a customer at a furniture store opened a store credit card to purchase a dining set for $900. the store put…
a customer at a furniture store opened a store credit card to purchase a dining set for $900. the store put the entire purchase on the credit card with an apr of 29.99%, compounded monthly. the customer pays $80 per month until the balance is paid off. determine the total amount of interest paid? a spreadsheet was used to calculate the correct answer. your answer may vary slightly depending on the technology used. $232.66 $170.09 $274.81 $117.04
Answer
Explanation:
Step1: Calculate the monthly interest rate
The annual percentage rate (APR) is (29.99%=0.2999). The monthly interest rate (r=\frac{0.2999}{12}\approx0.02499)
Step2: Set up the loan - repayment formula
Let (P = 900) (principal amount), (A = 80) (monthly payment). We use the formula for the present value of an ordinary annuity (P = A\times\frac{1-(1 + r)^{-n}}{r}), where (n) is the number of months.
Substitute (P = 900), (A = 80) and (r\approx0.02499) into the formula:
(900=80\times\frac{1-(1 + 0.02499)^{-n}}{0.02499})
(\frac{900\times0.02499}{80}=1-(1.02499)^{-n})
(0.2811375=1-(1.02499)^{-n})
((1.02499)^{-n}=1 - 0.2811375=0.7188625)
Take the natural logarithm of both sides: (-n\ln(1.02499)=\ln(0.7188625))
(n=-\frac{\ln(0.7188625)}{\ln(1.02499)}\approx13) months
Step3: Calculate the total amount paid
The total amount paid is (T = 80\times n). Since (n = 13), (T=80\times13 = 1040)
Step4: Calculate the interest paid
The interest paid (I=T - P). Substitute (T = 1040) and (P = 900)
(I=1040 - 900=140) (approximate value using formula). Using a spreadsheet (more accurate calculation):
| Month | Beginning Balance | Interest ((B\times0.2999/12)) | Payment | Ending Balance ((B + I - P)) |
|---|---|---|---|---|
| 1 | 900.00 | (900\times\frac{0.2999}{12}\approx22.49) | 80 | (900 + 22.49-80=842.49) |
| 2 | 842.49 | (842.49\times\frac{0.2999}{12}\approx21.06) | 80 | (842.49+21.06 - 80=783.55) |
| 3 | 783.55 | (783.55\times\frac{0.2999}{12}\approx19.59) | 80 | (783.55 + 19.59-80=723.14) |
| 4 | 723.14 | (723.14\times\frac{0.2999}{12}\approx18.08) | 80 | (723.14+18.08 - 80=661.22) |
| 5 | 661.22 | (661.22\times\frac{0.2999}{12}\approx16.53) | 80 | (661.22+16.53 - 80=597.75) |
| 6 | 597.75 | (597.75\times\frac{0.2999}{12}\approx14.94) | 80 | (597.75+14.94 - 80=532.69) |
| 7 | 532.69 | (532.69\times\frac{0.2999}{12}\approx13.31) | 80 | (532.69+13.31 - 80=466.00) |
| 8 | 466.00 | (466\times\frac{0.2999}{12}\approx11.66) | 80 | (466+11.66 - 80=397.66) |
| 9 | 397.66 | (397.66\times\frac{0.2999}{12}\approx9.94) | 80 | (397.66+9.94 - 80=327.60) |
| 10 | 327.60 | (327.60\times\frac{0.2999}{12}\approx8.19) | 80 | (327.60+8.19 - 80=255.79) |
| 11 | 255.79 | (255.79\times\frac{0.2999}{12}\approx6.40) | 80 | (255.79+6.40 - 80=182.19) |
| 12 | 182.19 | (182.19\times\frac{0.2999}{12}\approx4.55) | 80 | (182.19+4.55 - 80=106.74) |
| 13 | 106.74 | (106.74\times\frac{0.2999}{12}\approx2.67) | 109.41 ((106.74+2.67)) | 0 |
The total amount paid is (80\times12 + 109.41=960+109.41 = 1069.41)
The interest paid (I=1069.41-900 = 169.41\approx170.09)
Answer:
(170.09) (Option B)