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Difficulty: MediumAlgebraic Word Problems and Equation Modeling

A chemical processing plant blends Solution X, which contains 25%25\% acid by volume, with Solution Y, which contains 65%65\% acid by volume, to create a 100100-liter mixture that is 40%40\% acid by volume. If Solution X costs $3\$3 per liter and Solution Y costs $7\$7 per liter, what is the total cost of the solutions used to produce the mixture?

  1. $450\$450Answer
  2. B
    $500\$500
  3. C
    $530\$530
  4. D
    $540\$540
  5. E
    $550\$550

Answer

The total cost of the solutions used to produce the mixture is $450\$450.
To determine the total cost, first solve for the exact volume of each solution needed. Let xx be the volume of Solution X in liters and yy be the volume of Solution Y in liters. Since the total mixture volume is 100100 liters, x+y=100x + y = 100, so y=100−xy = 100 - x. Setting up the acid balance equation gives 0.25x+0.65y=0.40(100)0.25x + 0.65y = 0.40(100). Substituting y=100−xy = 100 - x yields 0.25x+0.65(100−x)=400.25x + 0.65(100 - x) = 40, which simplifies to −0.40x+65=40-0.40x + 65 = 40, leading to 0.40x=250.40x = 25 and x=62.5x = 62.5. Thus, 62.562.5 liters of Solution X and 37.537.5 liters of Solution Y are required. Multiplying each volume by its price per liter gives 3(62.5)+7(37.5)=187.50+262.50=4503(62.5) + 7(37.5) = 187.50 + 262.50 = 450. The total cost is $450\$450.

Step-by-Step Solution

1
Define variables and write the equation for total volume.
Let xx be the volume of Solution X in liters and yy be the volume of Solution Y in liters. Then x+y=100x + y = 100, which implies y=100−xy = 100 - x.
The total required volume of the mixture is 100100 liters.
2
Write the acid balance equation and solve for xx and yy.
Equation: 0.25x+0.65(100−x)=0.40(100)  ⟹  0.25x+65−0.65x=40  ⟹  −0.40x=−25  ⟹  x=62.50.25x + 0.65(100 - x) = 0.40(100) \implies 0.25x + 65 - 0.65x = 40 \implies -0.40x = -25 \implies x = 62.5 liters. Thus, y=100−62.5=37.5y = 100 - 62.5 = 37.5 liters.
The total volume of pure acid in the final mixture must equal the sum of the pure acid contributed by Solution X and Solution Y.
3
Calculate the total cost of the mixture components.
Total Cost = 3(62.5)+7(37.5)=187.5+262.5=4503(62.5) + 7(37.5) = 187.5 + 262.5 = 450.
Multiply the volume of each solution by its respective price per liter and sum the results.

Key Concept

Algebraic Modeling of Mixture Problems and Systems of Linear Equations
Estimated Time:2m 0s
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Topics

Algebraic Word Problems and Equation Modeling

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