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Difficulty: MediumLogical Venn Diagrams

A language institute conducted a survey among 400400 civil service aspirants to assess their proficiency in three foreign languages: French, German, and Spanish. The survey revealed the following data:
- 180180 aspirants are proficient in French.
- 150150 aspirants are proficient in German.
- 160160 aspirants are proficient in Spanish.
- 4040 aspirants are proficient in exactly French and German, but not Spanish.
- 3030 aspirants are proficient in exactly German and Spanish, but not French.
- 5050 aspirants are proficient in exactly French and Spanish, but not German.
- 7070 aspirants are not proficient in any of these three languages.

What is the number of aspirants who are proficient in all three languages?

Answer: 20 aspirants

Answer

20
By determining the union of the three sets (400−70=330400 - 70 = 330) and applying the inclusion-exclusion principle while correctly distinguishing between 'exactly two' and the full intersection of two sets, we find that 2020 aspirants are proficient in all three languages.

Step-by-Step Solution

1
Determine the number of aspirants proficient in at least one of the three languages.
n(F∪G∪S)=400−70=330n(F \cup G \cup S) = 400 - 70 = 330
The total population consists of those who speak at least one language and those who speak none.
2
Set up an equation using the Principle of Inclusion-Exclusion for three sets. Let xx be the number of aspirants proficient in all three languages.
n(F∩G)=40+xn(F \cap G) = 40 + x, n(G∩S)=30+xn(G \cap S) = 30 + x, and n(F∩S)=50+xn(F \cap S) = 50 + x
The total intersection of any two sets includes those in exactly those two sets plus those in all three sets.
3
Substitute all values into the union formula.
330=180+150+160−(40+x)−(30+x)−(50+x)+x330 = 180 + 150 + 160 - (40 + x) - (30 + x) - (50 + x) + x
The formula n(F∪G∪S)=n(F)+n(G)+n(S)−n(F∩G)−n(G∩S)−n(F∩S)+n(F∩G∩S)n(F \cup G \cup S) = n(F) + n(G) + n(S) - n(F \cap G) - n(G \cap S) - n(F \cap S) + n(F \cap G \cap S) accounts for all overlapping regions.
4
Simplify the equation and solve for xx.
330=490−120−2x  ⟹  330=370−2x  ⟹  2x=40  ⟹  x=20330 = 490 - 120 - 2x \implies 330 = 370 - 2x \implies 2x = 40 \implies x = 20
Basic algebraic simplification yields the final value for the intersection of all three sets.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Alternative Method

Instead of using the union formula, use a region-based approach in a Venn diagram. Let the central 'all three' region be xx. Calculate the 'only one' regions in terms of xx: Only French = 180−(40+50+x)=90−x180 - (40 + 50 + x) = 90 - x. Only German = 150−(40+30+x)=80−x150 - (40 + 30 + x) = 80 - x. Only Spanish = 160−(50+30+x)=80−x160 - (50 + 30 + x) = 80 - x. The sum of all disjoint regions inside the union is (90−x)+(80−x)+(80−x)+40+30+50+x=370−2x(90 - x) + (80 - x) + (80 - x) + 40 + 30 + 50 + x = 370 - 2x. Since the union is 400−70=330400 - 70 = 330, we have 370−2x=330370 - 2x = 330, which gives x=20x = 20.
Estimated Time:2m 0s
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Topics

Logical Venn Diagrams

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