Question
A market research group conducted a survey of 2000 consumers and reported that 1720 consumers like product $${P_1}$$ and 1450 consumers like product $${P_2}.$$ What is the least number that must have liked both the products ?
A.
1150
B.
2000
C.
1170
D.
2500
Answer :
1170
Solution :
Let $$U$$ be the set of all consumers who were questioned, $$A$$ be the set of consumers who liked product $${P_1}$$ and $$B$$ be the set of consumers who liked product $${P_2}.$$
It is given that
$$\eqalign{
& n\left( U \right) = 2000,\,\,n{\left( A \right)} = 1720,\,\,n\left( B \right) = 1450, \cr
& n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) \cr
& = 1720 + 1450 - n\left( {A \cap B} \right) \cr
& = 3170 - n\left( {A \cap B} \right) \cr
& {\text{Since,}}\,A \cup B \subseteq U\,\,\,\therefore \,\,n\left( {A \cup B} \right) \leqslant n\left( U \right) \cr
& \Rightarrow \,3170 - n\left( {A \cap B} \right) \leqslant 2000 \cr
& \Rightarrow \,3170 - 2000 \leqslant n\left( {A \cap B} \right) \cr
& \Rightarrow n\left( {A \cap B} \right) \geqslant 1170 \cr} $$
Thus, the least value of $$n\left( {A \cap B} \right)$$ is 1170.
Hence, the least number of consumers who liked both the products is 1170.