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Workers are grouped by their areas of expertise, and are

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New post 19 Oct 2016, 19:45
jaydevsachdeva wrote:
Answer is 62. Solved by both equations:

Hi, for all that are confused as to why there are 2 different answers from the given 2 equations,
There are two formulas for 3 overlapping sets:
Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither.

Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither

The answer is supposed to be 62 for both of them.

For 1st Equation, as it is a sum of 2 group overlaps, it will also include the common(all three) 4 part with it.
Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither
Total= 20+30+40-((5+4)+(9+4)+(6+4))+4+0=62
=90-32+4
=62
For 2nd Equation, as it is a sum of 2 Exactly group overlaps, it will not include the common 4 part with it.

Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither
=20+30+40-(5+9+6)-2*4+0
=90-20-8
=62

I hope I am right, and I hope everyone got it.



I completely agree with you in regards to this particular problem. See Below:

Reading through the history of this problem I can see why there is confusion. I understand applying Venn Diagrams and the formulas associated with such. My confusion initially resulted from the GMAT Club Math Book's formula explanation: Pay attention to the description on what it means to sum the intersections - it states to add "g" to every intersection (d, e, and f)...then you subtract from 90, then you add "g" again.
Total = A+B+C - (sum of 2-group overlaps) + (All 3) + Neither
"In the formula above, Sum of 2-group overlaps = AnB+AnC+BnC, where AnB means intersection of A and B (sections d, and g), AnC means intersection of A and C (sections e and g), and BnC means intersection of B and C (sections f and g).
Now, when we subtract AnB (d and g), AnC (e and g), and BnC (f and g) from A+B+C, we are subtracting sections "d, e, and f" ONCE BUT section "g" THREE TIMES (and we need to subtract section "g" only twice), therefore we should add only section "g", which is intersection of A, B, and C (AnBnC) again to get: Total = A+B+C - (sum of 2-group overlaps) + (all three) + Neither


Due to this, I initially obtained 62 as well because I did as it states: I summed AnB+AnC+BnC by adding d and g, e and g, and f and g...THAT is the key...although I now understand the concept, not just memorizing, I did not when I started learning this concept. It did not mention to only sum d, e, and f.....But that is what I had to do to obtain 74. Thus, by adding (5+4)+(6+4)+(9+4) we get 32....subtracting it from 90 = 58, + 4 = 62. Now, in the Mathbook, the second formula is explained well in that you only add what which is common for A and C, which is "e", for example, not "e and g".

I love that mathbook, it has been my favorite prep for formulas and methods, I am eternally grateful....

Going back through I realize now that although it may say to specifically sum d+g, e+g, f+g, I know that when the question asks for "both A and B" I know not to add the "all 3" in it as well because anything shared amongst all 3 will be included in those 2. I may have read to much into the Mathbook but by understanding the concept it just makes sense now. Bunuel did a great job explaining, thank you again.
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Re: Workers are grouped by their areas of expertise, and are  [#permalink]

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New post 12 Oct 2017, 17:44
BarneyStinson wrote:
Workers are grouped by their areas of expertise, and are placed on at least one team. 20 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 Vision teams, and 4 workers are on all three teams. How many workers are there in total?

A. 62
B. 78
C. 74
D. 66
E. 72



The thing with this question is that it doesn't say " Only 5 workers are on both the marketing and sales...only 6 are on both the sales and visions" if it said that then the answer would be 62
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New post 30 May 2018, 09:12
alexjoh89 wrote:
I have a question. If the formula is:

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

why is it then + 4 instead of -2*4?


I got this wrong the first time too. I would advise drawing the venn diagram to see it. The problem is when we subtract the overlap between the 3 groups, we already subtracted the center piece thrice. So we need to add a center piece to get what we want. Your reasoning would be correct if the overlapping piece doesn't cover the center piece, in which case we do need to subtract 2*4.
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New post 15 Jul 2018, 22:45
alexjoh89 wrote:
I have a question. If the formula is:

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

why is it then + 4 instead of -2*4?

Never try to memorize the formula! Always try to find the rationale behind any problem in both verbal and math. In these--overlapping set problems, the relatively tricky part is to understand what overlaps what. Precisely, when you draw the diagrams, understand what part overlaps and what part not.
For example, in this question, when the author says that 4 members are on three teams, you should understand that there can be members who are both in two teams and in three teams. When you think in that way, you eliminate the additional overlaps and find the real number of team members.
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New post 07 Jun 2019, 21:08
There are two formulas for 3 overlapping sets:
Total=A+B+C−(sum of 2−group overlaps)+(all three)+NeitherTotal=A+B+C−(sum of 2−group overlaps)+(all three)+Neither.

Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+NeitherTotal=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither.



Alright, I'm really really confused on this topic. I'd really appreciate if someone could help me understand the difference between the two.

Bunuel VeritasKarishma IanStewart chetan2u
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New post 07 Jun 2019, 22:03
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Sourav700 wrote:
There are two formulas for 3 overlapping sets:
Total=A+B+C−(sum of 2−group overlaps)+(all three)+NeitherTotal=A+B+C−(sum of 2−group overlaps)+(all three)+Neither.

Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+NeitherTotal=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither.



Alright, I'm really really confused on this topic. I'd really appreciate if someone could help me understand the difference between the two.

Bunuel VeritasKarishma IanStewart chetan2u


Let us look at both the formulas..

A=a+d+e+f
B=b+d+f+g
C=c+e+f+g

Main difference is the definition and contents of sum of 2−group overlaps and sum of EXACTLY 2−group overlaps
sum of 2−group overlaps means (d+f)+(e+f)+(g+f) and sum of EXACTLY 2−group overlaps means d+e+f

(I) Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither
Here sum of 2−group overlaps are (d+f)+(e+f)+(g+f).. So, you can see the (all three, that is f, is getting added thrice , the net result being 0, so you add one back .
all three = f
Neither =0
So Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither=A=a+d+e+f+b+d+f+g+c+e+f+g-((d+f)+(e+f)+(g+f))+(f)

(II) Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither
sum of EXACTLY 2−group overlaps means d+e+f..
so you are not subtracting all three even once, and they have been added thrice in A+B+C, so subtract it 2 times
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New post 07 Jun 2019, 22:22
Hi chetan2u

Thanks very much for the explanation. Appreciate it.

i'd like to follow up with another question here.

Now i get the difference between the two:
(sum of 2−group overlaps) = (sum of exactly 2) + 3f. correct?

My question is:

when I'm provided with individual values of d, e, f, and g. why do I get an incorrect answer when i use the second formula?

For example:

Example 1: Workers are grouped by their areas of expertise, and are placed on at least one team. 20 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 Vision teams, and 4 workers are on all three teams. How many workers are there in total?

The answer here is 74 and not 62.

1st equation: 20 + 30 + 40 - ..................

wait, i think i got it.

When we are given the value for d e and g we must subtract the value of f from each of these values for them to be valid in the second equation.

1st equation: 30 + 40 + 20 - (5+9+6) + 4 = 74
2nd equation: 30 + 40+ 20 - (1+5+2) - 2(4) = 74

Thanks a lot chetan2u :)

P.S. If you were in Calcutta, i'd buy you pizza :p
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New post 07 Jun 2019, 22:40
Sourav700 wrote:
Hi chetan2u

Thanks very much for the explanation. Appreciate it.

i'd like to follow up with another question here.

Now i get the difference between the two:
(sum of 2−group overlaps) = (sum of exactly 2) + 3f. correct?

My question is:

when I'm provided with individual values of d, e, f, and g. why do I get an incorrect answer when i use the second formula?

For example:

Example 1: Workers are grouped by their areas of expertise, and are placed on at least one team. 20 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 Vision teams, and 4 workers are on all three teams. How many workers are there in total?

The answer here is 74 and not 62.

1st equation: 20 + 30 + 40 - ..................

wait, i think i got it.

When we are given the value for d e and g we must subtract the value of f from each of these values for them to be valid in the second equation.

1st equation: 30 + 40 + 20 - (5+9+6) + 4 = 74
2nd equation: 30 + 40+ 20 - (1+5+2) - 2(4) = 74

Thanks a lot chetan2u :)

P.S. If you were in Calcutta, i'd buy you pizza :p



Yes, you are correct..
what is given is d+f as 5, so you have to subtract f from it.

I think I can let that Pizza go, even if it means shifting to calcutta. ;)
All the best
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New post 08 Jun 2019, 00:03
Sourav700 wrote:
There are two formulas for 3 overlapping sets:
Total=A+B+C−(sum of 2−group overlaps)+(all three)+NeitherTotal=A+B+C−(sum of 2−group overlaps)+(all three)+Neither.

Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+NeitherTotal=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither.



Alright, I'm really really confused on this topic. I'd really appreciate if someone could help me understand the difference between the two.

Bunuel VeritasKarishma IanStewart chetan2u


Check below.

19. Overlapping Sets


[*]Theory
Overlapping Sets Made Easy!
How to draw a Venn Diagram for problems
ADVANCED OVERLAPPING SETS PROBLEMS
Formulae for 3 overlapping sets
[*]Questions
The E-GMAT Sets Triad: 3 Exciting Sets Questions!
The Word “Or” in GMAT Math
DS Questions
PS Questions
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New post 08 Jun 2019, 10:06
Sourav700 wrote:
Alright, I'm really really confused on this topic. I'd really appreciate if someone could help me understand the difference between the two.


I've never used those formulas once on a GMAT question, and in my experience, a lot of test takers find them confusing, and often use the wrong one, so they aren't very reliable. Even when using a formula is an option, these overlapping set problems tend almost always to be easier and faster to solve just by using a Venn diagram anyway, as Paresh does on the first page of this thread, and since the Venn diagram can easily be adapted to situations where the formulas don't apply (the types of situations you encounter a lot on the GMAT), it's the better method to learn in general.
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Re: Workers are grouped by their areas of expertise, and are   [#permalink] 08 Jun 2019, 10:06

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