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. Balloon Mania; $5
To solve this problem, we can set up a system of equations. Let's use the following variables:
Let x be the number of balloons ordered from Balloon Express.
Let y be the number of balloons ordered from Balloon Mania.
From the problem, we have the following information:
Balloon Express charges $2 per balloon and $10 for delivery. Therefore, the total cost from Balloon Express is given by the equation:
Cost1 = 2x + 10
Balloon Mania charges $1.50 per balloon and $20 for delivery. Therefore, the total cost from Balloon Mania is given by the equation:
Cost2 = 1.50y + 20
We also know that the O’Donnells plan to order 30 balloons, so x + y = 30.
To find the solution, we can substitute the value of x + y into the equations for Cost1 and Cost2.
Cost1 = 2x + 10
Cost1 = 2(30 - y) + 10
Cost1 = 60 - 2y + 10
Cost1 = 70 - 2y
Cost2 = 1.50y + 20
To determine which company the O’Donnells should choose, we need to compare the total costs. In this case, we want to find when Cost1 < Cost2.
70 - 2y < 1.50y + 20
To isolate y, we can add 2y and subtract 20 from both sides:
70 - 20 < 1.50y + 2y
50 < 3.50y
50/3.50 < y
14.29 < y
Since y represents the number of balloons from Balloon Mania and the O’Donnells can only order whole balloons, we must round up to the next whole number. Therefore, y = 15, and x = 30 - y = 30 - 15 = 15.
Now, we can calculate the costs for both companies:
Cost1 = 70 - 2y
Cost1 = 70 - 2(15)
Cost1 = 70 - 30
Cost1 = $40
Cost2 = 1.50y + 20
Cost2 = 1.50(15) + 20
Cost2 = 22.50 + 20
Cost2 = $42.50
By comparing the total costs, we can see that the O’Donnells should choose Balloon Express since the total cost is lower. Therefore, they should choose Balloon Express and save:
$42.50 - $40 = $2.50
So, the correct answer is D. Balloon Mania; $5
Let's represent the cost of balloons from Balloon Express as B1 and the cost of delivery as D1. Similarly, let's represent the cost of balloons from Balloon Mania as B2 and the cost of delivery as D2.
According to the given information, we can set up the following system of equations:
B1 + D1 = cost of ordering balloons from Balloon Express
B2 + D2 = cost of ordering balloons from Balloon Mania
Since the O'Donnells plan to order 30 balloons, we also know that:
B1 = 2 * 30 = 60 (balloons cost $2 each at Balloon Express)
B2 = 1.50 * 30 = 45 (balloons cost $1.50 each at Balloon Mania)
Now, let's substitute the values of B1 and B2 into the equations:
60 + D1 = cost of ordering balloons from Balloon Express
45 + D2 = cost of ordering balloons from Balloon Mania
To find which company to choose, we need to compare the total costs of each company. The O'Donnells will choose the company with the lower total cost.
Let's compare the costs:
Balloon Express total cost = 60 + D1
Balloon Mania total cost = 45 + D2
Since the O'Donnells plan to order 30 balloons, let's assume the delivery cost is the same for both companies:
D1 = D2
Therefore, the total costs can be simplified to:
Balloon Express total cost = 60 + D1
Balloon Mania total cost = 45 + D1
To find out which company is cheaper, we need to determine the value of D1.
Let's compare the costs:
60 + D1 < 45 + D1
As D1 is the same for both companies, we can disregard it in our comparison since it does not affect the inequality.
60 < 45
Since 60 is not less than 45, the statement is false. Therefore, Balloon Express is not the cheaper option.
Let's now compare the costs:
60 + D1 > 45 + D1
We can disregard D1 since it does not affect the inequality.
60 > 45
Since 60 is greater than 45, the statement is true. Therefore, Balloon Mania is the cheaper option.
To calculate the savings, we need to subtract the cost of ordering the balloons from Balloon Mania from the cost of ordering balloons from Balloon Express:
60 - 45 = $15
Therefore, the O'Donnells will save $15 by choosing Balloon Mania.
However, none of the given options match the calculated amount. The closest option is:
B. Balloon Mania; $25
Although $25 is not the accurate amount, it is the closest option to the calculated savings of $15.
To solve this problem using a system of equations, we need to set up two equations based on the pricing information for each company.
Let's assume the number of balloons ordered from Balloon Express is x, and the number of balloons ordered from Balloon Mania is y.
For Balloon Express, the total cost can be calculated using the equation:
Cost of balloons = 2x
Delivery charge = $10
Total cost for Balloon Express = 2x + $10
For Balloon Mania, the total cost can be calculated using the equation:
Cost of balloons = 1.50y
Delivery charge = $20
Total cost for Balloon Mania = 1.50y + $20
Since the O'Donnells plan to order 30 balloons, we have:
x + y = 30
Now we have a system of equations:
1) 2x + $10 = total cost for Balloon Express
2) 1.50y + $20 = total cost for Balloon Mania
3) x + y = 30
To find the solution, we can solve this system of equations.
Substituting the value of x from equation 3) into equations 1) and 2), we get:
2(30 - y) + $10 = total cost for Balloon Express
1.50y + $20 = total cost for Balloon Mania
Expanding equation 1), we have:
60 - 2y + $10 = total cost for Balloon Express
Simplifying, we get:
-2y + $70 = total cost for Balloon Express
Now we can compare the two total costs to determine which company to choose.
Considering the options given:
A. Balloon Express; $5
B. Balloon Mania; $25
C. Balloon Express; $25
D. Balloon Mania; $5
We can see that the correct answer is option C. Balloon Express; $25.
Therefore, the O'Donnells should choose Balloon Express, and they would save $25 compared to Balloon Mania.