2000 tickets were sold for a play generating 19,700. The prices of the tickets were $5 for children, $10 for students and $12 for adults. There were 100 more adult tickets sold than students. Find the number of each type ticket sold.
Find the number of each type sold.
Thanks for your help.
student tickets ---- x
adult tickets ------- x+100
children tickets ---- 2000 -x - (x+100) = 1900-2x
5(1900-2x) + 10x + 12(x+100) = 19700
9500 - 10x + 10x + 12x + 1200 = 19700
12x = 9000
x = 750 student tickets
x+100 = 850 adult tickets
leaving 400 children tickets
check:
5(400) + 10(750) + 12(850) = 19700
To solve this problem, we can set up a system of equations.
Let's denote the number of children's tickets as "C", the number of student tickets as "S", and the number of adult tickets as "A".
From the given information, we can write three equations:
1) The total number of tickets sold is 2000:
C + S + A = 2000
2) The total revenue generated is $19,700:
5C + 10S + 12A = 19700
3) The number of adult tickets sold is 100 more than the number of student tickets:
A = S + 100
Now we can solve this system of equations.
Using equation 3, we can substitute (S + 100) for A in equations 1 and 2:
C + S + (S + 100) = 2000
5C + 10S + 12(S + 100) = 19700
Simplifying equation 1:
2S + C = 1900 (Equation 4)
Simplifying equation 2:
5C + 10S + 12S + 1200 = 19700
5C + 22S + 1200 = 19700
5C + 22S = 18500 (Equation 5)
Now we have two equations (equations 4 and 5) with two variables (C and S). We can solve this system of equations using substitution or elimination.
Let's use elimination to solve this system:
Multiplying equation 4 by 5 and equation 5 by 2, we get:
10S + 5C = 9500 (Equation 6)
10C + 44S = 37000 (Equation 7)
Subtracting equation 6 from equation 7:
10C + 44S - (10S + 5C) = 37000 - 9500
10C + 44S - 10S - 5C = 27500
5C + 34S = 27500 (Equation 8)
Now we have a new equation (equation 8) with two variables (C and S). We can solve this equation by isolating one variable and substituting it into the other equation.
Let's isolate C in equation 4:
C = 1900 - 2S
Now substitute this value of C in equation 8:
5(1900 - 2S) + 34S = 27500
9500 - 10S + 34S = 27500
24S = 18000
S = 750
Now that we have the value of S, we can substitute it back into equation 4 to find C:
C + 750 = 1900
C = 1150
Finally, we can substitute S = 750 into equation 3 to find A:
A = 750 + 100
A = 850
Therefore, the number of each type of ticket sold is:
Children's tickets (C) = 1150
Student tickets (S) = 750
Adult tickets (A) = 850