You are selling tickets for a high school concert. Student tickets cost $4 and general admission tickets cost $6. You sell 450 tickets and collect $2340. How many of each type of ticket did you sell?
A music store is selling compact discs for $11.50 and $7.50. You buy 12 discs and spend a total of $106. How many compact discs that cost $11.50 did you buy?
Are you learning algebra? Both of these questions are of the same type: two equations in two unknowns.
In the first problem, let S be the number of student tickets and G be the number of General Admisison tickets.
S = 450 - G
4(450-G) + 6G = 2340
2G = 2340 - 1800 = 540
G = 270
S = 450 - G = 180
Do the other one the same way.
To solve both of these problems, we can set up a system of equations.
Let's start by assigning variables to the unknown quantities.
For the first problem:
Let's say the number of student tickets sold is 's', and the number of general admission tickets sold is 'g'.
For the second problem:
Let's say the number of CDs costing $11.50 is 'x', and the number of CDs costing $7.50 is 'y'.
Now, let's set up the equations:
For the first problem:
1. The total number of tickets sold is 450, so we have: s + g = 450.
2. The total amount collected is $2340, so we have: 4s + 6g = 2340.
For the second problem:
1. The total number of CDs bought is 12, so we have: x + y = 12.
2. The total amount spent is $106, so we have: 11.5x + 7.5y = 106.
Now, we have a system of equations. We can solve it using any method of solving systems of equations, such as substitution or elimination.
Let's solve it using substitution:
First, rearrange the first equation of each problem to solve for one variable in terms of the other.
For the first problem:
s = 450 - g.
For the second problem:
x = 12 - y.
Now, substitute these expressions into the second equation of each problem:
For the first problem:
4(450 - g) + 6g = 2340.
For the second problem:
11.5(12 - y) + 7.5y = 106.
Simplify and solve each equation:
For the first problem:
1800 - 4g + 6g = 2340.
Combine like terms:
2g = 540.
Divide by 2:
g = 270.
Substitute this value back into the equation s = 450 - g:
s = 450 - 270,
s = 180.
Therefore, 180 student tickets and 270 general admission tickets were sold.
For the second problem:
11.5(12 - y) + 7.5y = 106.
Simplify:
138 - 11.5y + 7.5y = 106.
Combine like terms:
-4y = -32.
Divide by -4:
y = 8.
Substitute this value back into the equation x = 12 - y:
x = 12 - 8,
x = 4.
Therefore, 4 CDs costing $11.50 and 8 CDs costing $7.50 were bought.
In conclusion:
For the first problem, 180 student tickets and 270 general admission tickets were sold.
For the second problem, 4 CDs costing $11.50 and 8 CDs costing $7.50 were bought.