Please does the following problem requires three overlapping circles or just two to display the information in a Venn diagram? I need some hints.
A stock broker was presented with the detailed samples of companies that were either in the retail sector or quoted on the Stock Exchange or both. 200 companies were studied (set C). Of these two hundred companies, 140 were retail companies (set R) and 120 were quoted on the stock exchange (set Q). Months later, 90 of these two hundred companies failed (F forms the failed companies), even though only five of these failures were not in the retail sector.
a) Depict the information in a Venn diagram
b) What is the maximum and minimum number of companies studied that failed and were retail companies but not quoted on the Stock Exchange
c) Describe and show on the Venn diagram, the following
i. Quoted retail companies that were not studied
ii. Studied companies that were quoted but did not fail
iii. Studied companies that failed but were neither quoted nor retail companies
Best Answer
You can express the various overlaps with $3$ circles, since there are three states for each of the companies (elements of $C$):
Added: To start with, we have the following diagram:
We want to find that the maximum and minimum for that unknown value $x$. Since $5$ of the $90$ failed companies were not in retail, then $90-5=85$ were in retail. That is, of the studied companies, there were $85$ that failed and were retail companies. This tells us that $0\le x\le 85$ and that:
The two unfilled regions of $F$ must add up to $5$ (since $5$ of the failed companies were not in retail). Likewise, since $85$ of the $140$ retail companies failed, then $140-85=55$ of them did not, so the two unfilled regions of $R$ must add up to $55$. Hence:
where $0\le y\le 5$ and $0\le z\le 55.$ Now, all told, there are $120$ quoted companies, and we've accounted for $85-x+z+y$ of them so far. Thus:
Now, our total is $180+x-y-z,$ so:
Now, remember that $0\le x\le 85.$ We can refine this just a bit. Of the $200$ studied companies, $85$ of them were retail companies that failed and $120$ of them were companies quoted on the stock exchange. In order to make sure that there is "enough room" for these two sets of companies, we will need at least $120+85-200=5$ companies to be retail companies that failed and were quoted on the stock exchange, meaning $85-x\ge 5,$ meaning $80\ge x.$ Hence, our bounds on $x$ are $0\le x\le 80.$ At this point, we're all set to go. We know that $0\le x\le 80,$ so we have upper and lower bounds. To see whether they are the maximum and minimum, we must see if it is possible for those bounds to be attained (that is, see if it is possible for $x=80$ or $x=0$). The substitution $x=80$ gives us:
Recall that $0\le z\le 55.$ Of the $200$ companies, $140$ are in retail and $120$ were quoted, so to make sure there is "enough room" for those sets of companies, we need at least $140+120-200=60$ retail companies that were quoted. That is, $z+5\ge 60,$ so $z\ge 55.$ Hence, we need $z=55,$ and so:
Similarly, we need at least $90+120-200=10$ companies to be quoted companies that failed, meaning $5+y\ge10,$ so $y\ge 5,$ and since $0\le y\le 5,$ then $y=5,$ so:
Therefore, $80$ is the maximum number attainable for $x$.
Now, let's check and see if $x=0$ can be obtained. Substitution gives us:
Since $y\ge 0$ and $35-y-z\ge 0,$ then $35-z\ge 35-y-z\ge 0,$ so $z\le 35,$ so our new bounds of $z$ (having specified $x$) are $0\le z\le 35.$ Likewise, $y\le 35,$ but this doesn't refine the constraints $0\le y\le 5$ that we already had. Let's just pick a value of $z$ and see what happens. Say $z=10.$ Then:
That didn't introduce any issues, either, nor did it narrow our constraints for $y$ at all, so we can still pick any $0\le y\le 5$ and get an appropriate Venn diagram. Thus, $0$ is the minimum number attainable for $x$.