GMAT Practice Question: All 15 members of a foreign language club speak one or more of three languages-Spanish, French,...
Question
All 15 members of a foreign language club speak one or more of three languages-Spanish, French, and German. If \frac{1}{3} of the members speak Spanish, \frac{2}{5} of the members speak French, \frac{2}{3} of the members speak German, and 1 member speaks all three of the languages, how many members speak exactly two of the languages?
- 0
- 1
- 2
- 3
- 4
Topics: word problems, sets & groups
Solution
Step 1
We define variables for each language count and compute their values based on the fractions given.
S = \frac{1}{3} \times 15 = 5 F = \frac{2}{5} \times 15 = 6 G = \frac{2}{3} \times 15 = 10 T = 1
Step 2
We apply the inclusion-exclusion formula for three sets to relate the total to the individual and pairwise counts.
15 = 5 + 6 + 10 - (SF + SG + FG) + 1
Step 3
We isolate the sum of the pairwise intersections by solving the equation from the previous step.
SF + SG + FG = 5 + 6 + 10 + 1 - 15 = 7
Step 4
We calculate the number of members who speak exactly two languages by subtracting the triple count from the sum of pairwise intersections.
x = 7 - 3 \times 1 = 4
Answer
E
