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Can this question be answered? (Venn Diagrams, Sets)?
Out of a group of 275 YA! participants:
90 answer math questions,
70 answer chemistry questions,
10 answer math and biology questions,
8 answer biology and physics questions,
5 answer math, biology and chemistry questions,
3 answer math, chemistry and physics questions,
1 answer questions from all four categories,
80 answer biology questions,
60 answer physics questions,
9 answer math and chemistry question,
7 answer chemistry and physics questions,
4 answer math, biology and physics questions,
2 answer biology, chemistry and physics questions,
How many answer no questions in these categories?
Can this question be answered?:
"How many answer only math questions?"
If yes, then how many?
If no, then why not?
By the first question I mean: "How many answer no questions in any of these categories." I don't mean to give the number for each category.
Erbo B, I agree with all your numbers except maybe the number for only physics and math. How did you get that to be 0?
Hey! Vikram,
From your diagram can you tell how many people answer both math and physics and only math and physics?
Same question for biology and chemistry?
It appears that not all the areas of intersection can be represented by four circles. At least one of the sets must be represented by a different shape as Dr. D showed.
(This leads me to a number of other questions that I'll state later.)
I adapted this question from a Mensa puzzle book. It says that there are 8 that answer none of the categories.
I thought that there is something missing. Dr. D confirms this for me.
If information is not given then the answer is indeterminate; it is not correct to assume that they are zero.
Other questions:
In a Venn diagram that shows all the regions of intersections:
A symmetric diagram of congruent circles does not show the intersection of the opposite circles.
If one of the circles is shifted and/or enlarged then one other intersection can be shown, but I don't think all the intersecting regions can be shown with four circles.
Can the fourth set be represented by a convex shape?
How many sets can be represented by convex shapes and all intersection regions be shown?
How many sets can be represented by closed shapes and all intersection regions be shown?
3 réponses
- Dr DLv 7il y a 1 décennieRéponse favorite
It's indeterminate because not enough information is given. We don't know how many answer Math and Physics, for example. Same is true for Bio & Chem.
See the Venn diagram below. Note that every intersection must be represented. Let me know if I made an error.
http://i40.tinypic.com/30kstq8.jpg
*EDIT Vickram*
I think the best you can do with the info given is set possible limits for the answers. For example
0 ≤ x ≤ 47
0 ≤ y ≤ 57
So "math only" can be anywhere from 29 to 76.
"No categories" can range from 8 to 112.
We can say for sure that AT LEAST 8 persons answer in questions in NONE of the given categories, and AT MOST 76 answer in math only.
- Vikram PLv 5il y a 1 décennie
Refer to the following Venn Diagram First (as working is not shown below)
http://vcpandya.googlepages.com/VennDiagram2.JPG
Ans 1: 275 - [76 + 66 + 57 + 47 + 2 + 2 +3 + 3 + 1 + 2 + 3 + 4 + 1]
= 275 - 267 = 8
Ans 2: 76 Answers only math questions (90 - [ 2+ 2+ 2 + 3 + 4 +1])
Any questions can be answered with help of the Venn diagram.
I hope I have not gone wrong anywhere :-)
============
Both Math & Physics (Math Union Physics)
= 76 + 2 +2 +4 +1 +2 +3 +3 +1 +3 +47
= 144
Only Math & Physics (i.e. No overlapping of Math & Physics with any other category)
= 0
(Because of the word "only" answer is zero)
==============
Both Biology & Chemistry = 66 + 2 + 3+ 3 + 4 +1 +1 +2 +2 +3 + 57 = 144
Only Bio & Chem = 0
(Because of the word "only" answer is zero)
=======================
@ Dr. D:
I agree with you that the all intersections should be represented but If you want to "solve" the question by using the given information then I see my Venn diagram as the only possible solution.
(Which assumes unknown things as either zero or part of the overlapping region)
From the given information It is not possible for any one to show intersection for all categories since no information regarding the same is given.
If one draws them next to each other then you are correct, this question is not solvable but if you construct your Venn diagram as I have constructed and if the question is how many answer Math & Physics (without the word "only") then the answer would be
=1 + 2 + 3 = 6
However I may be wrong in assuming the other unknown things as zero or as a part of the overlapping region.
Jon, can you please throw some more light on this.
@ Dr. D:
I completely agree with your last answer.
@ Jon:
You are right it is not possible to represent more than three intersections with the help of circles.
I will think about your last two questions, but it seems that they are really difficult to answer as you are asking for maximum intersections possible . :)
- Erbo BLv 5il y a 1 décennie
If I understand this correctly the first question is in fact 4 questions:
1a. how many people did not answer math?
275-90 = 185
1b how many people did not answer chemistry?
275-70 = 205
1c how many people did not answer biology?
275 - 80 = 195
1d how many people did not answer physics?
275-60 = 215
2 How many only answer math questions?
76
Here is the logic:
your universe size is 90 (people that answer math)
WITHIN THIS UNIVERSE (people that answer math)
======================================
there are
persons that answer Phys+chem+bio = 1
persons that answer phys + bio but not chem = 3
persons that answer phys + chem but not bio = 2
persons that answer chem + bio but not phys = 4
persons that answer only chem = 2
persons that answer only bio = 2
persons that answer only phys = 0
So from all 90 people that answer math questions there are 1+3+2+4+2+2+0 = 14 persons that answered also questions besides math.
This means that there are 90-14=76 people that answer only math questions.
