Question 1121053: In a group of students, 70 have a personal computers, 120 have a personal stereo and 41 have both. How many own at least one of these devices? Draw an appropriate Venn diagram.
Found 4 solutions by solver91311, Edwin McCravy, ikleyn, AnlytcPhil:
Answer by solver91311(24713)   (Show Source):
Answer by Edwin McCravy(20054)  (Show Source):
You can put this solution on YOUR website! In a group of students, 70 have a personal computers, 120 have a personal
stereo and 41 have both. How many own at least one of these devices? Draw an
appropriate Venn diagram
There are two overlapping circles. We pretend that the red circle contains
all 70 students who have a personal computer. We pretend that the blue
circle contains all 120 students who have a personal stereo. The
overlapping part is in both circles and we pretend that the overlapping
part contains only the 41 students who have both a computer and a stereo. So
we write 41 in the overlapping part:
The red circle contains all 70 students who have a personal computer. and
the 41 who have both are part of the 70, so that leaves 70-41 or 29 in the
left part of the red circle, who only have a personal computer. So we write
29 in the left part of the red circle:
The blue circle contains all 120 students who have a personal stereo, and
the 41 who have both are part of the 120, so that leaves 120-41 or 79 in the
right part of the blue circle, who only have a personal stereo. So we write
79 in the right part of the blue circle:
Notice that there are 29 in the red circle that are not in the blue circle.
So 29 students have a computer but no stereo. Notice that there are 79 in
the blue circle that are not in the red circle. So 79 students have a
stereo but no computer. And as we said earlier, the 41 in the middle that
are in both circles are the 41 that have both a computer and a stereo.
So to determine how many students have one or the other (or both), that is,
at least one, we add the three numbers together:
29 students have a computer but no stereo
41 students have both a computer and a stereo
+79 students have a stereo but no computer
-------------------------------------------------
149 students have at least one of the two devices
Edwin
Answer by ikleyn(52780)  (Show Source):
You can put this solution on YOUR website! .
Actually, the standard way to solve such problem is in one line as it is shown below:
The number of those who own at least one of these devices = 70 + 120 - 41 = 149.
Intuitively, it is clear to you why I added 70 and 120, and only one aspect requires explanation: why I subtracted "41" ?
It is because when I added 70 and 120, I counted the common part twice.
After subtracting it, everything is in its right place.
Actually, you need Venn diagram only if you solve such problem at the first time.
After that, the short solution presented above is totally enough.
----------------
See the lesson
- Counting elements in sub-sets of a given finite set
in this site.
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Hello, Edwin !
Thanks for your question.
Regarding my preferences and answering your question, I prefer to use
|A U B| = |A| + |B| - |AnB| for two sets A and B and
|A U B U C| = |A| + |B| + |C| - |AnB| - |AnC| - |BnC| + |AnBnC| for 3 sets A, B and C.
It is how I teach the students to solve the corresponding problems in my lessons
- Counting elements in sub-sets of a given finite set
- Advanced problems on counting elements in sub-sets of a given finite set
in this site.
Also, it is how I solved numerous problems of this kind at this forum
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1116804.html
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1118494.html
Everybody has his or her own taste and style in explaining such problems.
I came from the culture (Math education culture) where the notion "Venn diagram" was totally unknown and was never used;
NEVERTHELESS, the students (at least, advanced students) were able to solve such problems leaning on their logical skills,
and I really prefer the students understand logic FIRST behind such problems.
Then they will be able to solve the problems for four, five and so on sets and subsets (where Venn diagrams become useless).
It does not mean that I negate using Venn diagrams on the everyday basis.
See THIS solved problem as another example.
What I try to explain to students is the logic behind Venn diagram and the shortest and straightforward way to solve the problem.
Thanks again for your question and the opportunity for me to highlight my point.
===================
P.S. I also love and prefer compact formulas that speak by themselves and tell the entire and whole Math stories, as those listed above.
Answer by AnlytcPhil(1806)  (Show Source):
You can put this solution on YOUR website! I see that Ikleyn prefers the combinatorial sieve formula:
N(A or B) = N(A)+N(B)-N(A and B)
to the Venn diagram with two sets.
However, I'll bet she doesn't prefer the combinatorial sieve formula:
N(A or B or C) = N(A)+N(B)+N(C)-N(A and B)-N(A and C)-N(B and C)+N(A and B and C)
to the Venn diagram with three sets.
Right, Ikleyn? J
AnlytcPhil (aka Edwin McCravy)
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