At an international conference of 200 delegates, every delegate speaks at least one of thr… — GMAT

At an international conference of $200$ delegates, every delegate speaks at least one of three languages: French, German, or Spanish. Exactly $115$ delegates speak French, $85$ speak German, and $67$ speak Spanish. Furthermore, exactly $15$ delegates speak all three languages. If the number of delegates who speak only one language is four times the number of delegates who speak exactly two languages, how many delegates speak exactly two of these three languages?

Answer: 37 delegates

Answer

The number of delegates who speak exactly two of the three languages is 37.

By dividing the 200 delegates into three disjoint groups—those speaking exactly 1 language (

E_1

), exactly 2 languages (

E_2

), and all 3 languages (

E_3 = 15

)—we establish that

E_1 + E_2 + 15 = 200

, giving

E_1 + E_2 = 185

. Given

E_1 = 4E_2

, substituting yields

5E_2 = 185

, which gives

E_2 = 37

Step-by-Step Solution

Define exact-count variables for the overlapping sets

Let

E_1

be the number of delegates speaking exactly one language,

E_2

be the number speaking exactly two languages, and

E_3 = 15

be the number speaking all three languages.

Categorizing elements into mutually exclusive subsets (exactly 1, exactly 2, exactly 3) simplifies 3-set inclusion-exclusion problems.

Set up the total population equation

E_1 + E_2 + E_3 = 200 \implies E_1 + E_2 + 15 = 200 \implies E_1 + E_2 = 185

Since every delegate speaks at least one language, the sum of all mutually exclusive regions equals the total number of delegates.

Apply the given relationship between

E_1

and

E_2

Substitute

E_1 = 4E_2

into

E_1 + E_2 = 185

to get

4E_2 + E_2 = 185 \implies 5E_2 = 185

The problem establishes that the number of single-language speakers is four times the number of dual-language speakers.

Solve for the target quantity

E_2

E_2 = 37

Dividing 185 by 5 yields the exact count of delegates speaking exactly two languages.

Key Concept

Three-Set Inclusion-Exclusion Principle using Exact-Overlap Subsets

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