Group of Groups [ Statistics ] GMAT, GRE

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?

A)	60%
B)	50%
C)	25%
D)	15%
E)	10%

1. Algebraic Method:
   
   1. 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.
   2. 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.
   3. 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%.
   4. 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%.
   5. 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%.
   6. 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.
   7. Now plug these values of f and m into the first equation:
      m+f+b=60%
      50%-b + 25%-b + b = 60%.
   8. Combine like terms to simplify:
      75% - b = 60%.
   9. Add b to both sides:
      75%= 60% + b.
   10. Subtract 60% from both sides:
       15%= b.
       Thus the correct answer is D.
2. Statistical Method:
   1. 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.
   2. 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
   3. Fill in what we know:
      P(M) = 50%
      P(F) = 25%
   4. 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.
   5. 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%
   6. 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%
      

   
   

   

   
   

D.