Question

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 cash advances. Assuming Paul wants to pay the painting off over 12 months, which of the following is true?
a.
Paul will pay a total of $696, over 12 months, if he purchases the painting with his credit card.
b.
Paul will pay a total of $653.28, over 12 months, if he purchases the painting with his credit card.
c.
Paul will pay a total of $782.40, over 12 months, if he purchases the painting with a cash advance for discounted purchase price.
d.
Paul will pay a total of $708.96, over 12 months, if he purchases the painting with a cash advance for discounted purchase price.

Answer 1

Use the usual loan/amortization formula for equal monthly payments: payment = P·r / (1 − (1+r)^−n), with r = APR/12 and n = 12.

1) Charge $600 at 16% APR:
- r = 0.16/12 = 0.0133333333
- monthly payment ≈ 600·0.01333333 / (1 − (1.01333333)^−12) ≈ $54.37
- total paid ≈ 54.37·12 ≈ $652.47

2) Cash advance of $570 (5% discount) at 32% APR:
- r = 0.32/12 = 0.0266666667
- monthly payment ≈ 570·0.02666667 / (1 − (1.02666667)^−12) ≈ $56.10
- total paid ≈ 56.10·12 ≈ $673.22

So paying with a normal credit-card purchase is cheaper. Among the given choices, (b) ($653.28) is the one that matches the correct conclusion (it’s the closest to the correctly computed ~$652–653 total for a credit-card purchase). Thus answer: b.