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Question

Difficulty: EasySet Theory Concepts and Venn Diagrams

In a department of 9090 employees, 5555 speak French, 4545 speak Spanish, and 1010 speak neither French nor Spanish. How many employees speak both French and Spanish?

Answer: 20 employees

Answer

20
Subtracting the 10 employees who speak neither language from the total of 90 leaves 80 employees who speak at least one language. By the principle of inclusion-exclusion, Total(At least one) = French + Spanish - Both. Substituting the known values gives 80 = 55 + 45 - Both, which simplifies to 80 = 100 - Both, so Both = 20.

Step-by-Step Solution

1
Find the number of employees who speak at least one of the languages
80 employees
Subtract the 10 employees who speak neither language from the total department size of 90.
2
Sum the total counts for each language group
100
Add the number of French speakers (55) to Spanish speakers (45).
3
Calculate the overlap (intersection) using inclusion-exclusion
20 employees
Subtract the count of employees speaking at least one language (80) from the sum of the individual language groups (100).

Key Concept

Principle of Inclusion-Exclusion for Two Sets
Estimated Time:45s
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Topics

Set Theory Concepts and Venn Diagrams

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