how many ticket of each were sold?

Question

A theatre contains 455 seats and the ticket prices were $49 for adults and $15 the total proceeds were 13,326 for a sold-out, how many of each type of ticket were sold?

Answer 1

Hello.

 

You have a problem which is very common and you can quickly learn how to conquer it by charting.

 

First, I assume you mean $15 for children's tickets? Assuming that, I worked the problem and got a fractional number of tickets. A little experimenting showed me that the $49 should probably be $48, which gives an integral number of tickets.

 

To make tracking facts easy and to make it clear which step you need to take next, use charting. You have the price for 2 types of tickets, the number of seats, and the total income received.

 

How do those numbers relate? We know the $13,326 was the sum of adult and children ticket sales. If 100 adult tickets were sold at $48 each, how much money was received by the theater from adult sales? It's simply 100 times 48, or $4800, right? (Not our numbers, just an example.)

 

So multiplying the cost of an adult ticket times the number of adult tickets sold gives the total revenue from adults. Ditto the same process for children.

 

We also know that the adult ticket sales plus children's gives the total $13,326 which we will put in our chart.

 

We don't know how many adult and children's tickets were sold, but we know that their total number was 455. A technique used frequently in algebra problems is assigning a variable to one unknown value, and then expressing the related unknown value with that same variable.

 

Think about how adult and children's ticket numbers relate. If there were 100 adult tickets sold (example, not our real number), then there must have been 355 children's tickets sold, since adult plus children equals 455 tickets. How did we get 355? Adult + children = 455, so 100 + children = 455, and children = 455 - 100 = 355. It helps to think about how the arithmetic works, but now let's do the same process with Algebra.

 

Let A = the number of adult tickets sold. Then the number of children's tickets must be 455 - A, which we will use in our chart.

 

Let's put it all together now. Make a chart with columns to systematize what you know and see what you still need. The amounts in columns 1 and 2 multiple together to give the product in column 3. The amounts in rows 1 and 2 add to give the total in row 3.

 

 

 

               $ EACH TICKET    # TICKETS OF THIS TYPE   $ TOTAL FROM THIS TICKET TYPE

 

ADULT               48                               A                                48 TIMES A

 

CHILDREN         15                           455 - A                          15 TIMES (455 - A)

 

TOTAL         NOT NEEDED           A + 455 - A = 455           48 A + 15(455 - A) = 13,326

 

 

We know the algebraic expression for the total revenue 48 A + 15 (455 - A) and we know what that total is (13,326), so set them equal and solve.

 

We get that 197 adult tickets were sold and 258 children's tickets.