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