How to Be an Expert in Solving GMAT Overlapping Sets?

GMAT and the quant section of GMAT have been one of the most addressing topics during the last few years. The quant section of GMAT has varied topics that are majorly based on the high syllabuses. Students who are proficient in maths or were good during their high schools in maths are stated to be dealing with the GMAT quant quite properly. As we already have seen that the students are majorly grouped between ‘good in verbal reasoning’ and ‘good in quant’ according to their traits. 

Talking about the GMAT Quant today, one very prominent topic among others is the sets. Sets, a topic that has been predominantly seen throughout the school days and for few even during the graduation. But if you notice carefully we have mentioned the syllabus are alike high school and not the content. And if you are also somebody whose face turns pale listening to sets and then moving on particularly overlapping sets then – here we have you covered.

What exactly are overlapping sets?

To understand this let us take an example.

In a box, there are 12 balls having black lines and white dots. 24 balls contain black lines but no white dots. If the box has 40 balls, all of which either have black lines or white dots or both, how many balls contain white dots but not black lines?

The question has overlapping sets. As we can see that few balls have black lines while few have white dots and few have both. Now when both are concerned, we can tell that it is an overlapping set. 

Let us calculate that. 

Now, add 12+24 = 36

40 – 36 = 4 and that is the answer.

Now, if you find there are more than two categories mentioned then simply continue with the tables. If you put the right content in the right box then the table won’t mislead you and therefore no wrong answers. 

The Overlapping Sets Formula

This formula can be used to solve the problems easily, Total = group 1 + group 2 – both + neither.

To solve this let us take another example: In an office, there are 40 employees, 25 of who are men, and 12 of whom play football. If 8 of the men play football, how many women do not play football?

 Solution: Here 40 is the total amount, 25 in group 1 and 12 in group 2, so it can be stated that the clubbing of G1 + G2 means the total of all the people doing one or the other activity, but that sum double counts the number ‘both’ so we subtract both. 

Three Overlapping Sets

Till now we dealt with two overlapping sets and now it is time we learn three overlapping sets. Though this is not frequently witnessed in the questions but practicing them will never fail anyone. If you are considering this particular heading to be very difficult then let us tell you, it is not, it just needs a slight extra dose of concentration. 

Example: Of the outfit stores in India, 30 carry tee shirts, 40 carry jeans, 25 carry skirts. If each store carry one of the brands, 32 of the stores carry two of the three outfit brands, and no stores carry all of these, how many outfit stores are there in the city?

Solution: 

Total = group1 + group2 + group3 – (sum of 2-group overlaps) – 2*(all three) + neither

Already confused? But wait, this is almost similar to the previous formula – group1 + group2 + group3 is the sum of all the stores that carry these brands, only a few are double-counted. Since 32 of the stores comprise of both the brands, we have double counted that and subtracted exactly 32, the sum of two groups overlap. Now, the confusing part is the second last term or 2*(all three). Though we do not really witness that often because it pops up in the data sufficiency questions. 

If one of the outfit stores in Mumbai carried all the three sorts of outfits then it would have been counted three times or once in the 30s, 40s, and 25s but eventually the count will still be a single store. So, when it is presented thrice we need to deduct it twice. In this case, we are seeing that the ‘all three’ term is equivalent to zero.

So, total = 30 + 40 + 25 - 32 - 2*0 + 0 = 63

(Neither denotes zero because each individual city stores comprise of at least one of the brands)

We know that the three overlapping sections are pretty unnerving but once you get your hold on this then you do not have to panic contemplating the fact that you left behind one topic in your home.

Explaining the two 3-set Venn diagrams

Now, students pursuing the GMAT quant section are well aware of the topic sets. And the above explanations show that we have covered one of the most difficult parts of the quant section of GMAT that is overlapping sets. Let us now delve into another part of sets that is the Venn diagram.

The two 3-set Venn diagrams that we are aware of are – 

1) Total = n (No Set) + n (Exactly one set) + n (Exactly two sets) + n (Exactly three sets)

2) Total = n (A) + n (B) + n(C) – n (A and B) – n (B and C) – n (C and A) + n (A and B and C) + n (No Set)

3) Total = n (No Set) + n (At least one set) – from this point, we can derive n (at least a single set) = total –n (no set) 

Thus, here we have covered a few of the major portions of the GMAT quant section. Students will now find it relatively easier to go through the Sets topic without fretting much.