Question

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

Answer 1

Hint:First define the number of people who likes tea and the number of people who likes coffee and the total number of peoples in the group, and then use the formula given below:
$n\left( {C \cup T} \right) = n\left( C \right) + n\left( T \right) - n\left( {C \cap T} \right)$
Substitute the given values and get the desired result.

Complete step-by-step answer:
It is given that 37 people like coffee, 52 people like tea in a group of 70 people and each person likes one of the two drinks.
The goal is to find the number of people who like both tea and coffee.
Given,
Number of people who like tea,$n\left( T \right) = 52$,
Number of people who likes coffee$n\left( C \right) = 37$,
Total number of peoples$n\left( {C \cup T} \right) = 70$
We have to find the number of people who like both tea and coffee which is denoted as $n\left( {C \cap T} \right)$.
Now, we can use the formula which gives the relation between the union of sets and the intersection of the sets given as:
$n\left( {C \cup T} \right) = n\left( C \right) + n\left( T \right) - n\left( {C \cap T} \right)$
Now, substitute the values into the equation:
$70 = 37 + 52 - n\left( {C \cap T} \right)$
Now, solve the equation for$n\left( {C \cap T} \right)$,
$n\left( {C \cap T} \right) = \left( {37 + 52} \right) - 70$
$n\left( {C \cap T} \right) = 89 - 70$
$ \Rightarrow n\left( {C \cap T} \right) = 19$
Thus, the value of $n\left( {C \cap T} \right)$is $19$, it means that the number of people who likes coffee and tea both is $19$, so there are 19 peoples out of the group of $70$ who like both coffee and tea.

Note:The union of two sets contains all the elements that are the element of at least one of the two sets and the intersection of two sets is the set that contains the elements that exist in both of the sets.