Question

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?
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