Question

A movie theater manager wants to know how many adults and how many children pay admission to a particular movie. The theater charges $10.00 for adult tickets and $5.00 for child tickets. At a showing where 210 tickets were sold the theater collected $1,355.00. Write a system of equations and determine the number of adult tickets sold and the number of child tickets sold.

What would be the best algebraic method to solve given how the equations are set up?
Elimination or Substitution?
What is the cost of an adult ticket?

Answer 1

To solve the problem, let's define the two variables:

- Let \( a \) be the number of adult tickets sold.
- Let \( c \) be the number of child tickets sold.

From the problem statement, we can set up the following system of equations based on the information provided:

1. The total number of tickets sold:
\[
a + c = 210
\]

2. The total amount of money collected:
\[
10a + 5c = 1355
\]

Now we have a system of linear equations:

\[
\begin{align*}
1) & \quad a + c = 210 \\
2) & \quad 10a + 5c = 1355
\end{align*}
\]

### Determining the Best Method
Both elimination and substitution methods can be used to solve this system. Here, substitution might be a good choice since the first equation is already solved for \( c \):

From equation 1, solve for \( c \):
\[
c = 210 - a
\]

Now substitute this expression for \( c \) into equation 2:
\[
10a + 5(210 - a) = 1355
\]

### Solving the Equations
Now, let's solve it step-by-step:

1. Substitute:
\[
10a + 5(210 - a) = 1355
\]
\[
10a + 1050 - 5a = 1355
\]

2. Combine like terms:
\[
5a + 1050 = 1355
\]

3. Subtract 1050 from both sides:
\[
5a = 305
\]

4. Now, divide by 5:
\[
a = 61
\]

Now that we have \( a \), we can find \( c \):
\[
c = 210 - a = 210 - 61 = 149
\]

### Conclusion
The number of adult tickets sold is \( 61 \) and the number of child tickets sold is \( 149 \).

### Cost of an Adult Ticket
The cost of an adult ticket is \( \$10.00 \).

So, the final answers are:
- **Adult tickets sold:** 61
- **Child tickets sold:** 149
- **Cost of an adult ticket:** $10.00