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How to draw a Venn Diagram for problems

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Ms. Big Fat Panda
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How to draw a Venn Diagram for problems [#permalink]

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26 Jul 2010, 09:01
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So. I realized I love Venn Diagrams (Yes, I'm a nerd, but who cares?). I've explained a lot of problems using Venn Diagrams and a forum user PMed me to explain how I drew my Venn Diagrams since he was getting confused with them. I am posting this as an informal and rough guide to how I visualize problems that make use of Venn Diagrams.

At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry and English. 53 students study math, 88 study chemistry and 58 study english. If 6 students take all 3 classes, how many take exactly 2 classes?
A. 37
B. 43
C. 45
D. 60

Step 1: Deconstruction

This is where you extract the given information from a problem. Use a symbol for each section given to you. I usually use the first letter in caps. Please remember at this point that any number specified without explicitly mentioning that it's "exactly" or "only" for a certain subject is not to be taken so.

For instance, in this case, it says 53 math students. This doesn't mean students who are taking only 1 subject (Math). This could include students who are taking Math and Chemistry or Math and English or all three too. So now break down the numbers given to us.

Total = 150
M (Total) = 53
C (Total) = 88 [Why the hell are people studying more Chemistry than Math?]
E (Total) = 58
MCE = 6

We are asked to find MC + CE + EM.

Step 2: Drawing the diagram

ALWAYS start from the center of the Venn Diagram wherever information is available. This will make life infinitely easier.

In this case, the center is the intersection of all three circles, i.e MCE = 6

So fill that in to the diagram you've drawn.

Attachment:

VD1.jpg [ 21.73 KiB | Viewed 47609 times ]

Now that you've gotten that, let's start filling in a variable for each section not known to us. Here, consider each letter to represent only that specific section and not the entire circle or a larger portion.

Attachment:
File comment: Updated

VD1.jpg [ 27.41 KiB | Viewed 47585 times ]

So now that you have the parts filled in, what you need to do is write down what you have in the diagram in terms of numbers. So we are given the totals for each subject.

Look at math first. There are four types of people taking math (each group of these people mutually exclusive, and not in common with any other group)
1. Only math: a
2. Math and Chemistry: y
3. Math and English: x
4. All three: 6

So now represent this as a sum and you get

$$a+x+y+6 = 53$$ and hence $$a+x+y = 47$$

Similarly for the other subjects you get:
$$b+x+z+6 = 58$$ and hence $$b+x+z = 52$$

$$c+y+z+6$$ = 88 and hence $$c+y+z = 82$$

And then you have the total:
$$x+y+z+a+b+c+6=150$$ and hence $$x+y+z+a+b+c = 144$$

Step 3: Solution

This is perhaps the most intuitive part, but in my experience the first part of this step is the same in all overlapping sets problem. It's only what's asked for that's different.

Add all the individual equations together to get a combined equation with all the variables and a number.

So you get:
$$2(x+y+z) + a+b+c = 47+52+82 = 171$$

Rearranging this to get $$a+b+c$$we get $$a+b+c = 171 - 2(x+y+z)$$

Substitute this into the total equation we derived earlier saying $$x+y+z+a+b+c = 144$$ so you get:

$$x+y+z+171 - 2(x+y+z)= 144$$

Upon rearranging this you get:

$$x+y+z = 171-144 = 37$$which is option A the right answer.

Hope this helps some of you! Please post a reply if there's anything else you want to know about my explanation or anything else.
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Re: How to draw a Venn Diagram for problems [#permalink]

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26 Jul 2010, 09:16
Just what i wanted equations with respective areas in the 3 venn diagram

thanks a lot !
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Re: How to draw a Venn Diagram for problems [#permalink]

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29 Jul 2010, 04:22
Great work whiplash!! Kudos!

I wonder why not many have responded to this effort. Any how, thanks again!!
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Re: How to draw a Venn Diagram for problems [#permalink]

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29 Jul 2010, 04:35
great job whiplash2411!
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Re: How to draw a Venn Diagram for problems [#permalink]

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29 Jul 2010, 06:21
Thank you. Added to the list of resources in the Math Book Project
gmat-math-book-87417.html
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Re: How to draw a Venn Diagram for problems [#permalink]

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29 Jul 2010, 09:00
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Re: How to draw a Venn Diagram for problems [#permalink]

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29 Jul 2010, 13:53
Nice whip !!

Probably this can be added as well
formulae-for-3-overlapping-sets-69014.html#p729340
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Re: How to draw a Venn Diagram for problems [#permalink]

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13 Aug 2010, 16:10
This is a great example!

Thanks for posting.

Small note which got me confused for a bit. I believe the answer is 27 not 37 (unless I am missing something).

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Re: How to draw a Venn Diagram for problems [#permalink]

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13 Aug 2010, 21:28
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whiplash2411 wrote:
Upon rearranging this you get:

$$x+y+z = 171-144 = 37$$which is option A the right answer.

Hope this helps some of you! Please post a reply if there's anything else you want to know about my explanation or anything else.

First off, great post, this really does help! My first instinct on how to handle this was definitely off.

Secondly, I believe the answer would actually be 27. 171-144=27... not 37. Not sure why I'm the only one that caught that, so my apologies if I'm doing something wrong.

BB, can you correct this in the Math Book Project?
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Re: How to draw a Venn Diagram for problems [#permalink]

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19 Aug 2010, 09:40
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The answer is A, the mistake was in 47+52+82=181

181-144=37
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Re: How to draw a Venn Diagram for problems [#permalink]

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19 Aug 2010, 11:21
Nice Explanation..

Now I got a better understanding regarding Venn Diagrams.
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Re: How to draw a Venn Diagram for problems [#permalink]

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19 Aug 2010, 12:22
Thanks for that, whiplash. After a mechanical engineering degree, I can safely say that systems of equations with

# of equations >= # of variables

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Re: How to draw a Venn Diagram for problems [#permalink]

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19 Aug 2010, 13:48
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Re: How to draw a Venn Diagram for problems [#permalink]

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19 Aug 2010, 20:38
This formula, I assume could be substituted by this Venn Diagram method?

Overlapping Groups-
Total = Group 1 + Group 2 + Neither - Both

This is how I've been tackling these problems before but this seems to be an easier method?
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Re: How to draw a Venn Diagram for problems [#permalink]

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20 Aug 2010, 15:34
Kudos!

Big thanks for this excellent problem solver, it comes as a big help for me.
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Re: How to draw a Venn Diagram for problems [#permalink]

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30 Aug 2010, 14:24
thank you
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Re: How to draw a Venn Diagram for problems [#permalink]

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05 Oct 2010, 06:09
Kudos! Just what I wanted.
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Re: How to draw a Venn Diagram for problems [#permalink]

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09 Oct 2010, 06:49
Kudos! Great post whiplash2411, gonna help a lot during preparation. Added to the Quantitative strategies collection .
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02 Jul 2013, 04:33
hello everyone,

I have recently joined the forum to pursue my dream to crack GMAT.
I am just beginners..just brushing my basics..
Kudos!!!!!!
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Re: How to draw a Venn Diagram for problems [#permalink]

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11 Jul 2013, 00:11
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Re: How to draw a Venn Diagram for problems   [#permalink] 11 Jul 2013, 00:11

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