EXAMPLE 3.3.12 SOLUTION

The information refers to two overlapping categories: "wears dark glasses" and "smokes cigars." We can organize the information in the following two-circle Venn diagram:

 

 

Venn diagram

 

1. To answer the question "How many wear dark glasses and don't smoke cigars?", we can subtract.
Since 30 wear dark glasses, and 28 those who wear dark glasses also smoke cigars, the number who wear dark glasses but don't smoke cigars must be 30 minus 28 equals 2.

 

2. To answer the question "How many wear dark glasses or smoke cigars", we use the following fact (based on what we did in Unit 1 Module 3):
number who wear dark glasses or smoke cigars = number who wear dark glasses plus number who smoke cigars minus number who do both things
= 30 plus 32 minus 28
= 34

 

 

In general, for a counting problem that refers to overlapping categories A and B,
the number who are in category A or B = the number who are in category A plus the number who are in category B minus the number who are in the intersection of the two categories.