at a college function a total of 180 tickets were sold. the tickets were priced at $5,10,15 and the money collected that day was $1,900. the sum of the numbers of $5,and $15 tickets sold was twice the number of $10 tickets sold. find the number of each type of ticket sold for a college function
To find the number of each type of ticket sold, we can set up a system of equations based on the given information.
Let's assume:
- The number of $5 tickets sold = x
- The number of $10 tickets sold = y
- The number of $15 tickets sold = z
We have three unknowns, so we need three equations to solve for them.
Equation 1: The total number of tickets sold
x + y + z = 180
Equation 2: The total money collected
5x + 10y + 15z = 1900
Equation 3: The sum of $5 and $15 tickets sold is twice the number of $10 tickets sold
x + z = 2y
Now we can solve these equations simultaneously to find the values of x, y, and z.
First, let's re-arrange Equation 3 to y in terms of x and z:
y = (x + z) / 2
Substitute this equation into Equation 1:
x + (x + z) / 2 + z = 180
Multiply through by 2 to clear the fraction:
2x + x + z + 2z = 360
3x + 3z = 360
Simplify this equation:
x + z = 120 --> Equation 4
Now, substitute the value of y from Equation 3 (y = (x + z) / 2) into Equation 2:
5x + 10((x + z) / 2) + 15z = 1900
5x + 5(x + z) + 15z = 1900
10x + 20z = 1900
Divide by 10 to simplify:
x + 2z = 190 --> Equation 5
Now we have a system of two equations (Equations 4 and 5) with two variables (x and z). We can solve them simultaneously.
Multiply Equation 4 by 2 and subtract Equation 5 from it:
2(x + z) - (x + 2z) = 240 - 190
2x + 2z - x - 2z = 50
x = 50
Substitute the value of x in Equation 4:
50 + z = 120
z = 120 - 50
z = 70
Now substitute the values of x and z back into Equation 3 to find y:
y = (x + z) / 2
y = (50 + 70) / 2
y = 120 / 2
y = 60
Therefore, the number of $5 tickets sold is 50, the number of $10 tickets sold is 60, and the number of $15 tickets sold is 70.
number of $5 tickets --- x
number of $10 tickets --- y
number of $15 tickets ---- 180-x-y
x + (180-x-y) = 2y
x + 180 - x - y = 2y
180=3y
y = 60
income equation:
5x + 10y + 15(180-x-y) = 1900
5x + 600 + 15(120-x) = 1900
5x - 15x + 600 + 1800 = 1900
-10x = -500
x = 50
x=50
y=60
number of $5 tickets --- 50
number of $10 ---------60
numbr of $15 = 180-50-60 = 70