Question

out of hundred percent group 72 persons speak English and 43 persons speak French each one of the 100 person speak at least one language then how many speak only English? speak only French ?how many of them speak English and French both?

Answer 1

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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