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GMAT Quant — Applied Mathematics (Word Problems)

All 30 members of an international club speak one or more of three languages — Japanese, Italian, and Chinese. If \frac{1}{3} of the members speak Japanese, \frac{2}{5} of the members speak Italian, \frac{3}{5} of the members speak Chinese, and 2 members speak all three of the languages, how many members speak exactly two of the languages?

Explanation	Calculations	Help
$We define variables for each language count and compute their values based on the fractions given.$	$J = \frac{1}{3} \times 30 = 10 I = \frac{2}{5} \times 30 = 12 C = \frac{3}{5} \times 30 = 18 T = 2$	Theory & Tactics Method Card TRANS1 Click to view full details
$We apply the inclusion-exclusion formula for three sets to relate the total to the individual and pairwise counts.$	$30 = 10 + 12 + 18 - (J I + J C + I C) + 2$	Theory & Tactics Method Card SET2 Click to view full details
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$J I + J C + I C = 10 + 12 + 18 + 2 - 30 = 12$	Theory & Tactics Method Card FDPR0-N Click to view full details
$We calculate the number of members who speak exactly two languages by subtracting the contributions of the triple speakers from the sum of pairwise intersections.$	$x = 12 - 3 \times 2 = 6$	Theory & Tactics Method Card FDPR0-N Click to view full details

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C

GMAT Quant — Applied Mathematics (Word Problems)

All 30 members of an international club speak one or more of three languages — Japanese, Italian, and Chinese. If \frac{1}{3} of the members speak Japanese, \frac{2}{5} of the members speak Italian, \frac{3}{5} of the members speak Chinese, and 2 members speak all three of the languages, how many members speak exactly two of the languages?

Explanation	Calculations	Help
$We define variables for each language count and compute their values based on the fractions given.$	$J = \frac{1}{3} \times 30 = 10 I = \frac{2}{5} \times 30 = 12 C = \frac{3}{5} \times 30 = 18 T = 2$	Theory & Tactics Method Card TRANS1 Click to view full details
$We apply the inclusion-exclusion formula for three sets to relate the total to the individual and pairwise counts.$	$30 = 10 + 12 + 18 - (J I + J C + I C) + 2$	Theory & Tactics Method Card SET2 Click to view full details
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$J I + J C + I C = 10 + 12 + 18 + 2 - 30 = 12$	Theory & Tactics Method Card FDPR0-N Click to view full details
$We calculate the number of members who speak exactly two languages by subtracting the contributions of the triple speakers from the sum of pairwise intersections.$	$x = 12 - 3 \times 2 = 6$	Theory & Tactics Method Card FDPR0-N Click to view full details

Scroll horizontally to view all columns

C