Question

A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, then.

• A. $x = 39$
• B. $x = 63$
• C. $39 \leq x \leq 63$
• D. None of these

Answer 1

Answer: (C)

$39 \leq x \leq 63$

Answer 2

Let A denote the set of Americans who like cheese and let B denote the set of Americans who like apples.

Let Population of American be 100.

Then $n ( A ) = 63 , n ( B ) = 76$

Now, $n ( A \cup B ) = n ( A ) + n ( B ) - n ( A \cap B )$

$= 63 + 76 - n ( A \cap B )$

∴ $n ( A \cup B ) + n ( A \cap B ) = 139$

⇒ $n ( A \cap B ) = 139 - n ( A \cup B )$

But $n ( A \cup B ) \leq 100$

∴$- n ( A \cup B ) \geq - 100$

∴ $139 - n ( A \cup B ) \geq 139 - 100 = 39$

∴$n ( A \cap B ) \geq 39$ i.e., $39 \leq n ( A \cap B )$ .....(i)

Again,

∴ $n ( A \cap B ) \leq n ( A ) = 63$ and $n ( A \cap B ) \leq n ( B ) = 76$

∴$n ( A \cap B ) \leq 63$ …..(ii)

Then, $39 \leq x \leq 63$.