The Baynard High School Boosters are selling hot dogs and pretzels at the concession stand. They have 300 hot dogs and pretzels in stock and want to earn an income of $800. They plan to sell the hot dogs for $4 each and the pretzels for $2 each. How many hot dogs and how many pretzels do the boosters need to sell to reach their goal of $800?(1 point)%0D%0AResponses%0D%0A%0D%0A200 hot dogs and 100 pretzels%0D%0A200 hot dogs and 100 pretzels%0D%0A%0D%0A250 hot dogs and 50 pretzels%0D%0A250 hot dogs and 50 pretzels%0D%0A%0D%0A100 hot dogs and 200 pretzels %0D%0A100 hot dogs and 200 pretzels %0D%0A%0D%0A50 hot dogs and 250 pretzels
To find the solution, we need to set up equations based on the given information. Let's assume the number of hot dogs sold is H, and the number of pretzels sold is P.
From the information given, we know that the selling price of a hot dog is $4, and the selling price of a pretzel is $2. Therefore, the equation for the income earned from selling hot dogs can be written as:
Income from hot dogs = 4H
Similarly, the income earned from selling pretzels can be written as:
Income from pretzels = 2P
The boosters want to earn a total income of $800. Therefore, we can write the equation for the total income earned as:
Total income = Income from hot dogs + Income from pretzels = 4H + 2P
We also know that the boosters have a total of 300 hot dogs and pretzels in stock. Thus, we have the equation for the total number of items sold as:
Total items = H + P = 300
Now, we can solve the system of equations to find the values of H and P that satisfy all the given conditions.
Let's look at each response option and substitute the values of H and P into the equations to see if they meet the given conditions.
1. Option: 200 hot dogs and 100 pretzels
Substituting H = 200 and P = 100 into the equations:
Total income = 4H + 2P = 4(200) + 2(100) = 800 + 200 = 1000 (not equal to $800)
Total items = H + P = 200 + 100 = 300 (equal to 300)
Therefore, this option does not satisfy the goal of earning $800.
2. Option: 250 hot dogs and 50 pretzels
Substituting H = 250 and P = 50 into the equations:
Total income = 4H + 2P = 4(250) + 2(50) = 1000 + 100 = 1100 (not equal to $800)
Total items = H + P = 250 + 50 = 300 (equal to 300)
Therefore, this option also does not satisfy the goal of earning $800.
3. Option: 100 hot dogs and 200 pretzels
Substituting H = 100 and P = 200 into the equations:
Total income = 4H + 2P = 4(100) + 2(200) = 400 + 400 = 800 (equal to $800)
Total items = H + P = 100 + 200 = 300 (equal to 300)
Therefore, this option satisfies the goal of earning $800.
4. Option: 50 hot dogs and 250 pretzels
Substituting H = 50 and P = 250 into the equations:
Total income = 4H + 2P = 4(50) + 2(250) = 200 + 500 = 700 (not equal to $800)
Total items = H + P = 50 + 250 = 300 (equal to 300)
Therefore, this option also does not satisfy the goal of earning $800.
Based on the analysis above, the correct answer is:
Option: 100 hot dogs and 200 pretzels
To determine the number of hot dogs (H) and pretzels (P) the boosters need to sell to reach their goal of $800, we can use the following equation:
4H + 2P = 800
Based on the given information, we know that the boosters have a total of 300 hot dogs and pretzels in stock. Therefore, we can also assume that H + P = 300.
To solve this system of equations, we can use substitution or elimination method. Let's use the substitution method:
1. Solve one equation for one variable (preferably H or P) in terms of the other variable.
From the second equation, we can rewrite it as H = 300 - P.
2. Substitute the expression from step 1 into the other equation.
Rewrite the first equation as 4(300 - P) + 2P = 800.
3. Simplify and solve for the remaining variable.
Distribute 4: 1200 - 4P + 2P = 800.
Combine like terms: -2P = 800 - 1200.
Simplify: -2P = -400.
Divide by -2: P = -400 / -2 = 200.
4. Substitute the value of P into one of the original equations to solve for the other variable.
Using the first equation, H + 200 = 300.
Subtract 200 from both sides: H = 300 - 200 = 100.
Therefore, the boosters need to sell 100 hot dogs and 200 pretzels to reach their goal of $800.
To solve this problem, we need to set up an equation based on the given information.
Let's assume they need to sell x hot dogs and y pretzels to reach their goal of $800.
The price of a hot dog is $4, so the total revenue from selling hot dogs is 4x dollars.
Similarly, the price of a pretzel is $2, so the total revenue from selling pretzels is 2y dollars.
We know that the total revenue from selling both hot dogs and pretzels should be $800. Therefore, we can set up the equation:
4x + 2y = 800
Now, we also know that they have 300 hot dogs and pretzels in stock. So, the total number of items they have is x + y = 300.
We have a system of equations:
4x + 2y = 800 (Equation 1)
x + y = 300 (Equation 2)
To solve this system, we can use the substitution method:
From Equation 2, we can rewrite it as: y = 300 - x
Now, substitute this value of y in Equation 1:
4x + 2(300 - x) = 800
4x + 600 - 2x = 800
2x + 600 = 800
2x = 800 - 600
2x = 200
x = 200/2
x = 100
Substitute this value of x back into Equation 2 to find y:
100 + y = 300
y = 300 - 100
y = 200
So, the Boosters need to sell 100 hot dogs and 200 pretzels to reach their goal of $800.