Practice GMAT Problem Solving Question
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Walking across campus, a student interviewed a group of students. 25% of the students took a finance class last semester, 50% took a marketing class last semester, and 40% took neither a finance nor a marketing class last semester. What percent of the students in the group took both a finance and a marketing class?
Correct Answer: D
 There are two common ways of solving this problem. One involves algebra and the other involves statistical formulas.

Method 1: Use Algebra
 Assign variables to the groups of interest:
Let b = the percent of students who took both classes (what we are interested in).
Let f = the percent of students who only took a finance class.
Let m = the percent of students who only took a marketing class.  We know that 40% of the students did not take either class, so 60% (=100%  40%) must have taken either a finance class, a marketing class, or both.
 This 60% is made up of those three distinct groups: those who took a finance class only, those who took a marketing class only, and those who took both:
m+f+b=60%.  We know that 25% of the students took a finance class, which is made up of those who only took this class and those who took both classes:
f+b=25%.  Likewise, 50% of the students took a marketing class, made up of those who only took marketing and those who took both:
m+b=50%.  We are interested in finding the value of b (percent who took both classes). So solve these last two equations for f and m by subtracting b from both sides of each equation:
f=25%b.
m=50%b.  Now plug these values of f and m into the first equation:
m+f+b=60%
50%b + 25%b + b = 60%.  Combine like terms to simplify:
75%  b = 60%.  Add b to both sides:
75%= 60% + b.  Subtract 60% from both sides:
15%= b.
Thus the correct answer is D.
 Assign variables to the groups of interest:

Method 2: Use Statistical Formulas
 In general, the probability of event M or F occurring is P(M∪F) = P(M) + P(F)  P(M∩F) where P(M∩F) is the probability of M and F simultaneously occurring.
 In this problem:
P(M) = the probability of a student taking marketing
P(F) = the probability of a student taking finance
P(M∪F) = the probability of a student taking marketing or finance
P(M∩F) = the probability of a student taking marketing and finance; this is the variable we are trying to solve for  Fill in what we know:
P(M) = 50%
P(F) = 25%  An important insight into this problem is to realize that (the probability of a student taking marketing or finance) + (the probability of a student taking neither marketing nor finance) = 1 since these two events are complementary and complementary events must sum to one.
 The question tells us that "40% took neither a finance nor a marketing class last semester." As a result, we know that 40% + P(M∪F) = 100%
Consequently: (M∪F) = 60%  Filling all that we know into the fundamental equation:
P(M∪F) = P(M) + P(F)  P(M∩F)
60% = 50% + 25%  P(M∩F)
15% =  P(M∩F)
P(M∩F) = 15%
Thus the correct answer is D.
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