Two friends are making lemonade to sell at their lemonade stand. First, they make 4 liters of lemonade containing 31% lemon juice, but that doesn't taste quite right to them. They decide to add some store-bought lemonade that is 26% lemon juice until they have a mixture that is 27% lemon juice. How much of the store-bought lemonade should they add?

Question

Write your answer as a whole number or as a decimal rounded to the nearest tenth.
liter

Answer 1

To solve the problem, let's define the variables involved:

• Let \( x \) be the amount of store-bought lemonade they will add, in liters.
• The original 4 liters of lemonade contains 31% lemon juice, which can be calculated as: \[ 0.31 \times 4 = 1.24 \text{ liters of lemon juice.} \]

The store-bought lemonade is 26% lemon juice, so the amount of lemon juice in the store-bought lemonade they add (which is \( x \) liters) is: \[ 0.26x \text{ liters of lemon juice.} \]

After adding \( x \) liters of store-bought lemonade, the total volume of the mixture becomes: \[ 4 + x \text{ liters.} \]

The total amount of lemon juice in the mixture becomes: \[ 1.24 + 0.26x \text{ liters of lemon juice.} \]

They want their new mixture to be 27% lemon juice. Thus, we set up the equation: \[ \frac{1.24 + 0.26x}{4 + x} = 0.27. \]

Now we will cross-multiply to eliminate the fraction: \[ 1.24 + 0.26x = 0.27(4 + x). \]

Expanding the right side: \[ 1.24 + 0.26x = 1.08 + 0.27x. \]

Next, we will rearrange the equation to isolate \( x \): \[ 1.24 - 1.08 = 0.27x - 0.26x, \] which simplifies to: \[ 0.16 = 0.01x. \]

Now, solving for \( x \): \[ x = \frac{0.16}{0.01} = 16. \]

Thus, the amount of store-bought lemonade that needs to be added is: \[ \boxed{16} \text{ liters.} \]