1. Understand the Problem
We need to find the minimum number of people who speak all three languages. The information tells us about the number of people speaking *at least* one of the languages.
2. The Formula
The Principle of Inclusion-Exclusion helps us with this type of problem. For three sets (in this case, the languages), the formula is:
Total = A + B + C - (A ∩ B) - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C)
Where:
* A, B, C represent the number of people speaking each language.
* A ∩ B represents the number speaking both A and B.
* A ∩ C represents the number speaking both A and C.
* B ∩ C represents the number speaking both B and C.
* A ∩ B ∩ C represents the number speaking all three.
3. Applying the Formula
Let's plug in the values:
100 = 90 + 80 + 75 - (A ∩ B) - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C)
4. Minimizing the Intersection
To find the minimum number speaking all three languages (A ∩ B ∩ C), we need to maximize the other intersections:
* Maximize (A ∩ B): The maximum overlap between Spanish and Italian is 80 (since there are only 80 Italian speakers).
* Maximize (A ∩ C): The maximum overlap between Spanish and Mandarin is 75 (since there are only 75 Mandarin speakers).
* Maximize (B ∩ C): The maximum overlap between Italian and Mandarin is 75 (since there are only 75 Mandarin speakers).
5. Solving for (A ∩ B ∩ C)
Now we can plug these maximum values back into the equation:
100 = 90 + 80 + 75 - 80 - 75 - 75 + (A ∩ B ∩ C)
100 = 65 + (A ∩ B ∩ C)
(A ∩ B ∩ C) = 35
Answer: At least 35 people speak all three languages.