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?
A. Balloon Express; $5
B. Balloon Mania; $25
C. Balloon Express; $25
D. $25 Balloon Mania; $5
Let's represent the number of balloons as "b" and the delivery fee as "d".
For Balloon Express, the total cost would be 2b + 10d.
For Balloon Mania, the total cost would be 1.50b + 20.
We are given that the O'Donnells plan to order 30 balloons, so we can substitute b = 30 into the equations:
For Balloon Express: 2(30) + 10d = 60 + 10d
For Balloon Mania: 1.50(30) + 20 = 45 + 20 = 65
We need to find the company that offers the lowest total cost. Comparing the two equations, we see that Balloon Express charges 60 + 10d and Balloon Mania charges 65.
To find the delivery fee "d" that makes the two costs equal, we can set the two equations equal to each other and solve for "d":
60 + 10d = 65
10d = 5
d = 0.5
We can plug this value of "d" back into one of the equations to find the total cost:
Balloon Express: 2(30) + 10(0.5) = 60 + 5 = 65
Therefore, the O'Donnells should choose Balloon Mania, and they will save $65 - $60 = $5.
The correct answer is:
D. $25 Balloon Mania; $5
Let's set up a system of equations to solve this problem:
Let x be the number of balloons and y be the total cost.
For Balloon Express:
Cost of balloons = 2x
Delivery cost = 10
Total cost = 2x + 10
For Balloon Mania:
Cost of balloons = 1.5x
Delivery cost = 20
Total cost = 1.5x + 20
Given that the O’Donnells plan to order 30 balloons, we can substitute x = 30 into the equations:
For Balloon Express:
Total cost = 2(30) + 10 = 60 + 10 = 70
For Balloon Mania:
Total cost = 1.5(30) + 20 = 45 + 20 = 65
Comparing the total costs, we see that Balloon Mania charges a lower cost.
Thus, the O’Donnells should choose Balloon Mania, and they will save:
Cost difference = Cost with Balloon Express - Cost with Balloon Mania
Cost difference = 70 - 65 = 5
So, they would save $5.
Therefore, the answer is option D. $25 Balloon Mania; $5.
To solve this real-world problem using a system of equations, we first need to set up the equations based on the given information.
Let x be the number of balloons ordered from Balloon Express.
Let y be the number of balloons ordered from Balloon Mania.
According to the problem, the O’Donnells plan to order a total of 30 balloons, so we can write the first equation as:
x + y = 30
Next, we need to calculate the cost for each company based on the number of balloons ordered.
For Balloon Express, the cost can be calculated as:
Cost for Balloon Express = (2 * x) + 10
For Balloon Mania, the cost can be calculated as:
Cost for Balloon Mania = (1.5 * y) + 20
Now we have two equations:
x + y = 30
Cost for Balloon Express = (2 * x) + 10
Cost for Balloon Mania = (1.5 * y) + 20
To determine which company the O’Donnells should choose, we need to find the solution to the system of equations. We can solve this system using various methods such as substitution or elimination.
To make it easier, let's solve this system of equations using the substitution method:
From the first equation, we can express x in terms of y as:
x = 30 - y
Substituting this value of x into the cost equation for Balloon Express, we get:
Cost for Balloon Express = (2 * (30 - y)) + 10
Now, substitute the same value of x into the cost equation for Balloon Mania:
Cost for Balloon Mania = (1.5 * y) + 20
We can now compare the costs to determine which company to choose.
If we simplify the equation for each company, we get:
Cost for Balloon Express = 60 - 2y + 10 = 70 - 2y
Cost for Balloon Mania = 1.5y + 20
We can set these two costs equal to each other to find the value of y:
70 - 2y = 1.5y + 20
Simplifying this equation, we get:
70 = 3.5y + 20
Subtracting 20 from both sides, we get:
50 = 3.5y
Dividing both sides by 3.5, we find:
y = 14.2857
Since y represents the number of balloons, we can round it to the nearest whole number, giving us y = 14.
Substituting this value of y back into the first equation, we find:
x + 14 = 30
x = 30 - 14
x = 16
Therefore, the O’Donnells should choose to order 16 balloons from Balloon Express and 14 balloons from Balloon Mania.
To determine how much they will save, we need to calculate the difference in cost between the two companies.
Cost for Balloon Express = (2 * 16) + 10 = 32 + 10 = 42
Cost for Balloon Mania = (1.5 * 14) + 20 = 21 + 20 = 41
Therefore, the O’Donnells will save $1 by choosing Balloon Mania over Balloon Express.
The correct answer would be:
D. Balloon Mania; $1