Quarter Wit, Quarter Wisdom: Three Overlapping Sets

Today, let’s take a look at how to use Venn diagrams to solve questions involving three overlapping sets.

First, let me show you what the three overlapping sets diagram looks like.

Notice that the total comprises of the elements that do not fall in any of the three sets and the elements that are a part of at least one of the three sets.

The elements falling in the red, yellow or green region (region a, b or c) fall in only one set. The elements falling in region d, e or f fall in exactly two sets and the elements falling in region g fall in all three sets.

Now some quick questions to get a clear picture:

Question 1: Which regions represent the elements that belong to at least 2 sets?

Answer 1: d + e + f + g

Question 2: Which regions represent the elements that belong to at least 1 set?

Answer 2: a + b + c + d + e + f + g = Total – None

Question 3: Which regions represent the elements that belong to at most 2 sets?

Answer 3: None + a + b + c + d + e + f = Total – g

Hope there are no doubts up till now. Let’s look at a question to see how to apply these concepts.

Question: Three table runners have a combined area of 200 square inches. By overlapping the runners to cover 80% of a table of area 175 square inches, the area that is covered by exactly two layers of runner is 24 square inches. What is the area of the table that is covered with three layers of runner?

(A) 18 square inches
(B) 20 square inches
(C) 24 square inches
(D) 28 square inches
(E) 30 square inches

Solution: Let’s first try to understand what exactly is given to us. The area of all the runners is equal to 200 square inches.

Runner 1 + Runner 2 + Runner 3 = 200

In our diagram, this area is represented by
(a + d + g + e) + (b + d + g + f) + (c + e + g + f) = 200

(We need to find the value of g i.e. the area of the table that is covered with three layers of runner.)

Area of table covered is only 80% of 175 i.e. only 140 square inches. This means that if each section is counted only once, the total area covered is 140 square inches.
a + b + c + d + e + f + g = 140
So the overlapping regions are obtained by subtracting second equation from the first. We get d + e + f + 2g = 60
But d + e + f (area with exactly two layers of runner) = 24
So 2g = 60 – 24 = 36

g = 18 square inches

Note that you don’t need to make all these equations and can directly jump to d + e + f + 2g = 60. We wrote these equations down only for clarity. It is a matter of thinking vs solving. If we think more, we have to solve less. Let’s see how.

Combined area of runners is 200 square inches while area of table they cover is only 140 square inches. So what does the extra 60 square inches of runner do? It covers another runner!
Wherever there are two runners overlapping, one runner is not covering the table but just another runner. Wherever there are three runners overlapping, two runners are not covering the table but just the third runner at the bottom.
So can we say that (d + e + f) represents the area where one runner is covering another runner and g is the area where two runners are covering another runner?
Put another way, can we say d + e + f + 2g = 60?
We know that d + e + f = 24 giving us g = 18 square inches
This entire ‘thinking process’ takes ten seconds once you are comfortable with it and your answer would be out in about 30 sec!

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

6 Responses

  1. Hi Karishma,

    I am a regular reader of your blogs, and I happen to be blind. most of your explainations are well written so I don’t need any help in reading your posts, but his post with diagrams was not so clear.

    Usually you write equations with standard computer symbols, so my software is able to read those. It could help if this time also you had written formula for 3 sets.

    Similar to total = a + b – both + neither.

    Not sure, but I thingk the formula for 3 is: total = a+b+c- overlap of 2 groups + * 2all

    • Karishma says:

      Hey Dinesh,
      I wrote this post to discuss venn diagram approach for 3 overlapping sets. I don’t really like to use formulas as there are different variations with slight differences which means they could confuse you.

      As for figuring out the formula, the diagram can be used to do it (though I don’t know whether it works out for you)

      Using the diagram:
      Total = a + b + c + d + e + f + g
      You need to arrive at this region using the number of elements in A, B and C.

      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)

      or Total = n(A) + n(B) + n(C) – n only (A and B) – n only (B and C) – n only (C and A) – 2*n(A and B and C)

      n(A and B) represents the region common to A And B including the region they have common with C.
      n only (A and B) represents the region common only to A and B i.e. it should not be in C.

  2. Vaishno says:

    Hi Karishma,

    I like the conceptual thinking over the algebra usage, but since I was always taught to use algebra so I am trying to understand this question using the formula for 3 overlapping sets.

    Please correct me because I know I am missing something here:

    Total = A + B + C – ( sum of exactly 2-group overlaps) – 2(all three) + Neither

    200 = 175 – 24 – 2x(variable) + 25


    200 = 140 – 24 – 2x + 60 (20% of 175 not covered + 25)


    • Karishma says:

      Note that runners have overlap so they are the sets. The covered area of the table is the total.

      Total is 80% of 175 = 140 (area of the table covered)
      A + B + C = 200 (total area of runners)
      Neither = 0 because no part of the runner is such that it is not on the table.

      140 = 200 – 24 – 2x + 0
      x = 18

  3. zish says:

    Hi karishma

    Can you please explain why we do we have to add None to the other 6 regions in the following
    Question 3: Which regions represent the elements that belong to at most 2 sets?
    Answer 3: None + a + b + c + d + e + f = Total – g

    Does None constitute another region or set that we have to consider? If we are adding none to 6 regions dose it mean that ” None” is a different set? since the question asks us to find the number that belong to at most 2 sets. I assume at most 2 includes those who belong to 1 and those who belong to 2 sets thats y we are adding “None”to other regions because it is also a set. Am i correct?

    I thought that i could just subtract only g from the total to get those regions that belong to at most 2.

    • Karishma says:

      What do we mean by “elements belonging to at most 2 sets?”
      It means we need to consider the elements that belong to no set, to one set and to two sets.
      Elements belonging to no sets – None
      Elements belonging to 1 set – a, b and c
      Elements belonging to 2 sets – d, e and f

      Elements belonging to at most 2 sets – None + a + b + c + d + e + f

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