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 

1 answer

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.