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Question

# Reagan's school is selling tickets to a spring musical. On the first day of ticket sales, the school sold 3 senior citizen tickets and 9 child tickets for a total of $75. The school took in$67 on the second day by selling 8 senior citizen tickets and 5 child tickets. What is the price of each ticket?

Solution

Verified

Let $s$ be the price of each senior ticket and $c$ be the price of each child ticket.

On the first day, they sold 3 senior tickets so they earned $3s$ dollars for the senior tickets. They sold 9 child tickets so they earned $9c$ dollars for the senior tickets. The total amount they earned was then $3s+9c$ dollars. Since they earned $75 on the first day, then $3s+9c=75$. Dividing both sides by 3 then gives $s+3c=25$. On the second day, they sold 8 senior tickets so they earned $8s$ dollars for the senior tickets. They sold 5 child tickets so they earned $5c$ dollars for the senior tickets. The total amount they earned was then $8s+5c$ dollars. Since they earned$67 on the second day, then $8s+5c=67$.

Since the first equation can be easily solved for $s$, use the substitution method to solve the system of equations. Solving $s+3c=25$ for $s$ gives $s=25-3c$.

Substitute $s=25-3c$ into the second equation and solve for $c$:

\begin{aligned} 8s+5c&=67\\ 8(25-3c)+5c&=67&&\text{Substitute.}\\ 200-24c+5c&=67&&\text{Distribute.}\\ 200-19c&=67&&\text{Combine like terms.}\\ -19c&=-133&&\text{Subtract 200 on both sides.}\\ c&=7&&\text{Divide both sides by -19.} \end{aligned}

The price of each child ticket is then $c=\boxed{\7}$ and the price of each senior ticket is $s=25-3c=25-3(7)=25-21=\boxed{\4}$.

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