Question

An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows. 836 drank alcohol regularly, 624 smoked cigarettes, 176 used illegal drugs, 395 drank alcohol regularly and smoked cigarettes, 101 drank alcohol regularly and used illegal drugs, 106 smoked cigarettes and used illegal drugs. 85 engaged in all three behaviors, and 131 engaged in none of these behaviors. How many students were surveyed?

Answer 1

Solution:

Given that 836 drank alcohol regularly, 624 smoked cigarettes, 176 used illegal drugs, let

$\begin{gathered} n(A)\Rightarrow number\text{ of students that drank alcohol} \\ n(C)\Rightarrow number\text{ od student s that smoke cigarettes} \\ n(D)\Rightarrow number\text{ of students that used illegal drugs.} \end{gathered}$

This implies that

$\begin{gathered} n(A)=836 \\ n(C)=624 \\ n(D)=176 \end{gathered}$

If 395 drank alcohol regularly and smoked cigarettes, 101 drank alcohol regularly and used illegal drugs, 106 smoked cigarettes and used illegal drugs, we have

$\begin{gathered} n(A\cap C)=395 \\ n(A\cap D)=101 \\ n(C\cap D)=106 \end{gathered}$

85 engaged in all three behaviors, we have

$n(A\cap C\cap D)=85$

131 engaged in none of these behaviors.

we can represent these data in a venn diagram as follows:

To find how many students were surveyed, we have

$\begin{gathered} Total\text{ number of students surveyed = 425+310+208+85+16+21+54+131} \\ =1250\text{ students} \end{gathered}$

Hence, the total number of students that were surveyed is 1250.