A beverage producer creates two liquid mixtures, Mixture A and Mixture B. Mixture A consis… — GMAT

A beverage producer creates two liquid mixtures, Mixture $A$ and Mixture $B$ . Mixture $A$ consists of $30\%$ fruit concentrate by volume, and Mixture $B$ consists of $70\%$ fruit concentrate by volume. A lab technician combines $x$ liters of Mixture $A$ with $y$ liters of Mixture $B$ to prepare an $80$ -liter batch that contains $45\%$ fruit concentrate by volume. What is the value of $x$ ?

Answer: 50

Answer

The value of

x

is 50.

To find

x

, we construct two linear equations based on total liquid volume and total fruit concentrate. The total volume equation is

x + y = 80

, which gives

y = 80 - x

. The concentrate equation is

0.30x + 0.70y = 0.45(80) = 36

. Substituting

y = 80 - x

yields

0.30x + 0.70(80 - x) = 36

. Expanding gives

0.30x + 56 - 0.70x = 36

, so

-0.40x = -20

, which results in

x = 50

Step-by-Step Solution

Formulate a system of two linear equations representing total volume and total concentrate volume.

System equations:

x + y = 80

and

0.30x + 0.70y = 36

The sum of the component volumes equals the total mixture volume, and the sum of the pure concentrate from each component equals the total concentrate in the final mixture.

Substitute

y = 80 - x

into the concentrate equation to eliminate

y

0.30x + 0.70(80 - x) = 36

Substituting one variable reduces the system to a single linear equation in one variable.

Simplify the single-variable linear equation and solve for

x

0.30x + 56 - 0.70x = 36 \implies -0.40x = -20 \implies x = 50

Combining like terms isolates the variable

x

Key Concept

Solving systems of two linear equations formed by weighted mixture word problems.

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