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Overlapping Sets (Manhattan GMAT CAT

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Overlapping Sets (Manhattan GMAT CAT

by markmosby » Sun Dec 13, 2015 10:59 am
So I thought I had a good grasp of overlapping sets, until I encountered this problem on a practice CAT yesterday:

"In a group of 68 students, each student is registered for at least one of three classes - History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?"

A. 13
B. 10
C. 9
D. 8
E. 7

What is the quickest, most efficient way to solve this problem? I've tried both equations and the triple Venn Diagram, but quickly confused myself. Any tips/advice would be greatly appreciated!

-Mark
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by DavidG@VeritasPrep » Sun Dec 13, 2015 12:06 pm
markmosby wrote:So I thought I had a good grasp of overlapping sets, until I encountered this problem on a practice CAT yesterday:

"In a group of 68 students, each student is registered for at least one of three classes - History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?"

A. 13
B. 10
C. 9
D. 8
E. 7

What is the quickest, most efficient way to solve this problem? I've tried both equations and the triple Venn Diagram, but quickly confused myself. Any tips/advice would be greatly appreciated!

-Mark
You could just plug n' chug with the following equation:
Total = Group 1 + Group 2 + Group 3 - [the number in EXACTLY two groups] - 2[the number in all three groups] + neither

Here we get:
68 = 25 + 25 + 34 - [the number in EXACTLY two groups] - 2*[3] + 0
68 = 84 - [the number in EXACTLY two groups] - 6
68 = 78 -[the number in EXACTLY two groups]
10 = [the number in EXACTLY two groups]
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by [email protected] » Mon Dec 14, 2015 9:40 am
Hi markmosby,

3-Group Overlapping Sets questions are relatively rare on the Official GMAT (you likely will NOT see this version of Overlapping Sets on Test Day). However, there is a formula that you can use to solve it.

Total = (1st group) + (2nd group) + (3rd group) - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd) - 2(all 3 groups).

In overlapping sets questions, any person who appears in more than one group has been counted more than once. When dealing with groups of people, you're not supposed to count any individual more than once, so the formula 'subtracts' all of the extra times that a person is counted.

For example, someone who is in BOTH the 1st group and the 2nd group will be counted twice....that's why we SUBTRACT that person later on [in the (1st and 2nd) group].

In this prompt, we're given the Total, a number for each of the 3 individual groups and the number of people who appear in all 3 groups. The equation would look like this...

68 = 25 + 25 + 34 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)- 2(3)

68 = 84 - 6 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)

68 = 78 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)

(1st and 2nd) + (1st and 3rd) + (2nd and 3rd) = 10

Since the prompt asks for the total number of students that are in exactly 2 classes, we have our answer.

Final Answer: [spoiler]B; 10[/spoiler]

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by MartyMurray » Mon Dec 14, 2015 3:17 pm
Hi Mark.

If you can believe it, I think I saw three, if not there were at least two, three overlapping sets questions in the quant section once when I took the GMAT. So depending on how things go, you could see a few of them. I knew basically how to do them but didn't know how to do them quickly, and that weakness probably affected my score.

To learn a LOT about how to handle them, click the below link and look at all of the responses. There are two main formulas that people use. Rich provided one, above, and I provided the other in my response in the thread the link goes to. Also, in the thread I linked to, Mitch goes over a great way to work with triple Venn diagrams and another expert explains how to get from one of the formulas to the other.

Then find a bunch of them and practice until you are good at them. Getting the answers to them is pretty straightforward once you know how. So you may as well learn how so that if you see one or two on the test you can get some pretty sure points and not take too much time doing so.

https://www.beatthegmat.com/pizza-hoagie ... 85217.html
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by MartyMurray » Mon Dec 14, 2015 3:17 pm
Hi Mark.

If you can believe it, I think I saw three, if not there were at least two, three overlapping sets questions in the quant section once when I took the GMAT. So depending on how things go, you could see a few of them. I knew basically how to do them but didn't know how to do them quickly, and that weakness probably affected my score.

To learn a LOT about how to handle them, click the below link and look at all of the responses. There are two main formulas that people use. Rich provided one, above, and I provided the other in my response in the thread the link goes to. Also, in the thread I linked to, Mitch goes over a great way to work with triple Venn diagrams and another expert explains how to get from one of the formulas to the other.

Then find a bunch of them and practice until you are good at them. Getting the answers to them is pretty straightforward once you know how. So you may as well learn how so that if you see one or two on the test you can get some pretty sure points and not take too much time doing so.

https://www.beatthegmat.com/pizza-hoagie ... 85217.html
Marty Murray
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by ceilidh.erickson » Fri Feb 05, 2016 10:41 am
markmosby wrote: What is the quickest, most efficient way to solve this problem? I've tried both equations and the triple Venn Diagram, but quickly confused myself. Any tips/advice would be greatly appreciated!

-Mark
The triple Venn Diagram approach is really only needed is the question specifies an overlap of two categories, e.g. "three students are registered for both math and history." Then, you'd need 7 variables:
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On most of the triple-overlapping sets that exist in the OG / GMATPrep / what we've seen on actual tests, they won't get that specific. Instead, they'll just just set the categories as "only one class," "exactly two classes," and "all three classes." If that's the case, then you can just use 3 variables, as David and Rich showed. Much easier!

Good luck!
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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