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 Mania; $25
Balloon Mania; $25
Balloon Express; $25
Balloon Express; $25
Balloon Express; $5
To solve this problem using a system of equations, let's assign variables to the unknowns.
Let's say B represents the number of balloons ordered from Balloon Express and M represents the number of balloons ordered from Balloon Mania.
From the given information, we can form two equations:
1) Balloon Express charges $2 per balloon and $10 for delivery, so the total cost for ordering B balloons is 2B + 10.
2) Balloon Mania charges $1.50 per balloon and $20 for delivery, so the total cost for ordering M balloons is 1.50M + 20.
We are given that the O'Donnells plan to order 30 balloons, so B + M = 30.
Now we can set up the system of equations:
2B + 10 = 1.50M + 20 (equation 1)
B + M = 30 (equation 2)
To solve this system, we can use substitution or elimination. Let's use substitution:
We can isolate B in equation 2 and substitute it into equation 1:
B = 30 - M (equation 3)
Now substitute equation 3 into equation 1:
2(30 - M) + 10 = 1.50M + 20
Distribute and combine like terms:
60 - 2M + 10 = 1.50M + 20
70 - 2M = 1.50M + 20
Simplify:
-2M - 1.50M = 20 - 70
-3.50M = -50
Divide by -3.50 to solve for M:
M = (-50) / (-3.50)
M = 14.29
Since we cannot have a fraction of a balloon, we can round M up to the nearest whole number, which is 15.
Substitute M = 15 back into equation 3 to solve for B:
B = 30 - 15
B = 15
Therefore, the O'Donnells should choose Balloon Mania to order 15 balloons, and they will save $25 compared to ordering from Balloon Express.
So, the correct answer is:
Balloon Mania; $25
To solve this problem using a system of equations, let's define some variables:
Let B_E represent the number of balloons ordered from Balloon Express.
Let B_M represent the number of balloons ordered from Balloon Mania.
From the given information, we know the following equations:
For Balloon Express: B_E balloons cost 2*B_E dollars and delivery costs 10 dollars.
So, the total cost for Balloon Express can be expressed as: 2*B_E + 10.
For Balloon Mania: B_M balloons cost 1.5*B_M dollars and delivery costs 20 dollars.
So, the total cost for Balloon Mania can be expressed as: 1.5*B_M + 20.
Since the O'Donnells plan to order a total of 30 balloons, we can also write the equation: B_E + B_M = 30.
To find the optimal choice and the amount saved, we need to solve this system of equations.
We have the equations:
Equation 1: 2*B_E + 10 = C_E (Total cost for Balloon Express)
Equation 2: 1.5*B_M + 20 = C_M (Total cost for Balloon Mania)
Equation 3: B_E + B_M = 30 (Total number of balloons)
Now, let's solve the system of equations:
From Equation 3, we know that B_E + B_M = 30.
Rearranging, we get B_M = 30 - B_E.
Substituting this value into Equation 2, we have:
1.5*(30 - B_E) + 20 = C_M
45 - 1.5*B_E + 20 = C_M
65 - 1.5*B_E = C_M
Now we can compare the two costs:
If C_E < C_M, then the O'Donnells should choose Balloon Express and the amount saved is C_M - C_E.
If C_M < C_E, then the O'Donnells should choose Balloon Mania and the amount saved is C_E - C_M.
To determine which company offers more savings, we need to calculate the total cost for both companies. Substituting B_M = 30 - B_E:
C_E = 2*B_E + 10
C_M = 1.5*(30 - B_E) + 20
C_M = 45 - 1.5*B_E + 20
C_M = 65 - 1.5*B_E
Now let's compare the costs:
If C_E < C_M:
2*B_E + 10 < 65 - 1.5*B_E
3.5*B_E < 55
B_E < 15.71
Since the number of balloons must be an integer, B_E can be at most 15.
If C_M < C_E:
1.5*(30 - B_E) + 20 < 2*B_E + 10
45 - 1.5*B_E + 20 < 2*B_E + 10
1.5*B_E > 55
B_E > 36.67
Since B_E cannot be greater than 30 (since they are ordering 30 balloons in total), Balloon Mania is not the optimal choice.
Therefore, the O'Donnells should choose Balloon Express, and they will save $25 compared to Balloon Mania.
To solve this real-world problem using a system of equations, we can set up two equations to represent the total cost from each company.
Let's denote the number of balloons as "b", and the total cost from Balloon Express as "C1", and the total cost from Balloon Mania as "C2".
According to the problem, Balloon Express charges $2 per balloon and $10 for delivery. So the equation representing the total cost from Balloon Express would be:
C1 = 2b + 10
Similarly, Balloon Mania charges $1.50 per balloon and $20 for delivery. So the equation representing the total cost from Balloon Mania would be:
C2 = 1.50b + 20
Now, we need to determine which company to choose and how much they will save. Since we know the O'Donnells plan to order 30 balloons, we can substitute b = 30 into the equations for C1 and C2.
For Balloon Express:
C1 = 2(30) + 10
C1 = 60 + 10
C1 = 70
For Balloon Mania:
C2 = 1.50(30) + 20
C2 = 45 + 20
C2 = 65
Comparing the total costs, we see that Balloon Mania charges $65 while Balloon Express charges $70. Therefore, the O'Donnells should choose Balloon Mania, and they will save $70 - $65 = $5.