Chapter 11.1, Problem 75AYU

To determine

To calculate: The number of orchestra, main and balcony seats in a Broadway theater with a total capacity of 500 seats divided in to orchestra, main and balcony. The selling price for orchestra, main and balcony seats are $50, $35, and $25 respectively. The gross revenue is $17100 with all the seats have sold. The gross revenue is $14600 with all the main and balcony seats are sold, but only half of orchestra seats are sold.

Answer to Problem 75AYU

There are 100 orchestra, 210 main and 190 balcony seats in a Broadway theater with a total capacity of 500 seats.

Explanation of Solution

Given information:

It is given that A Broadway Theater has 500 seats, divided in to orchestra, main and balcony. The selling price for orchestra, main and balcony seats are $50, $35, and $25 respectively. The gross revenue is $17100 with all the seats have sold. The gross revenue is $14600 with all the main and balcony seats are sold, but only half of orchestra seats are sold.

Calculation:

Consider the given information that total capacity of 500 seats.

Now, let

  $\begin{array}{c}x=\text{ the number of orchestra seats}\\ y=\text{the number of main seats}\\ z=\text{ the number of balcony seats}\end{array}$

Since, the total numbers of seats are 500. So,

  $x+y+z=500$

Since, the total revenue is $17100. So,

  $50x+35y+25z=17100$

Since, the gross revenue is $14600 with all the main and balcony seats are sold, but only half of orchestra seats are sold So,

  $\begin{array}{c}50\left(\frac{1}{2}\right)x+35y+25z=14600\\ 25x+35y+25z=14600\end{array}$

Now, system of equation is,

  $\{\begin{array}{c}x+y+z=500\\ 50x+35y+25z=17100\\ 25x+35y+25z=14600\end{array}$

Now, solve first and second equation as,

  $\begin{array}{l}\underset{\_}{\begin{array}{c}\left(-\text{25}\right)x+\left(-\text{25}\right)y+\left(-\text{25}\right)z=\left(-\text{25}\right)500\\ 50x+35y+25z=17100\end{array}}\\ 25x+10y=4600\end{array}$

Now, subtract third equation from second equation in the initial system of equation as,

  $\begin{array}{l}\underset{\_}{\begin{array}{c}50x+35y+25z=17100\\ -25x-35y-25z=-14600\end{array}}\\ 25x=2500\\ x=100\end{array}$

Substitute x in the equation $25x+10y=4600$ as,

  $\begin{array}{c}25x+10y=4600\\ 25\left(100\right)+10y=4600\\ 10y=4600-2500\\ y=210\end{array}$

Next is,

  $\begin{array}{c}x+y+z=500\\ 100+210+z=500\\ z=500-100-210\\ z=190\end{array}$

Therefore, there are 100 orchestra, 210 main and 190 balcony seats in a Broadway theater with a total capacity of 500 seats.