A movie theater has a seating capacity of 337. 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 $ 2456, How many children, students, and adults attended?
children attended.
students attended.
adults attended.

194 children, 46 students and 97 adults attended.

Step-by-step explanation:

Let x represent the number children that attended.

Let y represent the number students that attended.

Let z represent the number adults that attended.

The movie theater has a seating capacity of 337. Assuming it was filled to capacity, It means that

x + y + z = 337- - - - - - - - - - -1

The theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. If the total ticket sales was $ 2456, it means that

5x + 7y + 12z = 2456- - - - - - - - -2

There are half as many adults as there are children. It means that

z = x/2

x = 2z

Substituting x = 2z into equation 1 and equation 2, it becomes

2z + y + z - 337

y + 3z = 337- - -- - - - - - -3

5 × 2z + 7y + 12z = 2456

10z + 7y + 12z = 2456

7y + 22z = 2456- - - - - - - -4

Multiplying equation 3 by 7 and equation 4 by 1, it becomes

7y + 21z = 2359

7y + 22z = 2456

Subtracting, it becomes

- z = - 97

z = 97

x = 2z = 2 × 97

x = 194

Substituting x = 194 and z = 97 into equation 1, it becomes

194 + y + 97 = 337

y + 291 = 337

y = 337 - 291

y = 46