Question

Hannah has to make 36 gallons of punch for a big party. The punch is made of soda and fruit drink. The cost of the soda is $2.03 per gallon and the cost of the fruit drink is $2.39 per gallon. Hannah's budget requires that the punch cost $2.14 per gallon. How many gallons of soda and how many gallons of fruit drink does she need?

Please leave an explination i am so confused

Answer 1

She needs 11 gallons of fruit drink and 25 gallons of soda.

Explanation:

• We will need a system of equation to determine the amount of soda and fruit drink in the punch.

In our system:

• S represents the gallons of soda in the punch,
• and F represents the gallons of fruit drink in the punch.

Step 1: Determine the total cost Hannah is willing to spend on punch ingredients:

Since Hannah wants the punch to cost $2.14 per gallon and she wants the punch to have 36 gallons, we can find the total amount she's willing to spend on the ingredients by multiplying 2.14 by 36:

Hannah's total budget = $2.14 * 36 gallons

Hanna's total budget = 77.04

Thus, Hannah is willing to spend no more than $77.04 on the punch ingredients.

• Now we can start compiling and solving the two equations in our system.

First equation:

We know that the sum of the amounts of soda and fruit drink equals the total amount of punch:

soda amount + fruit drink = total amount of punch

Since there are 36 gallons of punch in all, our first equation is given by:

S + F = 36

Second equation:

We also know that sum of the soda and fruit drink costs equals the total cost:

(price * soda amount) + (price * fruit drink amount) = total cost.

Since the soda costs $2.03 per gallon, the fruit drink cost $2.39 per gallon, and the total cost Hannah is willing to pay is $77.04, our second equation is given by:

2.03S + 2.39F = 77.04

Method to solve: Elimination:

Solving for F (the amount of fruit drink in the punch):

We can start by multiplying the first equation by -2.03, which will eventually allow us to eliminate S since -2.03S + 2.03S = 0:

-2.03(S + F = 36)

-2.03S - 2.03F = -73.08

Now we can add this equation to the second equation (i.e., 2.03S + 2.39F = 77.04) to eliminate S and solve for F:

-2.03S - 2.03F = -73.08

+

2.03S + 2.39F = 77.04

----------------------------------------------------------------------------------------------------------(-2.03S + 2.03S) + (-2.03F + 2.39F) = (-73.08 + 77.04)

(0.36F = 3.96) / 0.36

F = 11

Thus, she needs 11 gallons of fruit drink for the punch.

Solving for S (the amount of soda in the punch):

Now we can solve for S by plugging in 11 for F in S + F = 36:

(S + 11 = 36) - 11

S = 25

Thus, she needs 25 gallons of soda for the punch.