Find i (the rate per period) and n (the number of periods) for the loan at the given annual rate.
-Quarterly Payments of $3000 are made for 4 years to repay a loan at 10.2% compounded quarterly.
A) i = 0.102; n = 16
B) i = 0.0255; n = 16
C) i = 0.0255; n = 4
D) i = 0.006375; n = 4

Question

Quizplus · Accepted Answer

First, we need to find the periodic interest rate, which is the annual interest rate divided by the number of periods in a year. Since payments are made quarterly, there are 4 periods in a year. 
So, the periodic interest rate would be:
10.2% / 4 = 2.55%
Next, we need to convert the periodic interest rate into a decimal and use it to find the number of periods. 
i = 0.0255 (Choice B)
To find n, we can use the formula for the future value of an ordinary annuity: 
FV = Pmt * ((1 + i)^n - 1) / i
Where FV is the future value (which is the amount borrowed), Pmt is the payment amount, i is the periodic interest rate, and n is the number of periods. 
We know the payment amount is $3000 and the future value is the amount borrowed, so we can rewrite the formula as: 
$3000 * ((1 + 0.0255)^n - 1) / 0.0255 = amount borrowed
Since the loan is being fully repaid, the amount borrowed should be equal to the total of all payments made over the 4 years. 
$3000 * 16 = $48,000
So, we can solve for n: 
$3000 * ((1 + 0.0255)^n - 1) / 0.0255 = $48,000
((1 + 0.0255)^n - 1) / 0.0255 = 16
(1 + 0.0255)^n - 1 = 16 * 0.0255
(1 + 0.0255)^n = 1.408
n = log(1.408) / log(1 + 0.0255)
n = 15.93
Since n represents the number of payments, we round up to the next whole number, which is 16. Therefore, Choice B is the correct answer.

Find I (The Rate Per Period) and N (The Number