Solve this real-world problem using a system of equations. The O’Donnells are having a birthday party for their son, and they plan to have a clown bring balloons to the party. Balloon Express charges $2 per balloon and $10 for delivery. Balloon Mania charges $1.50 per balloon and $20 for delivery. If the O’Donnells plan to order 30 balloons, which company should they choose, and how much will they save?(1 point)
Responses
Balloon Mania; $25
Balloon Mania; $25
Balloon Mania; $5
Balloon Mania; $5
Balloon Express; $5
Balloon Express; $5
Balloon Express; $25
To solve this problem, we can set up a system of equations:
Let x be the number of balloons ordered from Balloon Express and y be the number of balloons ordered from Balloon Mania.
The total cost from Balloon Express is given by:
Total Cost = (2)(x) + 10
The total cost from Balloon Mania is given by:
Total Cost = (1.50)(y) + 20
The O'Donnells plan to order a total of 30 balloons, so the sum of x and y is 30:
x + y = 30
To find the company they should choose, we need to minimize the total cost.
By substituting x = 30 - y into the equation for Balloon Express, we get:
Total Cost Express = (2)(30 - y) + 10
Total Cost Express = 60 - 2y + 10
Total Cost Express = 70 - 2y
By substituting y = 30 - x into the equation for Balloon Mania, we get:
Total Cost Mania = (1.50)(30 - x) + 20
Total Cost Mania = 45 - 1.50x + 20
Total Cost Mania = 65 - 1.50x
To determine which company to choose, we need to find the y-value at which the total cost from Balloon Mania is less than the total cost from Balloon Express.
Setting the two total costs equal to each other:
70 - 2y = 65 - 1.50x
Rearranging the equation to isolate x:
2y - 1.50x = 70 - 65
2y - 1.50x = 5
Now we can solve the system of equations:
x + y = 30
2y - 1.50x = 5
Multiplying the first equation by -1.50, we get:
-1.50(x + y) = -1.50(30)
-1.50x - 1.50y = -45
Adding this equation to the second equation eliminates x:
-1.50x - 1.50y + 2y - 1.50x = -45 + 5
-3x = -40
x = 40/3
x ≈ 13.33
Substituting the value of x back into the first equation:
13.33 + y = 30
y = 16.67
The O'Donnells should choose Balloon Mania to save $5.
To solve this problem using a system of equations, let's assume that x represents the number of balloons ordered from Balloon Express and y represents the number of balloons ordered from Balloon Mania.
From the problem, we know that the O’Donnells plan to order 30 balloons, so we have the equation:
x + y = 30
We also know the cost per balloon and the delivery fee for each company:
Balloon Express: Cost per balloon = $2 and Delivery fee = $10
Balloon Mania: Cost per balloon = $1.50 and Delivery fee = $20
The total cost for Balloon Express is:
Cost of balloons from Balloon Express = 2x
Delivery fee from Balloon Express = $10
So, the total cost for Balloon Express is:
Total cost for Balloon Express = 2x + $10
The total cost for Balloon Mania is:
Cost of balloons from Balloon Mania = 1.5y
Delivery fee from Balloon Mania = $20
So, the total cost for Balloon Mania is:
Total cost for Balloon Mania = 1.5y + $20
Since the O’Donnells plan to order 30 balloons, we also have:
x + y = 30
To find out which company the O'Donnells should choose, we need to determine the values of x and y. We can solve this system of equations.
Solving the system of equations:
Since x + y = 30, we can rewrite the equation as:
x = 30 - y
Substituting x = 30 - y into the total cost equation for Balloon Express:
Total cost for Balloon Express = 2(30 - y) + $10
Simplifying, we get:
Total cost for Balloon Express = 60 - 2y + $10
Total cost for Balloon Express = 70 - 2y
Similarly, substituting x = 30 - y into the total cost equation for Balloon Mania:
Total cost for Balloon Mania = 1.5y + $20
Comparing the total costs, we want to find the company that will give us the lowest cost. So, we need to determine the values of y for which the total cost for Balloon Express is less than the total cost for Balloon Mania.
Setting the total costs equal to each other and solving for y:
70 - 2y = 1.5y + $20
Adding 2y to both sides:
70 = 3.5y + $20
Subtracting $20 from both sides:
50 = 3.5y
Dividing both sides by 3.5:
y = 14.2857
Since y represents the number of balloons and it cannot have a decimal value, we will round it down to 14.
Now, we can substitute y = 14 into the equation x = 30 - y to find the value of x:
x = 30 - 14
x = 16
So, the O’Donnells should choose Balloon Express and order 16 balloons. This will result in a total cost of $42.
To solve this real-world problem using a system of equations, we need to set up equations for each scenario and then compare the total costs.
Let's assume the number of balloons ordered is represented by the variable 'x'. We also know the costs for each option. Balloon Express charges $2 per balloon and $10 for delivery, so the total cost can be expressed as 2x + 10. Balloon Mania charges $1.50 per balloon and $20 for delivery, so the total cost for Balloon Mania can be expressed as 1.50x + 20.
Since the O'Donnells plan to order 30 balloons, we can substitute x = 30 into each equation.
For Balloon Express: total cost = 2(30) + 10 = 60 + 10 = 70
For Balloon Mania: total cost = 1.50(30) + 20 = 45 + 20 = 65
Comparing the total costs, we can see that Balloon Mania charges $65 while Balloon Express charges $70. Therefore, the O'Donnells should choose Balloon Mania, and they will save $70 - $65 = $5.
Therefore, the correct answer is: Balloon Mania; $5.