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GMAT Quant — Applied Mathematics (Word Problems)

A university's art appreciation club has 30 members. If \frac{1}{3} of the members study painting, \frac{1}{2} study sculpture, \frac{2}{5} study photography, and 2 members study all three art forms, how many members study exactly two of the art forms?

Explanation	Calculations	Help
$We define variables for each art form count and compute their values based on the fractions given.$	$P = \frac{1}{3} \times 30 = 10 S = \frac{1}{2} \times 30 = 15 P h = \frac{2}{5} \times 30 = 12 T = 2$	Theory & Tactics Method Card TRANS1 Click to view full details
$We apply the inclusion-exclusion formula for three sets to relate the total membership to the counts of each art form and their overlaps.$	$30 = 10 + 15 + 12 - (PS + PP h + SP h) + 2$	Theory & Tactics Method Card SET2 Click to view full details
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$PS + PP h + SP h = 10 + 15 + 12 + 2 - 30 = 9$	Theory & Tactics Method Card FDPR0-N Click to view full details
$We calculate the number of members who study exactly two art forms by subtracting the triple-count contributions from the sum of pairwise intersections.$	$x = 9 - 3 \times 2 = 3$	Theory & Tactics Method Card FDPR0-N Click to view full details

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B

GMAT Quant — Applied Mathematics (Word Problems)

A university's art appreciation club has 30 members. If \frac{1}{3} of the members study painting, \frac{1}{2} study sculpture, \frac{2}{5} study photography, and 2 members study all three art forms, how many members study exactly two of the art forms?

Explanation	Calculations	Help
$We define variables for each art form count and compute their values based on the fractions given.$	$P = \frac{1}{3} \times 30 = 10 S = \frac{1}{2} \times 30 = 15 P h = \frac{2}{5} \times 30 = 12 T = 2$	Theory & Tactics Method Card TRANS1 Click to view full details
$We apply the inclusion-exclusion formula for three sets to relate the total membership to the counts of each art form and their overlaps.$	$30 = 10 + 15 + 12 - (PS + PP h + SP h) + 2$	Theory & Tactics Method Card SET2 Click to view full details
$We isolate the sum of the pairwise intersections by solving the equation from the previous step.$	$PS + PP h + SP h = 10 + 15 + 12 + 2 - 30 = 9$	Theory & Tactics Method Card FDPR0-N Click to view full details
$We calculate the number of members who study exactly two art forms by subtracting the triple-count contributions from the sum of pairwise intersections.$	$x = 9 - 3 \times 2 = 3$	Theory & Tactics Method Card FDPR0-N Click to view full details

Scroll horizontally to view all columns

B