A movie theater has a seating capacity of 235. The theater charges $5 for children, $7 for students, and $12 for adults. There are half as many adults as there are children. If the total ticket sales was $1706, How many children, students, and adults attended?

61 adults, 122 children, and 52 students

Step-by-step explanation:

Let a = amount of adults, c = amount of children, and s = amount of students.

Let's go over what we know:

total amount: 235

twice as many children as adults

children: $5

student: $7

adult: $12

Using this information, we can determine the following:

a = 1/2c

235 = c + s + a

1706 = 5c + 7s + 12a

Since we know a = 1/2c, plug 1/2c in for a. The equations become:

235 = c + s + 1/2c

1706 = 5c + 7s + 6c

Now, solve the inequalities by substitution.

235 = c + s + 1/2c

235 = 1 1/2c + s

-s + 235 = 1 1/2c

-s = - 235 + 1 1/2c

s = 235 - 1 1/2c

Now, plug in the value for s into the other equation.

1706 = 5c + 7 (235 - 1 1/2c) + 6c

-5c + 1706 = 7 (235 - 1 1/2c) + 6c

-5c = - 1706 + 7 (235 - 1 1/2c) + 6c

-5c = - 1706 + 1645 - 21c/2 + 6c

-5c = - 61 - 21c/2 + 6c

-5c = - 61 - 9c/2

c = 122

Now, plug in the value for c into the value of s.

s = 235 - 1 1/2 (122)

s = 235 - 183

s = 52

Finally, plug in the value for c into the value of a.

a = 1/2 (122)

a = 61

To check, add up all values to make sure they equal 235.

61 + 52 + 122 = 235.

Therefore, the final answers are 61 adults, 122 children, and 52 students.