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Why does the following Set Theory Formula not Work for this

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Why does the following Set Theory Formula not Work for this

by pierce22884 » Wed Apr 21, 2010 7:22 pm
Workers are grouped by thier areas of expertise and are placed on at least one team. 20 workers are on the Marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Visions, and 4 workers are on all three teams. How many workers are there in total?

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Set formula does not work this problem... Why? Instead the Venn is what you use and its tedious process.

The answer to this is 74. When you use the Set Theory formula:

Tot = 20+30+40 - (5+6+9) - 2(4) + (0)
Tot = 90 - 20 - 8
Tot = 62

Any help would be greatly appreciated. Thanks in advance for your time.
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by rn4gmat » Wed Apr 21, 2010 7:52 pm
The Set Theory Formula to get the exact unique count in one set is :

P(A) + P(B) + P(C) - 2P(AnB) - 2P(AnC) - 2P(BnC) + 3P(AnBnC)

so if we put this to implementation we will get :

40 + 30 + 20 - 5 - 6 - 9 + 4 = 74.

Hope it helps.
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by Stuart@KaplanGMAT » Wed Apr 21, 2010 10:01 pm
pierce22884 wrote:Workers are grouped by thier areas of expertise and are placed on at least one team. 20 workers are on the Marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Visions, and 4 workers are on all three teams. How many workers are there in total?

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Set formula does not work this problem... Why? Instead the Venn is what you use and its tedious process.

The answer to this is 74. When you use the Set Theory formula:

Tot = 20+30+40 - (5+6+9) - 2(4) + (0)
Tot = 90 - 20 - 8
Tot = 62

Any help would be greatly appreciated. Thanks in advance for your time.
Hi,

the problem is with the bolded part of the formula:

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

When you apply this formula, the bolded part refers to members of exactly 2 groups. In the question that you posted, we know how many people are in at least two groups, but not only two.

If the stem had read:
5 workers are on only the Marketing and Sales teams, 6 workers are on only the Sales and Vision teams, [and] 9 workers are on only the Marketing and Visions
then you could have applied the formula as you did.

So, you want to use the alternative formula posted by rn4gmat, with 1 small correction:

Total = G1 + G2 + G3 - 2(people in at least 2 groups) + (people in all groups)

(the correction is the removal of the "3" in front of the last term; when rn4gmat did the math he (she?) just plugged in 4, not 3*4, for that last term, so the final sum was correct).
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by pierce22884 » Thu Apr 22, 2010 11:17 am
Total = G1 + G2 + G3 - 2(people in at least 2 groups) + (people in all groups)

So in using this formula for "At least problems" we multiply by (2) and replace "+neither" with "+ (people in all groups)" ?

Also what if the question was looking for how many people belonged to at least 1 of the teams? If you can't use the above example here is one that you can use:

There are three country clubs in town: Abacus, Bradley, and Claymore. Abacus has 300 members, Bradley 400, and Claymore has 450. 30 people belong to both Abacus and Bradley, 40 to both Abacus and Claymore, and 50 to both Bradley and Claymore. 20 people are members of all three clubs. How many people belong to at least 1 country club in town?
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by pharmxanthan » Mon May 17, 2010 2:18 pm
Hey pierce#, I think that it is better to draw the Venn Diagram and start from inside out. (MGMAT Guide 4)
It will take a bit longer than it would take to plug in the formula but you will be sure of the answer.

If an expert can list most or all of the formulae and lists scenarios when when they could be used, we all will appreciate that very much.
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by Stuart@KaplanGMAT » Mon May 17, 2010 2:48 pm
pharmxanthan wrote:Hey pierce#, I think that it is better to draw the Venn Diagram and start from inside out. (MGMAT Guide 4)
It will take a bit longer than it would take to plug in the formula but you will be sure of the answer.

If an expert can list most or all of the formulae and lists scenarios when when they could be used, we all will appreciate that very much.
There's a detailed discussion of 3-group formulae here:

https://www.beatthegmat.com/least-value- ... 46464.html
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by pharmxanthan » Mon May 17, 2010 3:48 pm
Thanks Stuart. I have seen some equations using "Neither". The post to which you referred does not use this term. Also, why do we need to "minimize the number of people who love exactly 1 of the 3"? If we have, for example, a larger A, we will have a smaller ABC.
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by Sue2010 » Sat May 22, 2010 10:00 am
Stuart: Thanks much for the detailed explanation.. Getting to understand it a lot better thanks to all your inputs..

Qne quick question: Just noticed the following in your previous post. Shouldn't the formula also be corrected to remove the "2" in front of the 4th term?

>>So, you want to use the alternative formula posted by rn4gmat, with 1 small correction:

Total = G1 + G2 + G3 - 2(people in at least 2 groups) + (people in all groups)

(the correction is the removal of the "3" in front of the last term; when rn4gmat did the math he (she?) just plugged in 4, not 3*4, for that last term, so the final sum was correct).Total = G1 + G2 + G3 - 2(people in at least 2 groups) + (people in all groups)
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by Stuart@KaplanGMAT » Sat May 22, 2010 10:52 am
pharmxanthan wrote:Thanks Stuart. I have seen some equations using "Neither". The post to which you referred does not use this term. Also, why do we need to "minimize the number of people who love exactly 1 of the 3"? If we have, for example, a larger A, we will have a smaller ABC.
Hi!

You can add "+ neither" to the formulae; however, in the vast vast majority of GMAT 3 set questions neither=0, so it's irrelevant (i.e. on GMAT 3 set questions, everyone usually belongs to at least 1 of the 3 groups). For GMAT 2 set questions, it's far more common for "neither" to be relevant.

We want to minimize the singles on that question because we have to account for the duplicates; the more people we put in single categories, the more are going to get shoved in the triple group due to space limitations.

Minimizing the singles is really just a consequence of maximizing the doubles. (On the GMAT, when we want to maximize one thing, we minimize everything else.)
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