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Out of 100 persons in a group, 72 persons speak English and 43 persons speak French. Each one out of 100 persons speak at least one language. Then how many speak only English? How many speak only French? How many of them speak English and French both?
Given :
Total number of people in a group $= 100$.
Number of people who speak English $= 72$
Number of people who speak French $= 43$
To do :
We have to find the number of people who speaks only English, only French and both English and French.
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.
This implies,
$n(A) = 72, n(B) = 43$ and $n(A ∪ B) = 100$.
We know that,
$n(A ∩ B) = n(A) + n(B) - n(A ∪ B)$
$n(A) = n(A - B) + n(A ∩ B) $
Therefore, number of people who speak both French and English is equal to,
$n(A \cap B) = 72 + 43 - 100$
$= 115 - 100$
$= 15$
Number of people who speak both French and English is 15.
Number of people who speak English and not French is equal to,
$n(A - B) = n(A) - n(A ∩ B)$
$= 72 - 15$
$= 57$
Number of people who speak French and not English is equal to,
$n(B - A) = n(B) - n(A ∩ B)$
$= 43 - 15$
$= 28$
Number of people speaking only English is 57 .
Number of people speaking only French is 28.
Number of people who speak both French and English is 15.
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