GMAT® Official Guide 2019 Quantitative Review
- Sets
If Sis the set of numbers 1, 2, 3, and 4, you can write S = [l, 2, 3, 4}. Sets can also be represented by
Venn diagrams. That is, the relationship among the members of sets can be represented by circles.
Example 1: Each of 25 people is enrolled in history, mathematics, or both. If 20 are enrolled in history
and 18 are enrolled in mathematics, how many are enrolled in both history and mathematics?Solution: The 25 people can be divided into three sets: those who study history only, those who study
mathematics only, and those who study history and mathematics. Thus a Venn diagram may be drawn
as follows, where n is the number of people enrolled in both courses, 20 - n is the number enrolled in
history only, and 18 - n is the number enrolled in mathematics only.MathematicsSince there is a total of 25 people, (20 - n) + n + (18 - n) = 25, or n = 13. Thirteen people are enrolled
in both history and mathematics. Note that 20 + 18 - 13 = 25, which is the general addition rule for
two sets (see section 4.1.9).Example 2: In a certain production lot, 40 percent of the toys are red and the remaining toys are green.
Half of the toys are small and half are large. If 10 percent of the toys are red and small, and 40 toys are
green and large, how many of the toys are red and large?Solution: For this kind of problem, it is helpful to organize the information in a table:/'
Red Green Total\Small 10% 50%Large 50%\. Total 40% 60% 100%The numbers in the table are the percentages given. The following percentages can be computed on the
basis of what is given:/'
Red Green Total\Small 10% 40% 50%Large 30% 20% 50%\. Total 40% 60% 100%Since 20% of the number of toys (n) are green and large, 0.20n = 40 (40 toys are green and large), or
n = 200. Therefore, 30% of the 200 toys, or (0.3)(200) = 60, are red and large.