A movie theater has a seating capacity of 317. 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 $ 2296, How many children, students, and adults attended?

There are 154 children, 77 adults and 86 students in attendance

Step-by-step explanation:

Given

Seat capacity = 317

Total tickets = $2296

Charges is as follows;

Children = $5.00

Students = $7.00

Adults = $12.00

Required

Number of children, students and adults

Let A, C and S represent adults, children and students respectively.

So,

From sales of tickets, we have the following:

12A + 5C + 7S = 2296 --- Equation 1

From attendance, we have

A + C + S = 317 --- Equation 2

Given that, there are half as many adults as there are children.

So, A = ½C

Substitute ½C for A in equation 1 and 2

12A + 5C + 7S = 2296 becomes

12 * ½C + 5C + 7S = 2296

6C + 5C + 7S = 2296

11C + 7S = 2296 Equation 3

A + C + S = 317

½C + C + S = 317

Multiply through by 2

2(½C + C + S) = 2 * 317

C +2C + 2S = 634

3C + 2S = 634 Equation 4

We'll solve equations 3 and 4, simultaneously.

First, write out the two equations.

11C + 7S = 2296 (3)

3C + 2S = 634 (4)

Using elimination method to eliminate S.

Multiply (3) by 2 and multiply (4) by 7.

2 (11C + 7S = 2296 )

22C + 14S = 4592 Equation 5

7 (3C + 2S = 634 )

21C + 14S = 4438 Equation 6

Subtract (6) from (5)

22C + 14S = 4592

21C + 14S = 4438

22C - 21C + 14S - 14S = 4592 - 4438

C = 154

Recall that A = ½C

So, A = ½ * 154

A = 77

Recall equation 2

A + C + S = 317

Make S the subject of formula

S = 317 - A - C

Substituton 77 for A and 154 for C

So,

S = 317 - 77 - 154

S = 86

Hence, there are 154 children, 77 adults and 86 students in attendance

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