# Lesson: 3-Criteria Venn Diagrams

## Comment on 3-Criteria Venn Diagrams

### Hi,

Hi,
Can someone explain the formula for three overlapping sets? i don't get it since the video seems to be exhaustive enough to tackle 3 overlapping sets:
Total = Group1 + Group 2 + Group 3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Ben

### I've never been a fan of

I've never been a fan of formulas for 3 overlapping sets, and I doubt that the test-makers would create a question that relied on a student memorizing the formula.

Actually, I should say formulaS. There are two:

#1) Total = Group 1 + Group 2 + Group 3 - (sum of all 2 or more overlaps) + (in all 3 groups) + none.

#2) Total = Group 1 + Group 2 + Group 3 - (sum of members in EXACTLY 2 groups) - 2(All three groups) + none

You are using a formula that incorrectly combines formula #1 and formula #2

In this question, we need to use formula #1

We get: 100 = 40 + 60 + 80 - (7 + 46 + 36) + 6 + none
Simplify: 100 = 97 + none

So, none = 3

### It looks like the equation 1)

It looks like the equation 1) mentioned here should be:
#1) Total = Group 1 + Group 2 + Group 3 - (sum of all 2 or more overlaps) - (in all 3 groups) + none.

Thanks

### I'm very reluctant to discuss

I'm very reluctant to discuss these formula, since these equations are more likely to confuse than enlighten. I think it's MUCH safer to use Venn diagrams in these cases.

The formula below is correct:
#1) Total = Group 1 + Group 2 + Group 3 - (sum of all 2 OR MORE overlaps) + (in all 3 groups) + none.

So, in the video question above, we're told that 7 students take Physics and Sociology, but we're NOT told that 7 students take ONLY Physics and Sociology. So, among these 7 students, it's possible that some of them take all 3 courses. This means we must use formula #1.

Cheers,
Brent

### I guess Bennaghmouch is

I guess Bennaghmouch is talking about the formula for 2 overlapping sets

x-z+z+y-z+n=total

@ alicia for me the algebraic approach was the easiest one

### Can you please explain this

Can you please explain this with venn diagram,, i tried but failed.

Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness. If all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects, how many of the subjects experienced only one of these effects?

A. 105
B. 125
C. 130
D. 180

### Hi santhosh1989,

Hi santhosh1989,

ceilidh.erickson provides a nice solution (using a Venn diagram) here: http://www.beatthegmat.com/plz-explain-official-guide-ps-q-178-t281640.html

Cheers,
Brent

### Amazing explanation Brent!

Amazing explanation Brent!

I just had a question in terms of the commonality of facing such a question on the GMAT. I have taken 4 practice tests to date and have never seen a question such as this one or one that requires a double-matrix. Do you know if questions like these are a rare occurrence on the test?

### 3-Criteria Venn Diagrams are

3-Criteria Venn Diagrams are quite rare on the GMAT.
Double Matrix questions, on the other hand, are quite popular. That said, there are times when it's difficult to recognize that a question is actually a Double Matrix question.

### https://gmatclub.com/forum

https://gmatclub.com/forum/there-are-a-total-of-400-students-at-a-school-which-offers-a-chorus-191977.html

### Every student at the

Every student at the Performing Arts Academy must take at least one of the two drama courses offered, Classical Theater or Improvisation. If 15% of the students who take Classical Theater also take Improvisation, how many students take both Classical Theater and Improvisation?

(1) Ten percent of the students who take Improvisation also take Classical Theater.

(2) The Performing Arts Academy has a total of 450 students

### Just so you know, this isn't

Just so you know, this isn't a 3-criteria question. In other words, we don't have 3 overlapping sets; we have 2 overlapping sets.

Cheers,
Brent

### If of the students in a

If 3/7 of the students in a room are seniors, and 7/25 of the other students are juniors, and there are x students in the room who are not juniors or seniors, how many students are in the room?

A) 175x/72
B) 175c/51
c)25x/7
d)25x/3
e) 10x

### Just like we did with a

Just like we did with a double matrix for two populations, is there a table for these kind of problems

### Great question!

Great question!

Unfortunately, when it comes to solving 3-criteria questions, there's no strategy equivalent to the Double Matrix method. If we were to try to create such a strategy, we'd need a complicated 3-dimensional matrix.

Cheers,
Brent

### Hi Brent,

Hi Brent,

In continuation to the above mentioned question, can we say that all 3-criteria questions need to be solved through Venn Diagram approach and 2 criteria by Double Matrix?

Thanks

### Yes, that's correct.

Yes, that's correct.

### Could you please explain me

Could you please explain me why in this question: "Last year 26 members of a certain club traveled to England, 26 members traveled to France, ...", people who traveled to ALL 3 countries are zero?

I do not understand, since it says that people who traveled to both UK and FR are zero, not people who traveled to UK, FR and IT...?

Thanks!

We're told that "Last year no members of the club traveled to both England and France."

This is NOT the same as saying that 0 members traveled to ONLY England and France last year (which is how I think you are interpreting the information)

Here's another way to think of it:
- If we asked the club members "Raise your hand if you traveled to Italy last year," then the question tells us that 32 members would raise their hands.

- If we asked the club members "Raise your hand if you traveled to France and Italy last year," then the question tells us that 11 members would raise their hands.

