# Fraction / Percentages Brainteaser

In my business math class, the instructor put up the following case study, and asked us to find any flaws in the logic, but, I have been unable to see what is wrong with the following argument:

"A restaurant manager runs a restaurant, and records the total number of people that show up each day, and the number of people that show up at 7:00 PM each day. On Monday at 7:00 PM, 100/200 total people showed up (where 200 is the total number of people all day), Tuesday at 7:00: 150/300 total people showed, Wed. at 7:00: 140/400 people showed up, Thursday at 7:00: 80/250 total people showed up, Friday at 7:00: 150/250 people showed up, Saturday at 7:00: 150/300 people showed up, and Sunday at 7:00: 50/150 people showed up.

Based on this, the restaurant manager concluded that for the week at 7:00 PM a total of $\frac{100+150+140+80+150+150+50}{200+300+400+250+250+300+150} = \frac{820}{1850} = 44.32\%$ of his restaurant's customers show up 7:00 PM. He then uses this number to estimate that in a future week, the estimated attendance for the whole week (all times) is 5000 people. He expects $0.4432 \times 5000 = 2216$ people to show up at 7:00 in total for all days.

The professor said that the flaw is not the obvious one that the $44.32\%$ figure cannot be expected to hold every week. He said there is something deeper that the manager is missing, and I am just not seeing it.

Any ideas on this extremely strange question? Thanks.

• tag with self study so readers know to give guidance appropriately as opposed to listing the answer. It would also help to know what your thoughts currently are. It sounds like you think the assumptions are reasonable. It would help if you could explain what you understand to be the assumptions the manger made. Then explain why you think they are reasonable. This may help you answer the question yourself, but if not then readers will know how to structure their response. May 1 '18 at 9:09
• This question is loaded with ambiguities. Could you clarify what it means for "a/b total people showed up" at a particular time? How did the manager "use this number" (of 44.32%, presumably) to estimate the total attendance in future weeks? Why does this number vary so greatly from the actual observed total of 1850 during one week?
– whuber
May 1 '18 at 12:26
• Hi @whuber Now you know how I feel! I made a slight edit to the question: when I say a/b total people showed up, a = number of people at 7:00 in the particular day, b = total number of people for the whole day. We are not told how the manager estimated total attendance in future weeks, it is simply given. We were told it doesn't have anything to do with the current problem. May 1 '18 at 15:07

Using your notation, let $b_1, b_2,\ldots, b_7$ be the number of patrons for Monday, Tuesday, ... , Sunday that the manager recorded for the prior week, and let $a_1, a_2,\ldots, a_7$ be the respective number of patrons that showed up at 7 PM. The manager is making two assumptions in his estimate for a future week in which $N$ patrons will show up:
1. The allocation of those $N$ patrons over the future seven days is proportional to the observed $b_1, b_2,\ldots,b_7$: He expects (roughly) $\frac NB b_i$ people to show up on day $i$, where $B:=\sum b_i$ is the total number observed to show up that prior week.
2. The fraction of patrons on day $i$ who arrive at 7 PM is the same fraction as observed the prior week: He expects (roughly) a fraction $a_i/b_i$ of the future day $i$ arrivals to show up at 7 PM.
$$\sum \frac {a_i}{b_i}\cdot\frac NB b_i=\frac NB\sum a_i=N\frac{\sum a_i}{\sum b_i}.$$