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GMAT Quant — Applied Mathematics (Word Problems)

A club of 20 members offers courses in three languages-Italian, Japanese, and Portuguese. If \frac{1}{4} of the members study Italian, \frac{3}{5} of the members study Japanese, \frac{1}{2} of the members study Portuguese, and 2 members study all three languages, how many members study exactly two languages?

Explanation	Calculations	Help
$We define variables for each language count and compute their values based on the fractions given.$	$I = \frac{1}{4} \times 20 = 5 J = \frac{3}{5} \times 20 = 12 P = \frac{1}{2} \times 20 = 10 T = 2$	Theory & Tactics Method Card TRANS1 Click to view full details
$We apply the inclusion-exclusion formula for three sets to relate the total to the individual and pairwise counts.$	$20 = 5 + 12 + 10 - (I J + I P + J P) + 2$	Theory & Tactics Method Card SET2 Click to view full details
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$I J + I P + J P = 5 + 12 + 10 + 2 - 20 = 9$	Theory & Tactics Method Card FDPR0-N Click to view full details
$We calculate the number of members who study exactly two languages by subtracting the triple count from the sum of pairwise intersections.$	$x = 9 - 3 \times 2 = 3$	Theory & Tactics Method Card FDPR0-N Click to view full details

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GMAT Quant — Applied Mathematics (Word Problems)

A club of 20 members offers courses in three languages-Italian, Japanese, and Portuguese. If \frac{1}{4} of the members study Italian, \frac{3}{5} of the members study Japanese, \frac{1}{2} of the members study Portuguese, and 2 members study all three languages, how many members study exactly two languages?

Explanation	Calculations	Help
$We define variables for each language count and compute their values based on the fractions given.$	$I = \frac{1}{4} \times 20 = 5 J = \frac{3}{5} \times 20 = 12 P = \frac{1}{2} \times 20 = 10 T = 2$	Theory & Tactics Method Card TRANS1 Click to view full details
$We apply the inclusion-exclusion formula for three sets to relate the total to the individual and pairwise counts.$	$20 = 5 + 12 + 10 - (I J + I P + J P) + 2$	Theory & Tactics Method Card SET2 Click to view full details
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$I J + I P + J P = 5 + 12 + 10 + 2 - 20 = 9$	Theory & Tactics Method Card FDPR0-N Click to view full details
$We calculate the number of members who study exactly two languages by subtracting the triple count from the sum of pairwise intersections.$	$x = 9 - 3 \times 2 = 3$	Theory & Tactics Method Card FDPR0-N Click to view full details

Scroll horizontally to view all columns

D