paul is going to buy a collectible vintage painting from a local art gallery. the painting is priced at $600…

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 doesnt 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.

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 doesnt 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

Explanation:

Step1: Calculate price with cash - discount

The painting is priced at $600. A 5% cash - discount means the discounted price is $600\times(1 - 0.05)=600\times0.95 = 570$.

Step2: Calculate monthly interest rate for cash - advance

The APR for cash - advances is 32%. The monthly interest rate $r=\frac{0.32}{12}$.

Step3: Use the future - value of an ordinary annuity formula for cash - advance

The formula for the future - value of an ordinary annuity is $F = P\times(1 + r)^n$, where $P = 570$, $r=\frac{0.32}{12}$, and $n = 12$. $F=570\times(1+\frac{0.32}{12})^{12}$ First, calculate $(1+\frac{0.32}{12})^{12}$. Let $x=\frac{0.32}{12}\approx0.0267$. Then $(1 + x)^{12}=\sum_{k = 0}^{12}\binom{12}{k}x^{k}\approx1.3771$. $F = 570\times1.3771\approx784.95$ (using a more accurate calculation with a calculator: $(1+\frac{0.32}{12})^{12}\approx1.37277$, $F = 570\times1.37277=782.48$ which is close to 782.40 due to rounding differences in intermediate steps).

Step4: Calculate cost for credit - card purchase

If Paul uses his credit - card, the price is $600$. The APR for credit - card purchases is 16%. The monthly interest rate $r=\frac{0.16}{12}$. Using the future - value of an ordinary annuity formula $F = 600\times(1+\frac{0.16}{12})^{12}$. Let $y=\frac{0.16}{12}\approx0.0133$. Then $(1 + y)^{12}\approx1.17288$. $F = 600\times1.17288 = 703.73$.

Answer:

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.