A beverage manufacturer creates a functional tea blend by combining Extract A, which conta… — GMAT

A beverage manufacturer creates a functional tea blend by combining Extract $A$ , which contains $12\%$ polyphenols by volume, and Extract $B$ , which contains $28\%$ polyphenols by volume. If the manufacturer needs to produce a $200$ -liter batch of tea with a total polyphenol concentration of $18\%$ by volume, how many liters of Extract $A$ must be included in the mixture?

Cevap: 125 liters

Cevap

The manufacturer must include

125

liters of Extract

A

in the mixture.

By setting up the linear system

x + y = 200

and

0.12x + 0.28y = 36

, substituting

y = 200 - x

leads directly to

-0.16x = -20

, giving

x = 125

liters.

Adım Adım Çözüm

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

x + y = 200

and

0.12x + 0.28y = 36

The sum of the individual extract volumes equals the total batch volume, and the sum of the pure polyphenols from each extract equals the total pure polyphenols in the final mixture.

Substitute

y = 200 - x

into the polyphenol equation to solve for

x

0.12x + 0.28(200 - x) = 36 \implies 0.12x + 56 - 0.28x = 36

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

x

Combine like terms and solve for

x

-0.16x = -20 \implies x = \frac{-20}{-0.16} = 125

Isolating

x

yields the required volume of Extract

A

Anahtar Kavram

Systems of Linear Equations in Applied Mixture Problems

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Adım Adım Çözüm

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