Question

An outdoor concert venue sells two types of tickets: balcony tickets cost $30 and main floor tickets cost $55. When the venue sells all 500 available tickets for a given show it earns a total of $20,000. How many main floor tickets does the venue offer for each show?

Answer 1

To find the number of main floor tickets the venue offers for each show, we can set up a system of equations based on the given information.

Let's denote the number of balcony tickets sold as "b" and the number of main floor tickets sold as "m".

From the problem, we know that the balcony tickets cost $30 each, and the main floor tickets cost $55 each. We also know that the total number of tickets sold for a given show is 500, and the total earnings from ticket sales is $20,000.

Using this information, we can set up the following equations:

Equation 1: b + m = 500 (since the total number of tickets sold is 500)

Equation 2: 30b + 55m = 20,000 (since the total earnings from ticket sales is $20,000)

To solve this system of equations, we can use substitution or elimination.

Let's solve using elimination:

Multiply Equation 1 by 30 to eliminate "b" and get:

30b + 30m = 15,000

Subtract Equation 2 from this new equation:

(30b + 30m) - (30b + 55m) = 15,000 - 20,000

30b + 30m - 30b - 55m = -5,000

-25m = -5,000

Divide both sides by -25:

m = -5,000 / -25

m = 200

Therefore, the venue offers 200 main floor tickets for each show.

Answer 2

200 main floor tickets

Explanation:

To determine how many main floor tickets the venue offers for each show, we can set up and solve a system of equations.

Let x be the number of balcony tickets sold.

Let y be the number of main floor tickets sold.

As the total number of tickets sold is 500, then:

`x + y = 500`

As the balcony tickets cost $30, the main floor tickets cost $55, and the total revenue from ticket sales is $20,000, then:

`30x + 55y = 20000`

Therefore, the system of equations is:

`\begin{cases}x + y = 500\\30x + 55y = 20000\end{cases}`

Rearrange the first equation to isolate x:

`x = 500 - y`

Substitute this expression for x into the second equation and solve for y:

`\begin{aligned}30(500 - y) + 55y &= 20000\\\\15000-30y+55y&=20000\\\\15000+25y&=20000\\\\25y&=5000\\\\y&=200\end{aligned}`

Therefore, the venue offers 200 main floor tickets for each show.