations. The correct amounts are approximately \( 703.33 \) for the credit card purchase and \( 781.71 \) for the cash advance on the discounted price.