Question

Draw a Venn Diagram to obtain the answers.In a survey of 85 cities, it was found that:24 had an Amusement Park55 had a Baseball Team38 had a Circus13 had an Amusement Park and Baseball Team10 had an Amusement Park and a Circus19 had a Baseball Team and a Circus7 had all three features

Answer 1

SOLUTION

The Venn Diagram showing the information:

a) How many cities had only a circus?

$\begin{gathered} \text{Let C be the number of cities that had only circus.} \\ C\text{ + 12+ 7 + 3 = 38} \\ C\text{ + 22 = 38} \\ C\text{ = 38 - 22} \\ C\text{ = 16} \end{gathered}$

b) How many cities had at least one of the features?

$\begin{gathered} \text{For the Amusement Park only, we have that:} \\ A\text{ + 6+ 7 + 3 = 24} \\ A\text{ + 16 = 24} \\ \text{ A = 24 - 16} \\ A\text{ = 8} \end{gathered}$$\begin{gathered} \text{For the Baseball Team only, we have that:} \\ B\text{ + 6 + 7 + 12 = 55} \\ B\text{ + 25 = 55} \\ B\text{ = 55 - 25} \\ \text{B = 30} \end{gathered}$

The total number that had at least one of the features =

$\begin{gathered} A\text{ + B + C + ( A n B ) + ( A n C ) + ( B n C ) + ( A n B n C ) =} \\ 8\text{ + 30 +16 + 6 + 3 + 12 + 7 = 82} \end{gathered}$

c ) How many cities had exactly two of these features?

$\begin{gathered} (\text{ A n B ) + ( B n C ) + ( A n C )} \\ =\text{ 6 + 12 + 3 } \\ =\text{ 21} \end{gathered}$

d ) How many cities had none of the features?

$85\text{ - 82 = 3}$

e) How many cities had AT LEAST 2 of the features?

$\begin{gathered} (\text{ A n B ) + ( A n C ) + ( B n C ) + ( A n B n C )} \\ =\text{ 6 + 3 + 12 + 7} \\ =\text{ 28} \end{gathered}$