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)
A. Balloon Mania; $25
B. Balloon Express; $25
C. Balloon Express; $5
D. Balloon Mania; $5
Let's solve this problem using a system of equations:
Let x be the number of balloons ordered from Balloon Express.
Let y be the number of balloons ordered from Balloon Mania.
From the given information, we can set up the following equations:
2x + 10 = total cost of Balloon Express
1.5y + 20 = total cost of Balloon Mania
x + y = 30 (since they plan to order a total of 30 balloons)
Simplifying the third equation to isolate one variable, we get:
x = 30 - y
Now we can substitute this expression for x into the first equation, and solve for the total cost of Balloon Express:
2(30 - y) + 10 = 60 - 2y + 10 = 70 - 2y
Similarly, we substitute x = 30 - y into the second equation, and solve for the total cost of Balloon Mania:
1.5y + 20
Now we have the two equations:
70 - 2y = total cost of Balloon Express
1.5y + 20 = total cost of Balloon Mania
To find out which company the O'Donnells should choose, we need to compare the total costs. We can set up the inequality:
70 - 2y < 1.5y + 20
Simplifying this inequality:
70 - 20 < 1.5y + 2y
50 < 3.5y
14.29 < y
Since y represents the number of balloons ordered from Balloon Mania, it cannot be a fraction. Therefore, the lowest whole number value that satisfies this inequality is 15.
Substituting y = 15 into the equation for the total cost of Balloon Mania:
1.5(15) + 20 = 22.5 + 20 = 42.5
Now we can compare the total costs:
Balloon Express: 70 - 2(15) = 70 - 30 = 40
Balloon Mania: 42.5
Since the total cost of Balloon Express is $40 and the total cost of Balloon Mania is $42.5, we can see that the O'Donnells should choose Balloon Express and save $2.50.
Therefore, the correct answer is:
D. Balloon Mania; $5
Let's set up a system of equations to solve the problem.
Let x be the number of balloons from Balloon Express and y be the number of balloons from Balloon Mania.
According to the problem, the O'Donnells plan to order 30 balloons, so we have the equation:
x + y = 30 ...(1)
The cost for Balloon Express is $2 per balloon plus $10 for delivery, so the total cost is:
2x + 10 ...(2)
The cost for Balloon Mania is $1.50 per balloon plus $20 for delivery, so the total cost is:
1.50y + 20 ...(3)
We need to find the values of x and y that satisfy both equations (2) and (3) and also satisfy equation (1).
To solve this system, we can use substitution or elimination method. Let's use the substitution method.
From equation (1), we have:
x = 30 - y
Substituting this value of x into equation (2), we get:
2(30 - y) + 10 = 1.50y + 20
Simplifying this equation, we get:
60 - 2y + 10 = 1.50y + 20
70 - 10 = 1.50y + 2y
60 = 3.50y
y = 60/3.50
y = 17.14 (approximately)
Now, we need to find the corresponding value of x from equation (1):
x = 30 - y
x = 30 - 17.14
x = 12.86 (approximately)
Now, let's calculate the cost for each company:
For Balloon Express:
Cost = 2x + 10
= 2(12.86) + 10
= 25.71 + 10
= $35.71
For Balloon Mania:
Cost = 1.50y + 20
= 1.50(17.14) + 20
= 25.71 + 20
= $45.71
Therefore, the O'Donnells should choose Balloon Express, and they will save:
Amount saved = Cost from Balloon Mania - Cost from Balloon Express
= $45.71 - $35.71
= $10.00
So, the correct answer is D. Balloon Mania; $5.
To solve this real-world problem using a system of equations, we need to set up equations for each company's pricing and then compare the total cost.
Let's define the variables:
Let x be the number of balloons ordered.
Let y be the cost for Balloon Express.
Let z be the cost for Balloon Mania.
For Balloon Express, the cost can be calculated as: y = 2x + 10.
For Balloon Mania, the cost can be calculated as: z = 1.5x + 20.
Since the O'Donnells plan to order 30 balloons (x = 30), we can substitute the value into the equations:
For Balloon Express:
y = (2*30) + 10
y = 60 + 10
y = 70
For Balloon Mania:
z = (1.5*30) + 20
z = 45 + 20
z = 65
Now we can compare the costs. Since Balloon Express (y) costs $70 and Balloon Mania (z) costs $65, the O'Donnells should choose Balloon Mania.
To determine the savings, we subtract the cost of Balloon Mania ($65) from the cost of Balloon Express ($70).
Savings = Cost of Balloon Express - Cost of Balloon Mania
Savings = $70 - $65
Savings = $5
Therefore, the correct answer is D. Balloon Mania; $5. By choosing Balloon Mania, the O'Donnells will save $5.