Fees for the late return of a rental item are equal to the normal hour

Question

Fees for the late return of a rental item are equal to the normal hourly rate on the day the item is due, charged for each hour after the deadline between the hours of 9:00 am and 5:00 pm, and three times the normal hourly rate for each hour between the hours of 9:00 am and 5:00 pm on each subsequent day. For such an item that was overdue, what were the late return fees last week?

(1) The regular hourly rental rate is $12.

(2) The item was 54 hours late.

(1) The regular hourly rental rate is $12.

(2) The item was 54 hours late.

Karmesh · Answer

Although it seems pretty straightforward, the catch here is that we are not given the deadline time.
e.g if the deadline was 3pm we would be charged the normal rate for 2 hrs and triple rate for the rest, if it was 4pm then 1hr normal...
This leads to multiple answers, Hence

Bunuel · Answer

Official Explanation

The question describes a somewhat elaborate policy for the overdue charge on a rental item. We set about to write an algebraic expression for the overage fee in terms of how many hours late the item is, but find that the expression would depend on when the item was due. Writing expressions would not be entirely possible, but algebra is not our friend on this question, so we will proceed by analysis by cases, evaluating the statements separately first.

Statement (1) gives us a piece of information that is necessary, but not sufficient, to determine the overage fees. For all we know, the thing could have been returned an hour late or a day late. Insufficient.

Statement (2), likewise, gives us necessary but insufficient info. Considered independently of Statement (1), it doesn't give us the normal hourly rate, so we can't calculate the overage fees. Insufficient.

When the statements are combined, we seem to be in the ballpark of sufficiency. We can be sure through case analysis. Case I: the item was due 9:00 am on Monday. This means that it was returned 54 = 48 + 6 hours later, so on Wednesday at 3:00 pm. That interval includes 8 working hours on Monday, 8 on Tuesday, and 6 hours on Wednesday, so 8 hours at the regular rate of $12 and 14 hours at the triple rate of $36, or 8*$12 + 14*$36. We can calculate an exact value for the late return fees. Case II: the thing was due later on Monday; say it was due 2:00 pm on Monday. In that case it was still returned 54 = 48 + 6 hours later, on Wednesday at 8:00 pm. The return fees add up to something different in this case: there are only 3 hours on the first day, at $12/hour, and 16 hours at the triple rate on Tuesday and Wednesday. Just to be quite sure, we subtract:

8*12 + 14*36
-3*12 - 16*36

Yielding

5*12 - 2*36

Which is -12, not 0, confirming the two fees are different. We have insufficient information to answer the question definitively.

The correct answer is (E).

Bunuel · Answer

Video Explanation

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bumpbot · Answer

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Fees for the late return of a rental item are equal to the normal hour

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Fees for the late return of a rental item are equal to the normal hour

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