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 1/3 of the members speak Spanish, 2/5 of the members speak French, 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?

GMAT Panda · Accepted Answer

E We define variables for each language count and compute their values based on the fractions given. S = 1/3 15 = 5 F = 2/5 15 = 6 G = 2/3 15 = 10 T = 1 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 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 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 1 = 4

Explanation	Calculations	Help
$We define variables for each language count and compute their values based on the fractions given.$	$S = \frac{1}{3} \times 15 = 5 F = \frac{2}{5} \times 15 = 6 G = \frac{2}{3} \times 15 = 10 T = 1$
$We apply the inclusion-exclusion formula for three sets to relate the total to the individual and pairwise counts.$	$15 = 5 + 6 + 10 - (SF + SG + FG) + 1$
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$SF + SG + FG = 5 + 6 + 10 + 1 - 15 = 7$
$We calculate the number of members who speak exactly two languages by subtracting the triple count from the sum of pairwise intersections.$	$x = 7 - 3 \times 1 = 4$

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