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GMAT Quant — Applied Mathematics (Word Problems)

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?

Explanation	Calculations	Help
$We define variables for each language count and compute their values based on the fractions given.$	$S = \frac{1}{3} \times 15 = 5 F = \frac{2}{5} \times 15 = 6 G = \frac{2}{3} \times 15 = 10 T = 1$	Theory & Tactics Method Card TRANS1 Click to view full details
$We apply the inclusion-exclusion formula for three sets to relate the total to the individual and pairwise counts.$	$15 = 5 + 6 + 10 - (SF + SG + FG) + 1$	Theory & Tactics Method Card SET2 Click to view full details
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$SF + SG + FG = 5 + 6 + 10 + 1 - 15 = 7$	Theory & Tactics Method Card FDPR0-N Click to view full details
$We calculate the number of members who speak exactly two languages by subtracting the triple count from the sum of pairwise intersections.$	$x = 7 - 3 \times 1 = 4$	Theory & Tactics Method Card FDPR0-N Click to view full details

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GMAT Quant — Applied Mathematics (Word Problems)

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?

Explanation	Calculations	Help
$We define variables for each language count and compute their values based on the fractions given.$	$S = \frac{1}{3} \times 15 = 5 F = \frac{2}{5} \times 15 = 6 G = \frac{2}{3} \times 15 = 10 T = 1$	Theory & Tactics Method Card TRANS1 Click to view full details
$We apply the inclusion-exclusion formula for three sets to relate the total to the individual and pairwise counts.$	$15 = 5 + 6 + 10 - (SF + SG + FG) + 1$	Theory & Tactics Method Card SET2 Click to view full details
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$SF + SG + FG = 5 + 6 + 10 + 1 - 15 = 7$	Theory & Tactics Method Card FDPR0-N Click to view full details
$We calculate the number of members who speak exactly two languages by subtracting the triple count from the sum of pairwise intersections.$	$x = 7 - 3 \times 1 = 4$	Theory & Tactics Method Card FDPR0-N Click to view full details

Scroll horizontally to view all columns

E