movie theater question

Question

A movie theater has a seating capacity of 207. The theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. There are half as many adults as there are children. If the total ticket sales was $ 1496, How many children, students, and adults attended?

Answer 1

In this problem, we are creating and solving for a system of equations. First, identify the variables for the problem.

c = # of children

s = # of students

a = # of adults

The first equation in the system of equations is about the ticket sales. The sales of all of the moviegoers needs to add up to $1496, so we create the following equation.

1) 1496= 5c + 7s + 12a

The second equation is about the seating in the movie theater. We know that there can not be more than 207 people in the theater in total, so we create this equation.

2) 207 = c + s + a

We also know that for there are half as many adults as there are children. Another way of phrasing this is by saying that for every one adult there will be two children. We can represent this with the equation:

3) 2a=c

We can then substitute this equation into the first two equations, replacing c with 2a. This gives us:

1) 1496 = 5(2a) + 7s + 12a

2) 207 = 2a + s + a

Simplified, these equations become:

1) 1496= 22a + 7s

2) 207 = 3a + s

From here, you can go about solving the system of equations using either substitution or elimination. For this problem, I will use substitution, first isolating s.

Equation 2 becomes:

207-3a = s

Then substituting (207-3a) for s in equation 1, equation 1 becomes:

1496 = 22a + 7(207 - 3a)

1496 = 22a + 1449 -21a

1496 = a + 1449

47 = a

Now that we have solved for a, we can substitute the value of 47 back into either equation. Substituting it into equation 2 gives us:

207 = 3(47) + s

207 = 141 + s

66 = s

From here we can substitute 47 for a in our 3rd equation regarding the relationship between number of children and adults in order to find the number of children.

2a = c

2(47) = c

94 = c

Now we have solved for all the variables with the final answer being:

Number of children = 94

Number of students = 66

Number of adults = 47