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Difficulty: Very hardSolving Linear Equations

A scientist is mixing two solutions in a laboratory. Solution A contains 14\frac{1}{4} active ingredient by volume, and Solution B contains 34\frac{3}{4} active ingredient by volume. The scientist needs to mix these to create a 2020-liter solution containing exactly 1120\frac{11}{20} active ingredient by volume.

Let xx represent the volume, in liters, of Solution A used. The correct relationship is represented by the linear equation:
14x+34(20−x)=11\frac{1}{4}x + \frac{3}{4}(20 - x) = 11
An assistant translates the problem description incorrectly, writing the equation as:
14x+34(x−20)=11\frac{1}{4}x + \frac{3}{4}(x - 20) = 11

What is the absolute difference, in liters, between the value of xx obtained from the assistant's incorrect equation and the value of xx obtained from the correct equation?

  1. A
    8
  2. B
    30
  3. 18Answer
  4. D
    26
  5. E
    44

Answer

The correct answer is 18, which is the absolute difference between the two solutions.
The correct answer is 18. Solving the correct equation 14x+34(20−x)=11\frac{1}{4}x + \frac{3}{4}(20 - x) = 11 by multiplying by 4 gives x+60−3x=44x + 60 - 3x = 44, which simplifies to −2x=−16-2x = -16 and yields x=8x = 8. Solving the assistant's incorrect equation 14x+34(x−20)=11\frac{1}{4}x + \frac{3}{4}(x - 20) = 11 in a similar manner gives x+3x−60=44x + 3x - 60 = 44, which simplifies to 4x=1044x = 104 and yields x=26x = 26. The absolute difference between these two values is ∣26−8∣=18|26 - 8| = 18.

Step-by-Step Solution

1
Solve the correct equation for xx.
Multiply the entire equation by 4 to clear the denominators: x+3(20−x)=44x + 3(20 - x) = 44. Distribute the 3: x+60−3x=44x + 60 - 3x = 44. Combine like terms: −2x+60=44-2x + 60 = 44. Subtract 60 from both sides: −2x=−16-2x = -16. Divide by -2: x=8x = 8.
This determines the correct volume of Solution A needed.
2
Solve the assistant's incorrect equation for xx.
Multiply the entire equation by 4 to clear the denominators: x+3(x−20)=44x + 3(x - 20) = 44. Distribute the 3: x+3x−60=44x + 3x - 60 = 44. Combine like terms: 4x−60=444x - 60 = 44. Add 60 to both sides: 4x=1044x = 104. Divide by 4: x=26x = 26.
This determines the incorrect volume of Solution A obtained by the assistant.
3
Calculate the absolute difference between the two solutions.
∣26−8∣=18|26 - 8| = 18.
The question asks for the absolute difference between the two values of xx.

Key Concept

Solving linear equations involving fractions, parentheses, and algebraic manipulation.
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Topics

Solving Linear Equations

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