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Question: In a committee, 50 people speak French, 20 speak Spanish, and 10 speak both French and Spanish. How ...

In a committee, 50 people speak French, 20 speak Spanish, and 10 speak both French and Spanish. How many speak at least one of these two languages?

Explanation

Solution

Use the formula of set theory for the number of items n(A∪B)=n(A)+n(B)−n(A∩B)n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) for any two items A and B. Also, note that the number of items for each type is given in the question which is only required for the solution.

Complete step by step answer:
Let say F be the set of people who speak French in the committee and let say S be the set of people who speak Spanish in the committee.
Number of people who speak French, n(F)=50n\left( F \right) = 50
Number of people who speak Spanish, n(S)=20n\left( S \right) = 20
Number of people who speak both Hindi and English, n(F∩S)=10n\left( {F \cap S} \right) = 10
People who speak at least one language, n(F∪S)n\left( {F \cup S} \right)
The formula of set theory for the number of items is,
n(A∪B)=n(A)+n(B)−n(A∩B)n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)
Replace A with F and B with S,
⇒n(F∪S)=n(F)+n(S)−n(F∩S)\Rightarrow n\left( {F \cup S} \right) = n\left( F \right) + n\left( S \right) - n\left( {F \cap S} \right)
Substitute the values,
⇒n(F∪S)=50+20−10\Rightarrow n\left( {F \cup S} \right) = 50 + 20 - 10
Simplify the terms,
⇒n(F∪S)=60\Rightarrow n\left( {F \cup S} \right) = 60

Hence, 60 people in the committee in at least one of the two languages.

Note: The concept used here is set theory. Union of two sets gives a set of all elements that are at least in one of the two sets. If A and B are 2 sets, then its union is written as A∪BA \cup B and it contains all the elements in A along with all the elements in B. Intersection of two sets gives the set of all elements that are in both the sets. If A and B are 2 sets, then its intersection is written as A∩BA \cap B and it contains all the elements that are both in A and B. For set A, n(A)n\left( A \right) represents the number of elements in the set, and for set B, n(B)n\left( B \right) represents the number of elements in the set.