Final answer:
To solve the movie theater ticket sales problem, we used a system of equations with the variables c (children), s (students), and a (adults), leading to the solution: 112 children, 67 students, and 56 adults attended.
Step-by-step explanation:
Step-by-Step Solution to the Movie Theater Problem
Let's break down the problem into a system of equations based on the given information about the movie theater's ticket sales. We're told that the theater charges $5.00 for children, $7.00 for students, and $12.00 for adults, and that there are half as many adults as there are children.
The total sales from the three types of tickets amount to $2112.
Let c be the number of children, s be the number of students, and a be the number of adults. Therefore, we have:
1 adult for every 2 children, so a = c/2
$5 × c + $7 × s + $12 × a = $2112
The total number of people is the sum of children, students, and adults, so c + s + a = 291
These three expressions give us a system of equations we can solve to find the exact numbers of children, students, and adults.
1. a = c/2
2. 5c + 7s + 12a = 2112
3. c + s + a = 291
We substitute a = c/2 into the other two equations and solve for c and s.
Upon solving, we get the following numbers of attendees:
Children (c): 112
Students (s): 67
Adults (a): 56
Now we know how many children, students, and adults attended the movie.