Ava is in charge of making 8 gallons of lemonade iced tea for her family reunion. There are two ingredients: lemonade, which costs $2.84 per gallon, and sweet tea, which costs $1.99 per gallon. She would like to make the lemonade iced tea with as much lemonade as possible, but she can only afford to spend $2.50 per gallon. How many gallons of each should she use?

Question

Write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. Do not round.
gallons of lemonade
gallons of sweet tea

Answer 1

To find the number of gallons of lemonade and sweet tea Ava should use, we need to set up a system of equations based on the information given.

Let:

• \( x \) = gallons of lemonade
• \( y \) = gallons of sweet tea

We know that:

1. \( x + y = 8 \) (the total volume of lemonade iced tea)
2. She wants to spend an average of $2.50 per gallon on the mixture.

The cost equations based on the prices per gallon are:

• Lemonade costs $2.84 per gallon
• Sweet tea costs $1.99 per gallon

The total cost for the mixture should be: \[ \frac{2.84x + 1.99y}{x + y} = 2.50 \]

Since \( x + y = 8 \), we can substitute \( y = 8 - x \) into the cost equation: \[ \frac{2.84x + 1.99(8 - x)}{8} = 2.50 \]

Now, we can multiply both sides by 8 to eliminate the denominator: \[ 2.84x + 1.99(8 - x) = 20 \]

Expanding the left side: \[ 2.84x + 15.92 - 1.99x = 20 \]

Combining like terms: \[ (2.84 - 1.99)x + 15.92 = 20 \] \[ 0.85x + 15.92 = 20 \]

Now isolating \( x \): \[ 0.85x = 20 - 15.92 \] \[ 0.85x = 4.08 \] \[ x = \frac{4.08}{0.85} \approx 4.8 \]

Now, substituting \( x \) back to find \( y \): \[ y = 8 - x = 8 - 4.8 = 3.2 \]

Thus, the solution is:

• Gallons of lemonade: \( 4.8 \)
• Gallons of sweet tea: \( 3.2 \)

Final answers:

• \( 4.8 \) gallons of lemonade
• \( 3.2 \) gallons of sweet tea