A basketball team sells tickets that cost $10, $20, or, for VIP seats, $30. The team has sold 3401 tickets overall. It has sold 133 more $20 tickets than $10 tickets. The total sales are$65960. How many tickets of each kind have been sold?
n = $10 , w = $20 , t = $30
n + w + t = 3401 ... 30 n + 30 w + 30 t = 102030
10 n + 20 w + 30 t = 65960
n + 133 = w
subtracting equations (to eliminate t) ... 20 n + 10 w = 36070
substituting ... 20 n + 10 (n + 133) = 36070
solve for n , then substitute back to find w and t
Sold X $10 tickets
Sold x+133 $20 tickets
Sold Y $30 tickets
x + x+133 + y = 3401
Eq1: 2x+y = 3268
10x + 20(x+133) + 30y = 65960
30x+30y = 63300
Eq2: x+y = 2110
Multiply Eq2 by 2 and subtract from Eq1:
2x+y = 3268
2x+2y = 4220
Diff: y = 952 VIP tickets
In Eq1, replace y with 952 and solve for x
2x+952 = 3268
X = 1158 tickets
x+133 = 1158+133 = 1291 tickets
To solve this problem, let's assign variables to the number of each type of ticket sold. Let's say:
x = number of $10 tickets sold
y = number of $20 tickets sold
z = number of $30 VIP tickets sold
Given the information, we can set up a system of equations:
1) The team has sold 133 more $20 tickets than $10 tickets:
y = x + 133
2) The total tickets sold is 3401:
x + y + z = 3401
3) The total sales are $65960:
10x + 20y + 30z = 65960
Now, let's solve these equations simultaneously to find the values of x, y, and z.
First, substitute equation 1) into equation 2):
x + (x + 133) + z = 3401
2x + z + 133 = 3401
2x + z = 3268
Second, substitute equation 1) into equation 3):
10x + 20(y) + 30z = 65960
10x + 20(x + 133) + 30z = 65960
10x + 20x + 2660 + 30z = 65960
30x + 30z = 63300
x + z = 2110
Now, we have a system of two equations:
2x + z = 3268
x + z = 2110
To solve it, subtract equation 2) from equation 1):
2x + z - (x + z) = 3268 - 2110
x = 1158
Plug the value of x back into equation 2):
1158 + z = 2110
z = 2110 - 1158
z = 952
Finally, substitute the values of x and z into equation 1) to find y:
y = x + 133
y = 1158 + 133
y = 1291
So, the number of each type of ticket sold is:
$10 tickets: 1158
$20 tickets: 1291
$30 VIP tickets: 952