Question

In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French?

Answer 1

Solution


Let A be the set of people who speak English. 


B be the set of people who speak French. 


A - B be the set of people who speak English and not French. 


B - A be the set of people who speak French and not English. 


A ∩ B be the set of people who speak both French and English. 


Given


n(A) = 72       n(B) = 43       n(A ∪ B) = 100 


Now, n(A ∩ B) = n(A) + n(B) - n(A ∪ B) 


                   = 72 + 43 - 100 


                   = 115 - 100 


                   = 15 


Therefore, Number of persons who speak both French and English

 = 15 


n(A) = n(A - B) + n(A ∩ B) 


⇒ n(A - B) = n(A) - n(A ∩ B) 


               = 72 - 15 


               = 57


and n(B - A) = n(B) - n(A ∩ B) 


                 = 43 - 15 


                 = 28 


Therefore, Number of people speaking English only = 57 


Number of people speaking French only = 28 

Answer 2

Let A --> the set of people who speak English. 

B --> the set of people who speak French. 

A - B --> the set of people who speak English and not French. 

B - A --> the set of people who speak French and not English. 

A ∩ B --> the set of people who speak both French and English. 

Given,

n(A) = 72       n(B) = 43       n(A ∪ B) = 100 
 
Now, n(A ∩ B) = n(A) + n(B) - n(A ∪ B) 

          = 72 + 43 - 100 
          = 15 
∴ Number of persons who speak both French and English 
        = 15 
n(A) = n(A - B) + n(A ∩ B) 

⇒    n(A - B) = n(A) - n(A ∩ B) 
                     = 72 - 15 
                     = 57
and n(B - A) = n(B) - n(A ∩ B) 
                     = 43 - 15 
                     = 28 
∴ Number of people speaking English only = 57 
Number of people speaking French only = 28


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