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; $25
Balloon Express; $25
Balloon Express; $5
To solve this problem, we can create a system of equations.
Let's say x represents the number of balloons from Balloon Express and y represents the number of balloons from Balloon Mania.
From the given information, we can write the following equations:
Cost from Balloon Express: 2x + 10
Cost from Balloon Mania: 1.5y + 20
The O'Donnells plan to order 30 balloons, so x + y = 30 (balloons)
Now we can solve the system of equations:
1) 2x + 10 = 1.5y + 20
2) x + y = 30
We can rearrange equation 2 to solve for x: x = 30 - y
Substituting x in equation 1:
2(30 - y) + 10 = 1.5y + 20
60 - 2y + 10 = 1.5y + 20
70 - 2y = 1.5y + 20
Add 2y to both sides:
70 = 3.5y + 20
Subtract 20 from both sides:
50 = 3.5y
Divide both sides by 3.5:
y = 50/3.5
y = 14.29
Since we cannot have a fraction of a balloon, the O'Donnells should choose either 14 or 15 balloons from Balloon Mania.
Substituting y = 14 into equation 2:
x + 14 = 30
x = 30 - 14
x = 16
The O'Donnells should choose 16 balloons from Balloon Express and 14 balloons from Balloon Mania.
Comparing the costs:
Cost from Balloon Express: 2(16) + 10 = 32 + 10 = $42
Cost from Balloon Mania: 1.5(14) + 20 = 21 + 20 = $41
Therefore, the O'Donnells should choose Balloon Mania and they will save $1.
To solve this problem using a system of equations, let's define the following variables:
Let x represent the number of balloons ordered from Balloon Express.
Let y represent the number of balloons ordered from Balloon Mania.
We can now create the system of equations based on the given information:
For Balloon Express:
Cost per balloon: $2
Number of balloons ordered from Balloon Express: x
Delivery cost from Balloon Express: $10
So the total cost for Balloon Express is: 2x + 10
For Balloon Mania:
Cost per balloon: $1.50
Number of balloons ordered from Balloon Mania: y
Delivery cost from Balloon Mania: $20
So the total cost for Balloon Mania is: 1.5y + 20
We also know that the total number of balloons ordered is 30. Therefore:
x + y = 30
Now we can solve the system of equations to find the values of x and y. Substituting the value of x from the first equation into the second equation, we get:
(30 - y) + y = 30
30 - y + y = 30
30 = 30
This equation is always true, which means that there are infinitely many solutions for x and y. We can choose any values for x and y as long as they satisfy the equation x + y = 30.
Let's choose x = 10 and y = 20. Now we can calculate the total cost for each company:
For Balloon Express:
Total cost = 2x + 10 = 2(10) + 10 = $20 + $10 = $30
For Balloon Mania:
Total cost = 1.5y + 20 = 1.5(20) + 20 = $30 + $20 = $50
Comparing the costs, we can see that Balloon Express is the cheaper option, with a total cost of $30. Therefore, the O’Donnells should choose Balloon Express and they will save $20 if they order 30 balloons from them.
To solve this problem using a system of equations, we need to set up two equations representing the cost of ordering balloons from each company. Let's assume x represents the number of balloons ordered from Balloon Express and y represents the number of balloons ordered from Balloon Mania.
For Balloon Express, the total cost can be calculated as:
Cost = 2x + 10 (2 dollars per balloon and 10 dollars for delivery)
For Balloon Mania, the total cost can be calculated as:
Cost = 1.5y + 20 (1.5 dollars per balloon and 20 dollars for delivery)
Since the O'Donnells plan to order 30 balloons in total, we can set up another equation representing the total number of balloons:
x + y = 30
Now we have a system of equations:
2x + 10 = Cost for Balloon Express
1.5y + 20 = Cost for Balloon Mania
x + y = 30 (total number of balloons)
To find the solution, we can solve this system of equations using substitution, elimination, or other methods.
By substituting x = 30 - y into the first two equations, we have:
2(30 - y) + 10 = Cost for Balloon Express
1.5y + 20 = Cost for Balloon Mania
Expanding and simplifying the equations, we get:
60 - 2y + 10 = Cost for Balloon Express
1.5y + 20 = Cost for Balloon Mania
Simplifying further, we have:
-2y + 70 = Cost for Balloon Express
1.5y + 20 = Cost for Balloon Mania
To determine which company the O'Donnells should choose and how much they will save, we need to compare the costs. We can set up an inequality to represent this:
Cost for Balloon Mania < Cost for Balloon Express
Substituting the expressions for the costs:
1.5y + 20 < -2y + 70
By rearranging the inequality, we get:
3.5y < 50
y < 50/3.5
y < 14.29
Since the number of balloons must be a whole number, we can conclude that the O'Donnells would order 14 balloons from Balloon Mania and the remaining 16 balloons from Balloon Express (to reach a total of 30 balloons).
Plugging this into the cost equations, we find:
Cost for Balloon Express: 2(16) + 10 = 42 dollars
Cost for Balloon Mania: 1.5(14) + 20 = 41 dollars
Therefore, the O'Donnells should choose Balloon Mania, and they would save 42 - 41 = 1 dollar.