Please show me how to solve the following problem: The ocncession stand is selling hot dogs and hamburger during the game. At half-time,they sold a total of 78 hamburgers and hot dogs and brought in $105.50. How many of each item did they sell if hamburgers sold at $1.50 and hot dogs sold at $1.25?
# of burgers --- x
#of hotdogs ---- 78-x
1.5x + 1.25(78-x) = 105.50
times 4
6x + 5(78-x) = 422
6x + 390 - 5x = 422
x = 32
32 hamburgers and 46 hotdogs
To solve this problem, you can use a system of equations approach. Let's define two variables to represent the number of hot dogs and hamburgers sold.
Let's say "x" represents the number of hot dogs sold, and "y" represents the number of hamburgers sold.
Based on the given information, we can form two equations:
1. The total number of hot dogs and hamburgers sold is 78:
x + y = 78
2. The total amount earned from selling hot dogs and hamburgers is $105.50:
1.25x + 1.50y = 105.50
Now that we have our two equations, we can solve them simultaneously to find the values of x and y.
One way to solve these equations is through the substitution method. You can solve the first equation for x (x = 78 - y) and substitute this expression into the second equation:
1.25(78 - y) + 1.50y = 105.50
Simplify the equation:
97.5 - 1.25y + 1.50y = 105.50
Combine like terms:
0.25y = 8
Now, isolate y by dividing both sides of the equation by 0.25:
y = 8 / 0.25 = 32
So, the concession stand sold 32 hamburgers. To find the number of hot dogs sold, substitute this value of y into the initial equation:
x + 32 = 78
x = 78 - 32 = 46
Therefore, the concession stand sold 46 hot dogs.
In conclusion, the concession stand sold 46 hot dogs and 32 hamburgers to generate a total revenue of $105.50.