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I am trying to figure out how to solve this Venn diagram problem for my Discrete Mathematics class. So the problem goes like this:

In a school there are 420 students. 300 of them have gone to school by car, 80 of them walking, 120 on a bicycle, 46 in a car and walking, 26 in a car and a bicycle, 36 walking and a bicycle, but 22 of them have neither used a car, bicycle or walked.

a) How many students come to school by car, walking and cycling? b) How many students come on a bicycle and walking but not car?

I just can't find a way to find the missing x in the middle I need to find

drleifz
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  • What have you sketched so far ? – callculus42 Sep 16 '14 at 10:15
  • I don't know how to put an image here in Math Stackexchange because I just have it on my computer. But i have 3 circles, intersecting at 4 places. The first intersection is 46 which mean car and walk. Second intersection is 36 which mean walk and cycle. Third intersection is 26 which is car and cycle and then i have the fourth intersection which is marked as X – drleifz Sep 16 '14 at 10:22
  • I could explain it better, if you upload the image. You can edit you original post. Above the input field there is a button for upload an image. – callculus42 Sep 16 '14 at 10:33
  • I and others added a picture-with explanation. If you have questions, just ask. – callculus42 Sep 16 '14 at 11:18
  • I can imagine dissolving a Venn diagram, but not solving one. – Marc van Leeuwen Sep 16 '14 at 12:16

3 Answers3

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The three cricles represent $398 (=420-22)$ students. This area is equal to the sum of the partial areas of the three circles. First add the whole three cirlces. $398=300+80+120...$. Now you have counted the intersections of two events twice. Thus you have to substract them. $398=300+80+120-46-36-26...$.

The intersection of all 3 circles has been first counted three times. After substracting the intersections of 2 circles the intersection is not counted anymore. Thus you have to add it.

$398=300+80+120-46-36-26+x$

enter image description here

Community
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callculus42
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Well, I got something like that. The b section is pretty easy from there. Hope I didn't mess up. My drawing

Some calculations. Calculations

Eugene Denisov
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I approached this from two extremes to start with:

First I assumed the missing x is 0. This leads to the number of people only walking being -2 and the total people being 414. This is clearly wrong!

Then I assumed the maximum x, which will be 26. This leads to a total of 440 people. This is also clearly wrong, but the answer is somewhere between!

However - I then realised that the difference between 440 and 414 is 26 - which is also the difference between the missing x of 0 and the second of 26. Logically therefore, and correctly, if you put x as 6 it works.

AndyL
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