When the women's soccer team won the state championship, the parent boosters welcomed the team back to school with a balloon bouquet for each of the 18 players. The parents spent a total of $161.28 (excluding tax) on foil balloons that cost $1.96 each and latex school-color balloons that cost $0.14 each. Each player received 12 balloons, and all the balloon bouquets were identical. How many of each type of balloon did each bouquet include?

Each bouquet included 8 latex balloons and 4 foil balloons

Step-by-step explanation:

Hi there!

Let x be the number of latex balloons and y the number of foil balloons. The total cost of the balloons will be the cost of each latex balloon times the amount (x) added to the cost of each foil balloon times the amount of these balloons (y). Then:

Cost of each latex balloon = 0.14

Amount of latex balloon = x

Cost of each foil balloon = 1.96

Amount of foil balloons = y

0.14 x + 1.96 y = 161.28

We also know that each player received 12 balloons. Since there are 18 players, the number of foil and latex balloons will be 18 · 12 = 216 balloons.

Then:

x + y = 216

So we have a system of equations that we can solve because we have two equations and two unknowns. Let's solve it!

0.14 x + 1.96 y = 161.28

x + y = 216

Let's solve the second equation for y:

x + y = 216

y = 216 - x

Now, let's replace y in the first equation and solve it for x:

0.14 x + 1.96 y = 161.28

0.14 x + 1.96 (216 - x) = 161.28

Apply distributive property:

0.14 x + 423.36 - 1.96 x = 161.28

-1.82 x + 423.36 = 161.28

-1.82 x = 161.28 - 423.36

-1.82 x = - 262.08

x = - 262.08 / - 1.82

x = 144

There are 144 latex balloons

Then:

y = 216 - 144 = 72

There are 72 foil balloons

If 144 latex balloons are distributed among the 18 players, each player will receive (144/18) 8 balloons.

In the same way, if the 72 foil balloons are distributed among the players, each will receive (72 / 18) 4 balloons.

Then, each bouquet included 8 latex balloons and 4 foil balloons.