Question

A movie theater has a seating capacity of 187. 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 $ 1356, How
many children, students, and adults attended?​

Answer 1

There were 69 children, 83.5 students, and 34.5 adults in attendance at the movie theater.

Step-by-step explanation:

To solve this problem, we can set up a system of equations.

Let's denote the number of children as 'c', the number of students as 's', and the number of adults as 'a'.

From the given information, we have the following equations:

c + s + a = 187 (equation 1)

5c + 7s + 12a = 1356 (equation 2)

We also know that there are half as many adults as children, so we have the equation:

a = (1/2)c (equation 3)

Substituting equation 3 into equations 1 and 2, we can solve for the variables.

Substituting (1/2)c for a in equation 1:

c + s + (1/2)c = 187

(3/2)c + s = 187 (equation 4)

Substituting (1/2)c for a in equation 2:

5c + 7s + 12((1/2)c) = 1356

5c + 7s + 6c = 1356

11c + 7s = 1356 (equation 5)

Now we can solve equations 4 and 5 simultaneously:

Multiplying equation 4 by 11:

(33/2)c + 11s = 2057/2

11c + 7s = 1356

Subtracting the second equation from the first:

(33/2)c - 11c = 2057/2 - 1356

(-1/2)c = -69/2

c = 69

Substituting c = 69 into equation 4:

(3/2)(69) + s = 187

103.5 + s = 187

s = 83.5

Finally, substituting c = 69 and s = 83.5 into equation 3:

a = (1/2)(69)

a = 34.5

Therefore, there were 69 children, 83.5 students (which is not a whole number, so it could be rounded to 84 students), and 34.5 adults (which could be rounded to 35 adults) in attendance at the movie theater.

Answer 2

Number of children in theater = 94

Number of students = 46

Number of adults in theater= 47

Step-by-step explanation:

Total seating capacity in the theater = 187

Let us assume the number of students in the theater = m

and assume the number of children in the theater = 2k

So, the number of adults in theater = Half of number of children = 2k/2 = k

⇒ Number of ( Adults + Children + students) = 187

⇒ k + 2k + m = 187, or 3k + m = 187

Cost of 1 adult ticket = $12

So, the cost of k adult tickets = 12 x (k) = $12k

Cost of 1 student ticket = $7

So, the cost of m student ticket = 7 x (m) = $7m

Cost of 1 children ticket = $5

So, the cost of 2k children tickets = 5 x (2k) = $10k

⇒ 12k + 10k + 7m = 1356,

or 22k + 7m = 1356

Now, the given equations are:

3k + m = 187

22k + 7m = 1356

Substitute m = 187 - 3 k in second equation ,we get

22k + 7m = 1356 ⇒ 22 k + 7 ( 187 - 3 k) = 1356

⇒ k = 47

⇒ m = 187 - 3 k = 187 - 3(47)= 46, or m = 46

Hence, the number of children in theater = 2 k = 2 (47) = 94

The number of students in the theater = m = 46

The number of adults in theater = k = 47