Sure, here is the step-by-step working for the solution:
Given information:
Seating capacity of theater = 177
Ticket price for children = $5
Ticket price for students = $7
Ticket price for adults = $12
Number of adults = half the number of children
Total ticket sales = $1276
Let's represent the number of attendees with variables:
C: number of children
S: number of students
A: number of adults
From the given information, we can write the following equations:
A = 0.5C (since there are half as many adults as there are children)
C + S + A ≤ 177 (since the seating capacity of the theater is 177)
5C + 7S + 12A = 1276 (since the total ticket sales is $1276)
We can substitute the first equation into the third equation to get an equation with two variables:
5C + 7S + 12(0.5C) = 1276
Simplifying this, we get:
6.5C + 7S = 1276
Now we have two equations with two variables, so we can solve for C and S. We can isolate S in the second equation:
S ≤ 177 - C - A
S ≤ 177 - C - 0.5C
S ≤ 176.5 - 1.5C
We can substitute this inequality into the first equation:
6.5C + 7(176.5 - 1.5C) = 1276
Simplifying this, we get:
-0.5C = -46.5
C = 93
Substituting this value of C into the inequality for S:
S ≤ 176.5 - 1.5(93)
S ≤ 49.5
Now we can solve for A using the first equation:
A = 0.5C
A = 0.5(93)
A = 46.5, which we can round up to 47.
Therefore, the solution is:
93 children attended
49 students attended
47 adults attended
We can check that this solution is correct by verifying that the total ticket sales is $1276:
5(93) + 7(49) + 12(47) = 465 + 343 + 564 = 1276