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: $$\boxed{{\displaystyle 16}}\text{ liters.}$$