how many tickets are sold

Question

a concert hall has 1,250 seats. Tickets are sold for $45 and $85. If all seats are sold, how many seats of each type should be sold to generate a revenue of $86250?

Answer 1

Let x = # of $45 tickets sold

Let y = # of $85 tickets sold

 

From the given information, we can readily conclude the following two math statements (equations):

 

x + y = 1250   (the total of $45 tickets and $85 tickets is 1,250 tickets for the 1,250 seats in the concert hall)

45x + 85y = 86,250 ($45 price times the number of unknown tickets sold at $45, plus the $85 price times the unknown number of tickets sold at $85, equals total ticket sale of:  $86,250

 

So, we have the following two equations:

 

x + y = 1250

45x + 85y = 86250

 

Let's make the first equation state in terms of y; so that for the second equation, whenever we see the 'y', we substitute in the value for 'y' instead, so:

 

x + y = 1250

Subtract x from both sides, to get:

 

y = 1250 - x

 

Go to the second equation, then substitute the y-value, in lieu of the 'y' variable, to get:

 

45x + 85(1250-x) = 86250

 

Distribute to get:

 

45x + 106250-85x = 86250

 

Next, collect like terms to get:

 

-40x + 106250 = 86250; next, subtract 106250 from both sides, to get:

 

-40x = -20000

 

Next, divide both sides by -40, to get the x-value ($45 tickets) of:  500

 

Go back to the original equation of:  x + y = 1250

 

Substitute 500 for 'x', to get:  500 + y = 1250

 

Then, subtract 500 from both sides to get:  y = 750

 

Next, check your work:

 

x + y = 1250;  500 + 750 = 1,250   [Check]

 

45*500 + 85*750 = 22,500 + 63750 = 86,250   [Check]

 

500 Tickets at $45.00 and 750 Tickets at $85.00 is the correct and final answer.

 

 

Answer 2

a + b = 1250

45a + 85b = 86250

 

Solve for either a or b in the 1st equation, substitute into the 2nd equation and solve for the one variable. Then, plug the found value for that variable into the first equation to solve for the other variable.