Simple interest on a principal amount P borrowed at rate r for period t is
... I = Prt
When a loan is discounted, the interest is paid out of the amount borrowed, and the borrower is given the remaining amount, the "proceeds." Then the value of the proceeds can be computed from
... proceeds = P - I = P(1 - rt)
When the time period is measured in days, "t" is a fraction, using the number of days in the year as its denominator. For "ordinary interest", that denominator is 360. For "exact interest," the denominator is 365. Here, we will assume ordinary interest.
33) Find P such that proceeds are $8305.64 when r=.09 and t=180/360.
Using the above formula
... 8305.64 = P(1 - .09×180/360) = P×0.955
... 8305.64/0.955 = P = 8697.01
The face value of Mark's loan must be $8,697.01.
(The interest amount on 8697 comes to something with half a penny. If you round the interest up before subtracting from the face value, then the proceeds come out 8305.63. If you round up the proceeds from the above formula, Mark only needs to borrow 8697.00.)
34) The above formula for proceeds can be used. The time is 28/52 of a year, since there are 52 weeks in the year.
... cost of T-bill = $100,000(1 - .056×28/52) = $96,984.62
35) We assume the "discount period" refers to the number of days remaining in the loan period. If it is something different, then adjust the calculation appropriately. When "ordinary interest" is used, the number of days in a month is presumed to be 30, so July 13 to Sept 9 represents 2 months less 4 days, or 56 days. Then the discount period would be
... 180 days - 56 days = 124 days
If "exact interest" is being used, then the number of calendar days between the given dates is 58 and the remaining period is 122 days.
36) For a given number of days, "exact interest" will result in a smaller amount of interest. Here, it would be
... 71,500×0.09×200/365 = 3526.03
instead of
... 71,500×.09×200/360 = 3575.00
Choosing exact interest on this loan will save about $48.97 in interest charges.