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.
 A: 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:


*

*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.

*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.
With these two assumptions the estimated total of patrons arriving a 7 PM is calculated as:
$$\sum \frac {a_i}{b_i}\cdot\frac NB b_i=\frac NB\sum a_i=N\frac{\sum a_i}{\sum b_i}.$$
How reasonable do you think those two assumptions are?
