Question

In a party, $70$guests were to be served tea or coffee after dinner. There were $52$guests who preferred tea while $37$preferred coffee. Each one of the guests liked one or the other beverage. How many liked both tea and coffee?
A) $15$
B) $18$
C) $19$
D) $33$

Answer 1

Hint: Try to use the formulas of set theory or it can be solved by a venn diagram. It is easy to find answers with set theory formulas. It is easier to solve using set theory formula than the venn diagram.

Complete step-by-step solution:
$70$guests were to be served tea or coffee, which means
Total number of guests = $70$
Let $T$and $C$ be guests who preferred tea and coffee.
$n(T \cup C) = 70$
$52$guests preferred tea
$n(T) = 52$
$37$guests preferred coffee
$n(C) = 37$
We have to find number of guests who liked both tea and coffee which is $n(T \cap C)$
So, according to set theory,
$n(T \cup C) = n(T) + n(C) - n(T \cap C)$
Put all the values,
$n(T \cap C) = n(T) + n(C) - n(T \cup C)$
$
   = 52 + 37 - 70 \\
   = 89 - 70 \\
   = 19 \\
 $
Hence, $n(T \cap C) = 19$
So, $19$ guests liked both tea and coffee

Note: Carefully read the statement where you have given intersection and where union, taking it opposite will lead to wrong answer. remember ‘and’ is used for intersection and ‘or’ is used for union.