Final answer:
By setting up a system of equations with the given information about seating capacity, ticket prices, and the ratio of adults to children, we can solve for the number of children, students, and adults that attended the movie theater. We find that 154 children, 74 students, and 77 adults attended.
Step-by-step explanation:
To solve how many children, students, and adults attended the movie theater, we need to set up a system of equations using the given information:
- The theater has a seating capacity of 305.
- The ticket prices are $5.00 for children, $7.00 for students, and $12.00 for adults.
- There are half as many adults as there are children.
- The total ticket sales was $2212.
Let's denote the number of children as c, the number of students as s, and the number of adults as a. Based on the information, we have the following equations:
- c + s + a = 305 (total seating capacity)
- 5c + 7s + 12a = 2212 (total ticket sales)
- a = 0.5c (half as many adults as children)
We can substitute a with 0.5c in the first two equations:
- c + s + 0.5c = 305
- 5c + 7s + 12(0.5c) = 2212
Combining like terms, we get:
1.5c + s = 305
- 11c + 7s = 2212
Now we can solve the system of equations using substitution or elimination. Let's use substitution:
- From the first equation we get s = 305 - 1.5c.
- Substitute s in the second equation: 11c + 7(305 - 1.5c) = 2212.
- Solve for c: 11c + 2135 - 10.5c = 2212, which simplifies to 0.5c = 77, so c = 154.
- We substitute c back into s = 305 - 1.5c to find s: s = 305 - 1.5(154), which gives us s = 74.
- Now find a using a = 0.5c: a = 0.5(154), giving us a = 77.
Therefore, 154 children, 74 students, and 77 adults attended the movie theater.