advita wrote:At a certain company, each employee has either an office or a cubicle. If each office or cubicle is occupied by exactly one employee, what percent of the employees have offices?
(1) The employees outnumber the cubicles 3 to 2.
(2) There are 50 offices.
Hi there,
Beautiful problem, beautiful discussion, beautiful (and correct) arguments, anshumishra´s in particular on the "empty offices" possibility.
Let me give you another alternative, pretty quick, for the solution but, before that, let me say something:
If the question stem were:
At a certain company, each employee has either an office or a cubicle. If each office AND each cubicle is occupied by exactly one employee
many students would be messed-up, but mathematically speaking, we would guarantee that no place was left unoccupied, no people would be without place to sit, and it would have one person per place (bijection).
Why?
I. "Each employee has either an office or a cubicle" means that no employee is left without place to sit.
II. "If each office AND each cubicle is occupied..." means that no office AND no cubicle was left without one person there.
All that understood, I really believe that the person who wrote the problem simply WRONGLY chose OR in the place of AND, that´s why you all got discussing. The fact is, "by luck" the question itself was not related to that, as anshumishra wisely explained. (The example was the "smoking gun" for sure, LOL.)
Well, now the problem itself, the way I see it!
First: let us call the number of employees as "E".
We know that alfa*E is the fraction of E that sits in an office and (1-alfa)*E is the fraction of E that sits in a cubicle, where 0<= alfa <= 1 is a real constant that makes the place for the fraction we are looking for!
(You see: we are not dealing with the first problem discussed, as anshumishra noticed. In both cases, this way of modelling the problem suits because all employees "sit", that is all that matter!)
Focus is on ALPHA, ok?!
(1) E over cubicles = 3:2 means "for every 3 employees, 2 sit in cubicles" (on average, I know) therefore "for every 3 employees, 1 sit in office" (again), therefore offices (occupied by employees)/cubicles is 1:2 and offices (occupied by employees)/E = 1/3 and that means alfa = 1/3.
(2) we know that alpha*E = 50 (suppose all offices are occupied, for instance), therefore:
If E = 100, then alpha = 1/2
If E = 200, then alpha = 1/4
I hope you like it all.
Regards,
Fabio.