Questions: Hannah has to make 25 gallons of punch for a potluck. The punch is made of soda and fruit drink. The cost of the soda is 1.79 per gallon, and the cost of the fruit drink is 2.49 per gallon. Hannah's budget requires that the punch cost 2.21 per gallon. How many gallons of soda and how many gallons of fruit drink does she need? Provide your answer below: soda: gallons fruit drink: gallons

Solution Steps

To solve this problem, we need to set up a system of equations based on the given information. Let \( x \) be the number of gallons of soda and \( y \) be the number of gallons of fruit drink. We have two equations: one for the total volume of the punch and another for the cost per gallon. The first equation is \( x + y = 25 \) (total gallons), and the second equation is \( 1.79x + 2.49y = 2.21 \times 25 \) (total cost). We can solve this system of equations to find the values of \( x \) and \( y \).

We define the variables:

• Let \( x \) be the number of gallons of soda.
• Let \( y \) be the number of gallons of fruit drink.

From the problem, we can establish the following equations:

1. Total volume of punch: \[ x + y = 25 \]
2. Total cost of the punch: \[ 1.79x + 2.49y = 2.21 \times 25 \] Simplifying the right side gives: \[ 1.79x + 2.49y = 55.25 \]

We solve the system of equations:

1. From \( x + y = 25 \), we can express \( y \) as: \[ y = 25 - x \]
2. Substituting \( y \) into the second equation: \[ 1.79x + 2.49(25 - x) = 55.25 \] Expanding and simplifying: \[ 1.79x + 62.25 - 2.49x = 55.25 \] Combining like terms: \[ -0.70x + 62.25 = 55.25 \] Rearranging gives: \[ -0.70x = -7 \] Thus: \[ x = \frac{7}{0.70} = 10 \]

Substituting \( x = 10 \) back into the equation for \( y \): \[ y = 25 - 10 = 15 \]

Final Answer