- If we asked the club members "Raise your hand if you traveled to England and France last year," then the question tells us that 0 members would raise their hands.

KEY CONCEPT: If a person traveled to all 3 countries last year, then that person would have traveled to England and France last year. So, if zero members traveled to England and France last year, then zero members traveled to all 3 countries.

Does that help?

Cheers,
Brent

### Thank you Brent!

Thank you Brent!

My problem is probably more related to the English language formulation of this concept; because i interpret UK+FR and UK+FR+IT as two different groups. Therefore, if somebody has not traveled to UK+FR it does not imply that he has not traveled to UK+FR+IT too...

### I agree; it's more related to

I agree; it's more related to the English language formulation.

There are many instances in which GMAT Quant questions have a Reading Comprehension component.

Cheers,
Brent

### Hi Brent,

Hi Brent,

could you please explain why all three is 15% in this example. I thought it would be 30%. I know that it is because we don't want to include the same factor various times, but I don't understand why it would be included twice in the first place.

https://gmatclub.com/forum/of-the-300-subjects-who-participated-in-an-experiment-using-virtual-re-134147.html#p1094334

Thank you!
BR Pia

### Hi Pia,

Hi Pia,

I'm not sure if you're referring to a specific solution, but Paresh provides a nice solution here: https://gmatclub.com/forum/of-the-300-subjects-who-participated-in-an-ex...

Take a look, and if you need me to elaborate on the solution, I'd be happy to do so.

Cheers,
Brent

### Hi Brent,

Hi Brent,

This question of official guide is is three Set Venn-Diag.

https://gmatclub.com/forum/of-the-300-subjects-who-participated-in-an-experiment-using-virtual-re-134147.html

I understood that we are asked to find the value of the sets that experienced only one Symptoms

So My thought Process was to Subtract those sets Exactly two, all three and none .

If

T= A+B+C - 2(exactly two) - 3(all three) -n

So what led me to think this way
A= (a+d+g+f). B=(b+d+g+e) , C= (c+e+f+g)
we are asked the value of a+b+c

T= a+d+g+f+ b+d+g+e+ c+e+f+g - 2(d+e+f)-3(g)-n

we would end up with a+b+c.
But there is something wrong in this approach . So can you help identify my gaps

There are two formulas you can use for 3 overlapping sets:

#1) Total = Group 1 + Group 2 + Group 3 - (sum of all 2 or more overlaps) + (in all 3 groups) + none.

#2) Total = Group 1 + Group 2 + Group 3 - (sum of members in EXACTLY 2 groups) - 2(All three groups) + none

I'm not a big fan of memorizing these. You can always break the various area into sections and assign each area a variable.

If you tell me what each variable represents in your solution, I can help.

Cheers,
Brent

There are two formulas you can use for 3 overlapping sets:

#1) Total = Group 1 + Group 2 + Group 3 - (sum of all 2 or more overlaps) + (in all 3 groups) + none.

#2) Total = Group 1 + Group 2 + Group 3 - (sum of members in EXACTLY 2 groups) - 2(All three groups) + none

I'm not a big fan of memorizing these. You can always break the various area into sections and assign each area a variable.

If you tell me what each variable represents in your solution, I can help.

Cheers,
Brent

### Hi Brent,

Hi Brent,

I got a little confused in the language of the question
https://gmatclub.com/forum/there-are-a-total-of-400-students-at-a-school-which-offers-a-chorus-191977.html

It mentioned that 220 students were either baseball or Italian, by this I thought that it is only referring to the regions that are not common for Italian and Baseball (that is either). Had the question mentioned either or and , it would still have made sense to include all the area in those two circles. Can you please help me here?

On the GMAT, "or" always means "either one or both"

So, if there are 10 students who speak either Spanish or French, then that group can include students who speak BOTH Spanish and French.

Cheers,
Brent

### Hey Brent,

Hey Brent,

in this Q:

https://gmatclub.com/forum/of-the-300-subjects-who-participated-in-an-experiment-using-virtual-re-134147.html

Why is ALL THREE deducted twice? I didn´t get it.

Philipp

I'm not a big fan of that formula, but the idea is that when we first add all 3 circles. In doing so, we end up adding the "ALL THREE" value 3 times, when we should have just counted it once. As such, we must subtract 2 of the "ALL THREE" values

### Hi Mr. Brent,

Hi Mr. Brent,

I hope this email found you well.

I have a question in regards to this overlapping set question.

3/8 of all students at Social High are in all three of the following clubs: Albanian, Bardic, and Checkmate. 1/2 of all students are in Albanian, 5/8 are in Bardic, and 3/4 are in Checkmate. If every student is in at least one club, what fraction of the student body is in exactly 2 clubs?

(A) 1/8
(B) 1/4
(C) 3/8
(D) 1/2
(E) 5/8

Could you please enlighten me on how you would approach this question without using the overlapping set formula?

Thank you

### Hi Brent -

Hi Brent -

I have this question: https://gmatclub.com/forum/in-a-class-of-50-students-20-play-hockey-15-play-cricket-and-11-play-131736.html

I tried solving it using the Venn Diagram. Called the overlap of 3 is x. Then combine them in an equation equals 50. However, the answer of x I got is -2. I wonder what went wrong here. Can you help?