Question

3. Scott's school is selling tickets to a choral performance. On the first day of ticket sales the school sold 6 adult tickets and 11 student tickets for a total of $139. The school took in $178 on the second day by selling 12 adult tickets and 2 student tickets.a) Write a system of equations to represent the situation. Be sure to define your variables.b) Find the price of an adult ticket and the price of a student ticket. You may use either substation or elimination.c) Scott is budgeting for next year's school's materials. How much money should be budget if on the third day, he sells 9 student tickets and 3 adult tickets. (Be sure to include all money made)

Answer 1

a) We have to write a system of equations to solve this problem.

The variables are the price of the adult ticket (A) and the price of the student ticket (S).

On the first day the sales where $139, which correspond to 6 adult tickets and 11 students tickets.

The total sales are equal to the sum of the sales of the adult tickets (price times number of tickets) and the sum of the sales of the students tickets (price times number of tickets).

We then can write:

$6\cdot A+11\cdot S=139$

We can use the same logic for the second day and obtain:

$12\cdot A+2\cdot S=178$

Then, we have a system of equations as:

$\begin{cases}6A+11S={139} \\ 12A+2S={178}\end{cases}$

b) We can find the prices by solving this system of equations.

We can use elimination, by substracting two times the first equation from the second equation:

$\begin{gathered} 12A+2S-2(6A+11S)=178-2(139) \\ 12A+2S-12A-22S=178-278 \\ -20S=-100 \\ S=\frac{-100}{-20} \\ S=5 \end{gathered}$

We can now use this information to find A as:

$\begin{gathered} 6A+11S=139 \\ 6A+11(5)=139 \\ 6A+55=139 \\ 6A=139-55 \\ 6A=84 \\ A=\frac{84}{6} \\ A=14 \end{gathered}$

Then, the price for an adult ticket is $14 and the price for a student ticket is $5.

c) He sells 9 student tickets and 3 adult tickets for the 3rd day, so we can calculate how much money this represents as:

$\begin{gathered} 9S+3A \\ 9(5)+3(14) \\ 45+42 \\ 87 \end{gathered}$

The total revenue is $87.

Now, if we calculate the budget as the sum of the revenue for the 3 days:

$\begin{gathered} B=R_1+R_2+R_3 \\ B=139+178+87 \\ B=404 \end{gathered}$

Answer:

a) The equations are:

6A + 11S = 139

12A + 2S = 178

where A: price of adult ticket, S: price of student ticket.

b) A = 14, S = 5

c) The budget has to be $404, the revenue of the three days.