Question

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

Answer 1

We can consider the people who like coffee as a set and people who likes tea as another set. Then the union of these sets are the people whole like at least one of the two and their intersection gives the set of people who like both tea and coffee. Then we can find the number of people who like both using the equation $n\left( {A \cap B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cup B} \right)$.

Complete step by step solution:
Consider C as the set of people who coffee. Then number of people who like coffee is given by, $n\left( C \right) = 37$
Consider T as the set of people who tea. Then number of people who like tea is given by, $n\left( T \right) = 52$
It is given that out of 70 peoples, all of them likes at least one of the two drinks.
$\Rightarrow T \cup C$ is the union of sets T and C. Their number is given by, $n\left( {T \cup C} \right) = 70$
We can draw a Venn diagram with the given details for better understanding.

Here, curves T and C represent the sets of people who like tea and coffee. The overlapping region is the intersection of the 2 sets and the union of the set is given by the region of both the circle.
Now we need to find the number of people who likes both tea and coffee. These people who likes both belong to the intersection of T and C which is represented by $T \cap C$. Then we need to find $n\left( {T \cap C} \right)$
We know that, $n\left( {T \cap C} \right) = n\left( T \right) + n\left( C \right) - n\left( {T \cup C} \right)$
Substituting the values in the equation, we get,
$n\left( {T \cap C} \right) = 52 + 37 - 70$
On solving we get,
$\Rightarrow n\left( {T \cap C} \right) = 19$

Therefore, the number of people who like both coffee and tea are 19.

Note:
The concept used here is set theory. A set is a well-defined collection of objects. Union of two sets gives a set of all elements that are at least in one of the two sets. The intersection of two sets gives the set of all elements that are in both the sets. A Venn diagram is used to represent sets. In a Venn diagram, each set is separated by a circle or a curve. The elements inside each curve belong to that particular set. The Venn diagram is enclosed in a rectangle which represents the universal set.