Question

A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read both newspapers, then what is the possible value of x?
A.37
B.29
C.65
D.55

Answer 1

Hint:First draw a Venn diagram of the stated problem. Then use the concept that B is less than \[A \cup B\] and \[A \cup B\] is less than \[U\], that is the universal set. Use the inequality to obtain the required result.

Complete step by step solution
Suppose that the universal set is \[n(U) = 100\] .
It is given that \[n(A) = 63,n(B) = 76\] and \[n(A \cap B) = x\] .
The Venn diagram of the stated problem is,

Image: Venn diagram of set A and set B
We can see that,
\[n(B) \le n(A \cup B) \le n(U)\]
Therefore,
\[76 \le 63 + 76 - x \le 100\]
\[76 \le 139 - x \le 100\]
\[76 - 139 \le - x \le 100 - 139\]
\[ - 63 \le - x \le - 39\]
\[63 \ge x \ge 39\]
In the given options only D satisfies the given inequality.
The correct option is D.

Additional information
In this type of problem, we use Venn diagrams to clear the concept we are going to use to solve the problem. From the given information it is not so difficult to draw the required image. With the help of the diagram use the required values and obtain the result.

Note Always remember that when inequality is multiplied by a negative sign, the greater than sign becomes less than, and the less than sign becomes greater than, which means the direction of the inequality changes.