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; $5
Balloon Mania; $5
Balloon Express; $25
Balloon Express; $25
Balloon Express; $5
Balloon Express; $5
Balloon Mania; $25
To solve this problem using a system of equations, let's denote the number of balloons as "x" and the total cost as "y" for each company.
For Balloon Express:
The cost for balloons would be $2 per balloon, so the cost for x balloons would be 2x.
The delivery cost is a flat fee of $10, so the total cost for Balloon Express would be 2x + 10.
For Balloon Mania:
The cost for balloons would be $1.50 per balloon, so the cost for x balloons would be 1.5x.
The delivery cost is a flat fee of $20, so the total cost for Balloon Mania would be 1.5x + 20.
The O'Donnells plan to order 30 balloons, so we can substitute x = 30 into the equations to find the total cost for each company:
For Balloon Express:
Total cost = 2(30) + 10 = 60 + 10 = $70
For Balloon Mania:
Total cost = 1.5(30) + 20 = 45 + 20 = $65
Therefore, the O'Donnells should choose Balloon Mania, and they would save $70 - $65 = $5.
To solve this problem using a system of equations, let's assume the number of balloons ordered from Balloon Express is x and the number of balloons ordered from Balloon Mania is y.
From the given information, we know:
Balloon Express charges $2 per balloon and $10 for delivery.
So the cost of ordering x balloons from Balloon Express would be: 2x + 10.
Balloon Mania charges $1.50 per balloon and $20 for delivery.
So the cost of ordering y balloons from Balloon Mania would be: 1.50y + 20.
The O’Donnells plan to order a total of 30 balloons, so we have the equation:
x + y = 30
We need to compare the costs for each company and see who offers a lower price.
For Balloon Express, the cost is 2x + 10.
For Balloon Mania, the cost is 1.50y + 20.
To find the company that offers a lower price, we need to compare the two expressions:
2x + 10 < 1.50y + 20
Now, let's solve the equations:
Rearranging equation 1, we get:
x = 30 - y
Substituting this into the inequality, we get:
2(30 - y) + 10 < 1.50y + 20
Simplifying the expression:
60 - 2y + 10 < 1.50y + 20
70 - 2y < 1.50y + 20
Adding 2y to both sides:
70 < 1.50y + 2y + 20
70 < 3.50y + 20
Subtracting 20 from both sides:
50 < 3.50y
Dividing by 3.50 on both sides:
y > 50/3.50
y > 14.29
Since the balloons cannot be fractioned, we round y up to the next whole number: y = 15.
Now, we can substitute the value of y back into equation 1 to find x:
x = 30 - 15
x = 15
Therefore, the O’Donnells should choose Balloon Mania and order 15 balloons.
To find the cost savings, we compare the costs of the two companies:
Cost at Balloon Express: 2x + 10 = 2(15) + 10 = 30 + 10 = $40
Cost at Balloon Mania: 1.50y + 20 = 1.50(15) + 20 = 22.50 + 20 = $42.50
The O’Donnells will save $40 - $42.50 = $2.50 by choosing Balloon Express.
To solve this problem using a system of equations, let's assume that the number of balloons ordered from Balloon Express is x, and the number of balloons ordered from Balloon Mania is y.
According to the given information, Balloon Express charges $2 per balloon and $10 for delivery. Therefore, the total cost from Balloon Express can be calculated using the equation:
Total cost from Balloon Express = 2x + 10
Similarly, Balloon Mania charges $1.50 per balloon and $20 for delivery. Therefore, the total cost from Balloon Mania can be calculated using the equation:
Total cost from Balloon Mania = 1.5y + 20
Since the O'Donnell's plan to order 30 balloons (x + y = 30), we can solve the system of equations:
2x + 10 = 1.5y + 20 (equation 1)
x + y = 30 (equation 2)
Simplifying equation 1, we get:
2x - 1.5y = 10
To solve this system of equations, we can use the substitution method or the elimination method. Let's use the elimination method:
Multiply equation 2 by 2 to make the coefficients of x in both equations the same:
2x + 2y = 60 (equation 3)
Now, subtract equation 3 from equation 1 to eliminate x:
2x - 1.5y - (2x + 2y) = 10 - 60
-3.5y = -50
y = 50/3.5
y = 14.286 (approximately)
Substitute the value of y into equation 2 to find x:
x + 14.286 = 30
x = 30 - 14.286
x = 15.714 (approximately)
Now, we need to check which company the O'Donnells should choose and how much they will save.
Substituting the values of x and y into the respective equations, we can calculate the total cost from each company:
Total cost from Balloon Express = 2x + 10
Total cost from Balloon Express = 2(15.714) + 10
Total cost from Balloon Express = 31.428 + 10
Total cost from Balloon Express = $41.428
Total cost from Balloon Mania = 1.5y + 20
Total cost from Balloon Mania = 1.5(14.286) + 20
Total cost from Balloon Mania = 21.429 + 20
Total cost from Balloon Mania = $41.429
Both companies have nearly the same total cost, but Balloon Mania is slightly cheaper. The O'Donnells will choose Balloon Mania and save approximately $0.001. Therefore, the correct answer is:
Balloon Mania; $0.001