Question

There were 36,000 people at a ball game in Charlotte. The day’s receipts were $304,000. How many people paid $14 for reserved seats and how many paid $6 for general admission. What is the receipts total if 23,500 of the $6 tickets and 12, 421 of the $14 tickets are sold? (Write a system of equations that represents this scenario. Clearly indicate what your variables represent in the context of the problem.)

Answer 1

`\begin{gathered} a)\text{ 25,000 for general tickets, 11,000 for reserved tickets} \\ b)\text{ \$314,894} \end{gathered}`

Step-by-step explanation:

a) Firstly, we start by assigning variables

Let the number of people who paid $14 for the reserved seats be r while the number of people who paid for general admission be g

The total number of tickets is 36,000

Thus:

`g\text{ + r = 36000}`

For the reserved seats, the total amount paid will be:

`14\text{ }* r\text{ = 14r}`

For the general seat, the total amount paid will be:

`6*\text{ g = 6g}`

Now, the total sum paid would be:

`14r\text{ + 6g = 304,000}`

This simply means we have two sets of equations as follows:

`\begin{gathered} g\text{ + r = 36000} \\ 14r\text{ + 6g = 304,000} \end{gathered}`

From equation i:

`r\text{ = 36000-g}`

Substitute this into equation ii as follows:

`\begin{gathered} 14(36000-g)\text{ + 6g = 304000} \\ 504000-14g\text{ + 6g = 304000} \\ 504000-304000\text{ = 14g-6g} \\ 8g\text{ = 200000} \\ g\text{ = }(200000)/(8) \\ g\text{ = 25,000} \end{gathered}`

Recall:

`\begin{gathered} r\text{ = 36000-g} \\ r\text{ = 36000-25000 = 11,000} \end{gathered}`

This means that 11,000 people paid for $14 reserved seats while 25,000 people paid for $6 general seats

b) We want to get the receipts total if 23,500 paid for $6 tickets and 12,421 paid for $14 tickets

Mathematically, we have that as:

`\begin{gathered} 6(23500)\text{ + 14\lparen12,421\rparen } \\ =\text{ \$314,894} \end{gathered}`