Two friends are making lemonade to sell at their lemonade stand. First, they make $4\,\text{liters}$ 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? $\newline$ Write your answer as a whole number or as a decimal rounded to the nearest tenth. $\newline$ $\newline$ ____ liters $\newline$

View step-by-step help

Home

Math Problems

Algebra 1

Weighted averages: word problems

Full solution

Q. Two friends are making lemonade to sell at their lemonade stand. First, they make

4\,\text{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?

\newline

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

\newline

\newline

____ liters

\newline

Calculate Lemon Juice Amount: Determine the amount of lemon juice in the initial $4$ liters of lemonade. $\newline$ Calculation: $4$ liters $\times$ $31\% = 4 \times 0.31$
Determine Lemon Juice in Mixture: Calculate the amount of lemon juice in the initial mixture. $\newline$ Calculation: $4 \text{ liters} \times 0.31 = 1.24 \text{ liters}$ of lemon juice
Find Store-Bought Lemonade Amount: Let $x$ be the amount of store-bought lemonade (in liters) that needs to be added. $\newline$ Expression: $x$ liters of store-bought lemonade
Calculate Total Lemon Juice: Determine the amount of lemon juice in the store-bought lemonade that will be added. $\newline$ Expression: $x$ liters $\times 26\% = x \times 0.26$
Set Up Equation: Set up an equation for the total amount of lemon juice in the final mixture. $\newline$ Expression: $1.24$ liters (from initial mixture) + $x \times 0.26$ (from store-bought lemonade) = $(4 \text{ liters} + x) \times 27\%$
Convert Percentage to Decimal: Convert the percentage to a decimal and set up the equation. $\newline$ Equation: $1.24 + x \times 0.26 = (4 + x) \times 0.27$
Distribute Equation: Distribute the $0.27$ on the right side of the equation. $\newline$ Equation: $1.24 + 0.26x = 4 \times 0.27 + x \times 0.27$
Calculate Lemon Juice in Final Mixture: Calculate the amount of lemon juice in the $4$ liters of the final mixture. $\newline$ Calculation: $4 \times 0.27 = 1.08$ liters of lemon juice
Substitute Calculated Amount: Substitute the calculated amount into the equation. $\newline$ Equation: $1.24 + 0.26x = 1.08 + 0.27x$
Rearrange Equation: Rearrange the equation to isolate $x$ on one side. $\newline$ Equation: $0.26x - 0.27x = 1.08 - 1.24$
Simplify Equation: Simplify the equation. $\newline$ Equation: $-0.01x = -0.16$
Solve for x: Solve for x, which represents the liters of store-bought lemonade to be added. $\newline$ Calculation: $x = \frac{-0.16}{-0.01}$
Calculate x Value: Calculate the value of x. $\newline$ Calculation: $x = 16 \, \text{liters}$