CLASS-8
WORD PROBLEM OF VENN DIAGRAM

WORD PROBLEM

Example.- A) Out of 50 students in a class someone like either Chocolate or Cake or both, 25 like Chocolate while 15 like both. Draw a Venn diagram and find the number of students who like 1) Cake, 2)  Only Cake and 3) Only Chocolate.

Answer) Let decide C = { students who likes Chocolate }, M = { students who like Cake }

Then  n(C) = 25

n (C U M) = number of students who like either chocolate or milk or both = 50 and  n(C ∩ M) = number of students who like both =  15       

Now, as per the rules  n (C U M) =  n(M) + n(C) - n(C ∩ M)

Or,   50  =  n(M)  + 25 – 15

Or   n(M) = 40

Number of students who like Cake n(M) = 40

Number of students who like both Chocolate and Cake  n(C ∩ M) =  15

The number of students who like only Chocolate  =  n(M) - n(C ∩ M)

                                                      =   40 – 15

                                                      =   25

And the number of students who like Cake =  n(C) - n(C ∩ M)

                                               =   25 – 15

                                               =  10


If we show this obtained figure in the Venn diagram, then it looks like -

In this diagram  -

Total students of class = 50

Number of Students who like Chocolate = 25


Number of students who like Cake = 10

Number of students who like both Chocolate & Cake = 15




Example.-B) Out of a class of 120 students, 80 students likes a have math, 70 students like Physics and all like either math or physics or both, find the number of students who like 1) both math physics & math, 2) only math, 3) only physics  

Answer)  Let us assume –

M = { students who like Math }, P = { students who like Physics }

M U P = { students who like either math or physics }

M ∩ P = { student who like both }

As per the given condition, n(M) = 80,  n(P) = 70, n(M U P) = 120

Let, n(M ∩ P) =  the number of students who like both =  B

Then the number of students who like only math = 80 – B

And the number of students who like only physics = 70 – B

So, the total number of students = 120 = (80 – B) + B + (70 – B)

Or,   120 =  150 – B

Or,    B  =  150 – 120 =  30

1) so, the number of students who like both math & physics = B = 30

2) the number of students who like only math =  80 – B 

                                                  =  80 – 30 = 50

3) the number of students who like only physics = 70 – B =  40 

If we show this obtained figure in the Venn diagram, then it looks like -

In this diagram –

Total number of students = 120

Number of students who like Math = 50


Number of Students who like physics = 40

Number of students who like both = 30



Example.-C) Out of a group of  2000 people, 250 like to watch Tennis, 500 like to watch football and 150 watch both, find the number of people who like to watch – 1) only tennis, 2) only football, 3) either tennis or football or both, 4) neither tennis nor football     

Answer)  let us assume –

Let, U = { the entire group }, T = { People who like to watch tennis },

 F = { people who like to watch football }

As per the given condition, n(T) = 250,  n(F) = 500, n(T ∩ F) = 150

1) the number of people who watch only Tennis =

 n(T) – n(T ∩ F)  = 250 – 150  = 100

 

2)  the number of people who watch only football =

       n(F) –  n(T ∩ F) 

      =  500 – 150 =  350

 

3)  the number of people who watch either tennis or football or both =

 n (T U F)  =  n(T) + n(F) - n(T ∩ F) 

             =  250 + 500 – 150  =  600


4) the number of people who watch neither tennis nor football

  n(U) - n (T U F)  = 2000 – 600 = 1400