Question

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If \[x\] denotes the percentage of them who like both coffee and tea. Which is not a value of \[x\]?
A. 63
B. 54
C. 38
D. 36

Answer 1

Hint:We know that \[n\left( U \right) \le n\left( {A \cup B} \right)\] , where \[U\] is the universal set. Then we will apply the formula \[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\] to calculate the least possible value of \[x\].

Formula Used:
\[n\left( U \right) \le n\left( {A \cup B} \right)\]
\[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\]

Complete step by step solution:
Let, \[C\] denotes the workers who like coffee.
Let, \[T\] denote the workers who like tea.
Assume that the total number of workers is 100.
The number of workers of like coffee is \[n\left( C \right) = 73\].
The number of workers of like tea is \[n\left( T \right) = 65\].
The number of workers who like both tea and coffee is \[x\].
Applying the formula \[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\] to calculate \[n\left( {C \cup T} \right)\]
\[n\left( {C \cup T} \right) = n\left( C \right) + n\left( T \right) - n\left( {C \cap T} \right)\]
Substitute the value of \[n\left( C \right)\], \[n\left( T \right)\] and \[n\left( {C \cap T} \right)\]
\[n\left( {C \cup T} \right) = 73 + 65 - x\]
Since the total number of workers is 100. So, \[n\left( {C \cup T} \right) \le 100\]
Therefore, \[73 + 65 - x \le 100\]
Add 73 and 65
\[ \Rightarrow 138 - x \le 100\]
Subtract 138 from both sides of the inequality
\[ \Rightarrow 138 - x - 138 \le 100 - 138\]
\[ \Rightarrow - x \le - 38\]
Multiply -1 on both sides and reverse the inequality sign
\[ \Rightarrow x \ge 38\]
The value of \[x\] is always greater than or equal to 38.
56 and 63 are greater than 38.
 But 36 is less than 38.
So 36 is not a solution to the inequality.

Hence option D is the correct option.

Note: Student often do a mistake to solve the inequality \[ - x \le - 38\]. They simply multiply -1 on both sides of the inequality but did not reverse the inequality sign. For this reason, they get an incorrect solution.