wo friends have built a lemonade stand, and they plan to start out by selling 24 liters of lemonade containing 27% lemon juice. To make this, they will mix some homemade lemonade containing 31% lemon juice with some store-bought lemonade that contains 26% lemon juice. How much of each type should the friends use?

Question

Write your answers as whole numbers or as decimals rounded to the nearest tenth.
liters of

Answer 1

Let's denote:

• \( x \) as the amount of homemade lemonade in liters,
• \( y \) as the amount of store-bought lemonade in liters.

We want to mix these two types of lemonade to get a total of 24 liters with 27% lemon juice.

From the problem, we have the following equations based on the total volume and the percentage of lemon juice:

1. The total volume equation: \[ x + y = 24 \]

2. The total lemon juice content equation: \[ 0.31x + 0.26y = 0.27 \times 24 \]

Calculating \( 0.27 \times 24 \): \[ 0.27 \times 24 = 6.48 \]

Now, we can rewrite the total lemon juice content equation as: \[ 0.31x + 0.26y = 6.48 \]

We now have a system of linear equations:

1. \( x + y = 24 \)
2. \( 0.31x + 0.26y = 6.48 \)

From the first equation, we can express \( y \) in terms of \( x \): \[ y = 24 - x \]

Now we can substitute this expression for \( y \) into the second equation: \[ 0.31x + 0.26(24 - x) = 6.48 \]

Distributing \( 0.26 \): \[ 0.31x + 6.24 - 0.26x = 6.48 \]

Combining like terms: \[ (0.31 - 0.26)x + 6.24 = 6.48 \] \[ 0.05x + 6.24 = 6.48 \]

Subtracting 6.24 from both sides: \[ 0.05x = 6.48 - 6.24 \] \[ 0.05x = 0.24 \]

Dividing both sides by 0.05: \[ x = \frac{0.24}{0.05} = 4.8 \]

Now, substituting \( x \) back into the equation for \( y \): \[ y = 24 - 4.8 = 19.2 \]

So the friends should use:

• \( 4.8 \) liters of homemade lemonade,
• \( 19.2 \) liters of store-bought lemonade.

Final Answer:

• Homemade lemonade: 4.8 liters
• Store-bought lemonade: 19.2 liters