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 Express; $5 Balloon Express; $5 Balloon Mania; $25 Balloon Mania; $25 Balloon Mania; $5 Balloon Mania; $5 Balloon Express; $25
To solve this problem using a system of equations, let's first define the variables.
Let x represent the number of balloons ordered from Balloon Express.
Let y represent the number of balloons ordered from Balloon Mania.
Based on the problem, we know that the total number of balloons ordered is 30. So we can set up the first equation:
x + y = 30
Next, we can calculate the total cost for Balloon Express:
2x + 10
And the total cost for Balloon Mania:
1.5y + 20
Since we want to find out which company the O'Donnells should choose and how much they will save, we need to set up a second equation:
Cost Balloon Express - Cost Balloon Mania = Savings
(2x + 10) - (1.5y + 20) = Savings
Now, we can solve the system of equations to find the values of x and y, and then determine which company the O'Donnells should choose and how much they will save.
Using either substitution or elimination method, we can solve the system of equations.
To solve this real-world problem, we can set up a system of equations.
Let's use the variables B and C to represent the number of balloons and the cost for delivery, respectively.
For Balloon Express, the cost equation would be:
2B + 10C = Total cost
For Balloon Mania, the cost equation would be:
1.50B + 20C = Total cost
Since the O'Donnells plan to order 30 balloons, we can substitute B = 30 into both equations:
For Balloon Express:
2(30) + 10C = Total cost
60 + 10C = Total cost
For Balloon Mania:
1.50(30) + 20C = Total cost
45 + 20C = Total cost
Now we can compare the two cost equations to determine which company offers a lower cost.
Let's solve the first equation:
60 + 10C = Total cost
And the second equation:
45 + 20C = Total cost
By simplifying both equations, we have:
10C + 60 = Total cost
20C + 45 = Total cost
Since the O'Donnells are only concerned about the total cost, we can ignore the specific values of Total cost.
Now, we can compare the coefficients of C, which represent the delivery cost:
For Balloon Express: 10C
For Balloon Mania: 20C
Since 20C > 10C, it means that Balloon Mania has a higher delivery cost.
Therefore, the O'Donnells should choose Balloon Express as it offers a lower delivery cost.
To calculate the savings, we need to compare the total costs:
Total cost for Balloon Express (using the equation 10C + 60 = Total cost):
10C + 60
Total cost for Balloon Mania (using the equation 20C + 45 = Total cost):
20C + 45
Substituting B = 30 into both equations:
Total cost for Balloon Express: 10C + 60 = 10(30) + 60 = 300 + 60 = 360
Total cost for Balloon Mania: 20C + 45 = 20(30) + 45 = 600 + 45 = 645
The O'Donnells will save the difference between the two costs:
Savings = Total cost for Balloon Mania - Total cost for Balloon Express
Savings = 645 - 360
Savings = $285
Therefore, the O'Donnells will save $285 by choosing Balloon Express over Balloon Mania.
Let's define two variables:
- Let x be the number of balloons ordered from Balloon Express.
- Let y be the number of balloons ordered from Balloon Mania.
We can set up the following equations based on the given information:
1) Balloon Express charges $2 per balloon and $10 for delivery:
2x + 10 = total cost for Balloon Express
2) Balloon Mania charges $1.50 per balloon and $20 for delivery:
1.5y + 20 = total cost for Balloon Mania
We know that the O'Donnells plan to order 30 balloons, therefore x + y = 30.
We can solve this system of equations to determine which company the O'Donnells should choose and how much they will save.
Let's solve it:
From equation 1: 2x + 10 = total cost for Balloon Express --> (equation 3)
From equation 2: 1.5y + 20 = total cost for Balloon Mania --> (equation 4)
From equation 3, solve for x:
2x + 10 = total cost for Balloon Express
2x = total cost for Balloon Express - 10
x = (total cost for Balloon Express - 10)/2
From equation 4, solve for y:
1.5y + 20 = total cost for Balloon Mania
1.5y = total cost for Balloon Mania - 20
y = (total cost for Balloon Mania - 20)/1.5
Now, substitute x + y = 30 into the equations for x and y:
(total cost for Balloon Express - 10)/2 + (total cost for Balloon Mania - 20)/1.5 = 30
Multiply through by 2 to get rid of the fraction:
total cost for Balloon Express - 10 + 2(total cost for Balloon Mania - 20)/1.5 = 60
total cost for Balloon Express - 10 + 4(total cost for Balloon Mania - 20)/3 = 60
total cost for Balloon Express - 10 + 4(total cost for Balloon Mania - 20) = 180
total cost for Balloon Express - 10 + 4 * total cost for Balloon Mania - 80 = 180
total cost for Balloon Express + 4 * total cost for Balloon Mania - 90 = 180
Combine like terms:
5 * total cost for Balloon Mania = 270
Solve for total cost for Balloon Mania:
total cost for Balloon Mania = 270 / 5 = $54
Now substitute this value into equation 3 to solve for total cost for Balloon Express:
total cost for Balloon Express = 2 * x + 10 = 2 * (30 - y) + 10 = 2 * (30 - 54) + 10 = 2 * (-24) + 10 = -48 + 10 = $-38
Based on the calculations, the O'Donnells should choose Balloon Mania since the total cost for Balloon Mania is $54, compared to a negative cost with Balloon Express.
To find how much they will save, we calculate the difference in total costs:
Amount saved = total cost for Balloon Express - total cost for Balloon Mania
Amount saved = $-38 - $54 = -$92
Therefore, the O'Donnells will save $92 by choosing Balloon Mania.