Set theory  Maxima / Minima Using Line Method

Set theory is a good friend for those who find it difficult to score in quant section. We need to arrange the given data based on the question and this can get complicated sometimes. There are some methods which will make our life easier while dealing with set based questions. We have shared two methods in the article, Line method and Sum of sets. If you know any thumb rule regarding when to use what or some other good methods to solve set based questions please share your thoughts.
Thanks Ankan Sengupta and Alok Moghe for your support in this article :)
Let's solve!
In a school of 100 students each of them play at least one sport among cricket, football, volleyball, basketball, hockey and tennis.It is known that exactly 90 play cricket, 80 play football, 70 play hockey, 60 play basketball , 40 play volleyball and 10 play tennis.
What is the maximum number of students who play exactly four of the six games?
Don't rush to venn diagrams. We will represent the given data as simple lines.
Now we are asked to find the max number of students who play EXACTLY 4 games. All what we need is to draw boxes which will contain EXACTLY 4 lines using the given 6 lines. Just ensure that all lines in the box has the same length and as the question is asked forĀ maximum we need to arrange the lines so that the length of the box ( basically overlapping part of the lines within the box) should be maximum.
We can draw 3 boxes as above. one has 4 lines of length 60 each and other 2 boxes having 5 lines of length 10 in each of them. Answer is 60 + 10 + 10 = 80. Easy right ?
What is the maximum number of students who play exactly five of the six games?
Again we will try to draw boxes which will contain EXACTLY 5 lines using the given 6 lines.
We can draw 2 boxes. one with 5 lines of length 40 each and other one with 5 lines of length 10 each. Our answser is 40 + 10 = 50.
What is the minimum number of students who play cricket, football and hockey?
Here we are asked for the minimum. So we need to arrange the lines so that we can have the box which has all 3 lines with MINIMUM length of lines. Remember we have 100 students in the class. Diagram will look like below
The length of the lines in the box is 100  ( 20 + 30 + 10 ) = 40, our answer :)
PS: you can also solve this one using below trick
Find the difference from total for each item and add all of them together.
For cricket, 100  90 = 10
For football, 100  80 = 20
For hockey, 100  70 = 30
Sum = 10 + 20 + 30 = 60
Again find the difference from the total, 100  60 = 40 :)We will solve another set now.
The following table gives the number of students who secured more than 90% marks in each of the five subjects from class 6th to 10th at a school in year 2006
class/subject english physics chemistry maths biology 6th 12 16 15 22 18 7th 15 22 22 21 15 8th 7 18 16 23 17 9th 10 19 15 22 18 10th 15 25 21 29 16 The number of students in the different classes in the year were as below
Class No. of students 6th 30 7th 35 8th 28 9th 36 10th 40 In the class 7th the number of students who scored more than 90% in at least two of the five subjects is at least
Sum of set method: Let a, b, c, d and e represents students who got more than 90% in exactly 1, 2, 3, 4 and 5 subjects respectively.
a + 2b + 3c + 4d + 5c = sum of all those who got above 90
= 15 + 22 + 22 + 21 + 15
= 95Now we need to minimize b, c, d and e ( atleast 2 subject cases ). We can do this by putting b = c = d = 0 and putting max value to e ( as we multiply e with 5 )
so a + 5e = 95
As atleast 15 students got 90% in a subject (check the table) e can have the minimum value of 15.
so e = 15 and a = 20 is the case.
and minimum atleast 2 case will be 15.This will satisfy total number of students too, 20 + 15 = 35.
For the curious ones line method given below
Line method : We need to find the minimum value of atleast 2. now atleast 2 consist of ( students with 90% in 2 subjects ) + ( students with 90% in 3 subjects ) + ( students with 90% in 3 subjects ) + ( students with 90% in 4 subjects ) + ( students with 90% in 5 subjects ). So to minimize "atleast 2" scenario we have to maximize atleast 1.
Diagram given below
We can see with atleast 1 case maximum ( means boxes which can have only one line ) we get min atleast 2 cases as 15.
Happy learning. Cheers!