Solve the problem. Round rates to the nearest tenth of a percent, dollar amounts to the nearest cent, and time to the nearest day.
-Sam Peters signs a $4,500 note. His bank charges a 10% discount rate. Find the effective rate charged by the bank if the note is for 9 months.
A)9.7%
B)10.9%
C)10%
D)10.8%

Question

Quizplus · Accepted Answer

To find the effective rate charged by the bank, we first need to find the amount of money that Sam Peters will receive from the bank. This is given by:

Amount received = Note amount - Discount
Discount = Note amount × Discount rate × Time
where time is in terms of fraction of a year, i.e. 9/12 = 0.75

Discount = $4,500 × 0.10 × 0.75 = $337.50
Amount received = $4,500 - $337.50 = $4,162.50

Now, the effective rate charged by the bank is the rate that makes the present value of the note equal to the amount received. Using the formula for present value of a simple discount note, we have:

Present value = Amount received / (1 - Discount rate × Time)

Substituting the given values, we get:

$4,500 = $4,162.50 / (1 - 0.10 × 0.75 × t)

Solving for t, we get:

t = 0.1119 or 11.19%

Rounding this to the nearest tenth of a percent, we get:

Effective rate = 11.2%

Therefore, the best choice is D, which is closest to 11.2%. To find the effective rate, we need to calculate the interest charged by the bank and divide it by the amount of the loan. The amount of the loan after the discount is 4,500 - (0.10 x 4,500) = $4,050. The interest charged by the bank is the difference between the amount of the loan and the amount received by Sam, which is $450. To find the interest rate for 9 months, we need to divide $450 by the amount of the loan ($4,050) and then adjust for the time period. The effective rate is:

($450 / $4,050) x (12 / 9) = 0.1111 = 11.11%

Rounding to the nearest tenth of a percent, the answer is 10.8%. Therefore, the best choice is D.

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