GRE Prep Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
GRE Quant - Overlapping Set Theory
[#permalink]
10 Jul 2020, 10:30
10 Jul 2020, 10:30
1
Expert Reply
3
Bookmarks
Re: GRE Quant - Overlapping Set Theory
[#permalink]
10 Jul 2020, 23:54
10 Jul 2020, 23:54
2
Expert Reply
2
Bookmarks
Overlapping Sets Made Easy!
A rare question involves an overlapping set for group members, These questions a fairly easy but they could result intimidating to those students who have never seen them before.
Say, there are a total of 100 people in a housing society. There are two clubs close to the society – A and B. You are given that of the 100 people of the housing society, 60 people are members of club A and 50 people are members of club B.
Question 1: How many people are members of both the clubs?
We are looking for the number of people in the green region. The answer here is not 10. It is ‘cannot be determined’ i.e. you cannot say how many people are members of both the clubs. The reason is that you do not know how many people belong to neither club.
Say, for all future questions (unless mentioned otherwise), you are given that 20 people belong to neither club. What can you say about the number of people who belong to both the clubs? Now, out of the pool of 100, 20 are out. Only 80 people are club members. Since 60 are members of club A and 50 people are members of club B which gives us a total of 110, there must be an overlap of 30 people i.e. 30 people must belong to both the clubs (80 = 60 + 50 – Both)
Question 2: How many people belong to only one club?
We found above that 30 people belong to both the clubs. So out of the 60 people of club A, 30 belong to only club A. Out of the 50 people of club B, 20 belong to only club B. So a total of 30+20 = 50 people belong to only one club, either A or B but not both. (60 – Both + 50 – Both = 30 + 20 = 50)
Question 3: Say, you don’t know the number of people who belong to neither club. What is the minimum number of people who must belong to both the clubs?
We know that there are a total of 100 people. 60 belong to club A and 50 belong to club B which adds up to 110. Therefore, AT LEAST 10 people must have membership of both the clubs. Now if you increase the number of people who do not belong to either club, the number of people who belong to both will increase by the same number. Think in terms of the Venn diagram. If the ‘Neither’ number increases, the number of people who are members decreases. Hence, the overlap increases to keep A = 60 and B = 50.
Let’s look at the promised question which will make this concept clear.
Question: Of the 400 members at a health club, 260 use the weight room and 300 use the pool. If at least 60 of the members do not use either, then the number of members using both the weight room and the pool must be between:
(A) 40 to 100
(B) 80 to 140
(C) 160 to 260
(D) 220 to 260
(E) 220 to 300
Solution: When we minimize “number of members who do not use either”, we are minimizing the “number of members who use both” as well.
Look at the equation:
Total = A + B – Both + Neither
Eventually, this is the only formula you may need to memorize to tackle the overlapping sets on the GRE
Since the total sum 400 is constant, if we increase the ‘Neither’ i.e. 60, we will have to increase the ‘Both’ term too to maintain the sum of 400 (Assuming A and B are constant which they are since they are given to us).
Least value of ‘number of members who use neither’ is 60. We will get the least value of ‘number of members who use both’ when we put ‘Neither’ = 60.
400 = 260 + 300 – ‘Minimum value of both’ + 60
Minimum value of both = 220
On the same lines, if we maximize “number of members who use neither”, we are maximizing the “number of members who use both” as well.
What is the maximum number of people who use neither? Out of a total of 400 people, 300 people use the pool. Hence at least 300 people use at least one of the two facilities. This means that there can be AT MOST 100 people (total 400 – 300 who use pool) who use neither facility.
400 = 260 + 300 – ‘Maximum value of both’ + 100
Maximum value of both = 260
Re: GRE Quant - Overlapping Set Theory
[#permalink]
11 Jul 2020, 00:34
11 Jul 2020, 00:34
1
Expert Reply
1
Bookmarks
Three Overlapping Sets
How a 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!
In the end, it's not that difficult of a concept to memorize if you simply extrapolate the formula for overlapping sets as follows:
Total = Group1 + Group2 + Group3 - Those in Exactly Two Groups - 2(Those in All Three Groups) + None
