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 Express; $25
B. Balloon Mania; $5
C. Balloon Mania; $25
D. Balloon Express; $5
Let's represent the number of balloons as "b" and the cost of delivery as "d".
For Balloon Express, the cost equation would be:
Cost of balloons = 2b
Cost of delivery = 10
Total cost = 2b + 10
For Balloon Mania, the cost equation would be:
Cost of balloons = 1.5b
Cost of delivery = 20
Total cost = 1.5b + 20
Now, we need to find the cost for 30 balloons for each company and compare them.
For Balloon Express:
Total cost = 2(30) + 10
Total cost = 60 + 10
Total cost = 70
For Balloon Mania:
Total cost = 1.5(30) + 20
Total cost = 45 + 20
Total cost = 65
The O'Donnells should choose Balloon Mania since they will save $5 compared to Balloon Express.
Therefore, the correct answer is:
D. Balloon Express; $5
Let's set up a system of equations to solve this problem.
Let's say x represents the number of balloons ordered from Balloon Express and y represents the number of balloons ordered from Balloon Mania.
Based on the information given, we have the following equations:
For Balloon Express:
Cost = 2x + 10 (since they charge $2 per balloon and $10 for delivery)
For Balloon Mania:
Cost = 1.50y + 20 (since they charge $1.50 per balloon and $20 for delivery)
We also know that the total number of balloons ordered is 30, so we have the equation:
x + y = 30
Now we can solve this system of equations to find the values of x and y.
We can solve the system by substitution or elimination. Let's solve it using the substitution method.
From the equation x + y = 30, we can rearrange it to get:
x = 30 - y
Substituting x = 30 - y into the Cost equation for Balloon Express:
Cost = 2(30 - y) + 10
= 60 - 2y + 10
= 70 - 2y
Substituting x = 30 - y into the Cost equation for Balloon Mania:
Cost = 1.50y + 20
Now we have two equations for the cost of each company.
To determine the company the O'Donnells should choose, we need to find out which option has a lower cost for 30 balloons.
Comparing the costs, we have:
Cost for Balloon Express: 70 - 2y
Cost for Balloon Mania: 1.50y + 20
Now we can solve for y and find the cost for each option.
To find the savings, we subtract the cost from the more expensive company from the cost of the less expensive company.
Let's calculate:
For the Balloon Express:
Cost = 70 - 2y
Cost = 70 - 2(30 - y)
Cost = 70 - 60 + 2y
Cost = 10 + 2y
For the Balloon Mania:
Cost = 1.50y + 20
Now, let's find the values of y and the corresponding costs.
When y = 15:
Cost for Balloon Express = 10 + 2(15) = 10 + 30 = 40
Cost for Balloon Mania = 1.50(15) + 20 = 22.50 + 20 = 42.50
When y = 20:
Cost for Balloon Express = 10 + 2(20) = 10 + 40 = 50
Cost for Balloon Mania = 1.50(20) + 20 = 30 + 20 = 50
When y = 25:
Cost for Balloon Express = 10 + 2(25) = 10 + 50 = 60
Cost for Balloon Mania = 1.50(25) + 20 = 37.50 + 20 = 57.50
From these calculations, we can see that when the O'Donnells order 30 balloons, they should choose Balloon Mania, as it has the lower cost of $50. The savings they would make compared to Balloon Express is $10.
Therefore, the correct answer is:
C. Balloon Mania; $25
To solve this real-world problem using a system of equations, we can represent the cost of each company's service using variables.
Let's assume:
x = number of balloons ordered from Balloon Express
y = number of balloons ordered from Balloon Mania
Given:
The O'Donnells plan to order 30 balloons.
We can set up two equations with the given information:
1. Balloon Express:
Cost of balloons + Cost of delivery = Total cost
2x + 10 = Total cost
2. Balloon Mania:
Cost of balloons + Cost of delivery = Total cost
1.5y + 20 = Total cost
To determine which company to choose, we need to compare the total costs for each company.
For Balloon Express:
2x + 10 = Total cost
For Balloon Mania:
1.5y + 20 = Total cost
Since the O'Donnells plan to order 30 balloons, we know that:
x + y = 30
Now, let's substitute the value of x or y from the third equation into the first two equations:
x = 30 - y
1. Balloon Express:
2(30 - y) + 10 = Total cost
60 - 2y + 10 = Total cost
70 - 2y = Total cost
2. Balloon Mania:
1.5y + 20 = Total cost
Now, we can compare the total costs:
Balloon Express: 70 - 2y
Balloon Mania: 1.5y + 20
To find which company to choose, we need to determine when the costs are equal.
70 - 2y = 1.5y + 20
Combine like terms:
70 - 20 = 1.5y + 2y
50 = 3.5y
Divide by 3.5:
y = 50 / 3.5
y ≈ 14.29
Since y represents the number of balloons ordered from Balloon Mania, we cannot order 14.29 balloons. We have to round it to the nearest whole number.
Set y = 14:
y = 14
x = 30 - y
x = 30 - 14
x = 16
So, to minimize costs, the O'Donnells should choose to order 16 balloons from Balloon Express and 14 balloons from Balloon Mania.
Now, let's calculate the total costs for both companies:
Balloon Express:
2x + 10 = 2(16) + 10 = 32 + 10 = $42
Balloon Mania:
1.5y + 20 = 1.5(14) + 20 = 21 + 20 = $41
So, the O'Donnells should choose Balloon Mania because they will save $1 compared to Balloon Express.
Therefore, the correct answer is:
D. Balloon Express; $5