Chapter 11.4, Problem 31E

a.

To determine

To organize: The given information in a two-way table.

Explanation of Solution

Given Information:

Number of tickets sold to play $=1809$

Number of tickets for main floor $=800$

Number of adult tickets on the main floor $=2x+y$

Number of child tickets on the main floor $=x-40$

Number of adult tickets in the balcony $=x+2y$

Number of child tickets in the balcony $=3x-y-80$

Table:

Creating the two-way table,

  $\begin{array}{llll}\hfill & MainFloor\hfill & Balcony\hfill & Total\hfill \\ Child\hfill & x-40\hfill & 3x-y-80\hfill & \hfill \\ Adult\hfill & 2x+y\hfill & x+2y\hfill & \hfill \\ Total\hfill & 800\hfill & 1009\hfill & 1809\hfill \end{array}$

b.

To determine

To find: The values of $x$ and $y$ .

Answer to Problem 31E

  $x=249$ and $y=93$ .

Explanation of Solution

Given Information:

Number of tickets sold to play $=1809$

Number of tickets for main floor $=800$

Number of adult tickets on the main floor $=2x+y$

Number of child tickets on the main floor $=x-40$

Number of adult tickets in the balcony $=x+2y$

Number of child tickets in the balcony $=3x-y-80$

Calculation:

Calculating the values with the help of equations formed through two-way table,

From column $1$ ,

  $\begin{array}{l}(x-40)+(2x+y)=800\\ 3x+y=840-(i)\end{array}$

From column 2,

  $\begin{array}{l}(3x-y-80)+(x+2y)=1009\\ 4x+y=1089-(ii)\end{array}$

Subtracting $(i)$ from $(ii)$ ,

  $x=249$

Putting this value of $x$ in $(i)$ ,

  $\begin{array}{l}3(249)+y=840\\ y=93\end{array}$

Thus, $x=249$ and $y=93$ .

c.

To determine

To find: The percentage of adult tickets.

Answer to Problem 31E

Percentage of adult tickets is about $57\%$ .

Explanation of Solution

Given Information:

Number of tickets sold to play $=1809$

Number of tickets for main floor $=800$

Number of adult tickets on the main floor $=2x+y$

Number of child tickets on the main floor $=x-40$

Number of adult tickets in the balcony $=x+2y$

Number of child tickets in the balcony $=3x-y-80$

Calculation:

Putting $x=249$ and $y=93$ to complete the two-way table,

  $\begin{array}{llll}\hfill & MainFloor\hfill & Balcony\hfill & Total\hfill \\ Child\hfill & 209\hfill & 574\hfill & 783\hfill \\ Adult\hfill & 591\hfill & 435\hfill & 1026\hfill \\ Total\hfill & 800\hfill & 1009\hfill & 1809\hfill \end{array}$

Using the information from completed two-way table,

Percentage of adult tickets $=\frac{1026}{1809}\simeq 0.57or57\%$

Thus, percentage of adult tickets is about $57\%$ .

d.

To determine

To find: The percentage of child tickets that are balcony tickets.

Answer to Problem 31E

Percentage of child tickets that are balcony tickets is about $73\%$ .

Explanation of Solution

Given Information:

Number of tickets sold to play $=1809$

Number of tickets for main floor $=800$

Number of adult tickets on the main floor $=2x+y$

Number of child tickets on the main floor $=x-40$

Number of adult tickets in the balcony $=x+2y$

Number of child tickets in the balcony $=3x-y-80$

Calculation:

Using the information from completed two-way table,

Total child tickets $=783$

Total child tickets in balcony $=574$

Percentage of child balcony tickets $=\frac{574}{783}\approx 0.73or73\%$

Thus, percentage of child tickets that are balcony tickets is about $73\%$ .