Question

Suppose that there are two types of tickets to a show: advance and same day. Advance tickets cost 30 and the same day tickets cost 20. For one performance there were 60 tickets sold in all and the total amount paid for them was $1600. How many tickets of each type were sold

Answer 1

Let A be the number of advance tickets sold and S be the total number of same-day tickets sold. The total amount of tickets is A+S, then:

`A+S=60`

The total earnings for A advanced tickets is 30A, while the total earnings for selling S same-day tickets is 20S. Then, the total amount of money for selling A advanced tickets and S same-day tickets, is 30A+20S, then:

`30A+20S=1600`

Solve the system of equations to find the total amount of tickets of each type that were sold. To do so, isolate A from the first equation and then substitute the resulting expression in the second one:

`\begin{gathered} A+S=60 \\ \Rightarrow A=60-S \end{gathered}`
`\begin{gathered} 30A+20S=1600 \\ \Rightarrow30(60-S)+20S=1600 \end{gathered}`

Solve for S:

`\begin{gathered} \Rightarrow1800-30S+20S=1600 \\ \Rightarrow1800-10S=1600 \\ \Rightarrow-10S=1600-1800 \\ \Rightarrow-10S=-200 \\ \Rightarrow S=-(200)/(-10) \\ \therefore S=20 \end{gathered}`

Substitute S=20 into the expression for A:

`\begin{gathered} A=60-S \\ =60-20 \\ =40 \end{gathered}`

Then, the solution for this system is:

`\begin{gathered} A=40 \\ S=20 \end{gathered}`