A theater has tickets for $6 for adults, $3.50 for students, and $2.50 for pre-school children. A total of 238 tickets were sold for one showing with a total revenue of $1142. If the number of adult tickets sold was 10 more than twice the number of student tickets, how many of each type ticket were sold for the showing?
a + s + t = 278
60a + 35s + 25t = 13000
a - 2s + 0 = -10
Use any method to get:
a = 150
s = 80
t = 48
To solve this problem, let's denote the number of adult tickets as "A", the number of student tickets as "S", and the number of pre-school tickets as "P".
According to the given information:
1. The price of an adult ticket is $6, so the revenue from adult tickets is 6A.
2. The price of a student ticket is $3.50, so the revenue from student tickets is 3.50S.
3. The price of a pre-school ticket is $2.50, so the revenue from pre-school tickets is 2.50P.
We are also given that the total number of tickets sold is 238 and the total revenue is $1142. Therefore, we have the following equations:
A + S + P = 238 (equation 1)
6A + 3.50S + 2.50P = 1142 (equation 2)
We are also told that the number of adult tickets sold was 10 more than twice the number of student tickets, which gives us the equation:
A = 10 + 2S (equation 3)
Now we can solve the system of equations (equations 1, 2, and 3) to find the values of A, S, and P.
1. Substitute equation 3 into equation 1:
(10 + 2S) + S + P = 238
10 + 3S + P = 238
2. Rearrange equation 2 to solve for P:
P = (1142 - 6A - 3.50S) / 2.50 (equation 4)
3. Substitute the rearranged equation 2 into equation 1:
10 + 3S + (1142 - 6A - 3.50S) / 2.50 = 238
4. Solve the above equation for S:
Multiply both sides of the equation by 2.50 to remove the fraction:
25 + 7.50S + (1142 - 6A - 3.50S) = 595
Simplify the equation:
25 + 7.50S + 1142 - 6A - 3.50S = 595
Combine like terms:
10S + 1167 - 6A = 595
Rearrange the equation:
10S - 6A = 595 - 1167
10S - 6A = -572 (equation 5)
5. Substitute equation 3 into equation 5:
10(10 + 2S) - 6A = -572
Distribute the multiplication:
100 + 20S - 6A = -572
Rearrange the equation:
20S - 6A = -672 (equation 6)
6. Now we have the system of equations:
10 + 3S + P = 238 (equation 1)
20S - 6A = -672 (equation 6)
6A + 3.50S + 2.50P = 1142 (equation 2)
You can solve this system of equations by substitution or elimination to find the values of A, S, and P.