Two laboratory solutions are being mixed: x milliliters of a 30% acid solution and y milliliters of a 60% acid...
GMAT Algebra : (Alg) Questions
Two laboratory solutions are being mixed: \(\mathrm{x}\) milliliters of a 30% acid solution and \(\mathrm{y}\) milliliters of a 60% acid solution. The inequality \(0.3\mathrm{x} + 0.6\mathrm{y} \leq 180\) gives the possible values of \(\mathrm{x}\) and \(\mathrm{y}\) for which the total amount of pure acid is less than or equal to 180 milliliters. Which statement is the best interpretation of \(\(\mathrm{x}, \mathrm{y}\) = \(150, 200\)\) in this context?
- If 150 mL of 60% solution and 200 mL of 30% solution are mixed, the pure acid content is less than or equal to 180 mL.
- If 150 mL of 30% solution and 200 mL of 60% solution are mixed, the pure acid content is less than or equal to 180 mL.
- If 150 mL of 60% solution and 200 mL of 30% solution are mixed, the pure acid content is greater than or equal to 180 mL.
- If 150 mL of 30% solution and 200 mL of 60% solution are mixed, the pure acid content is greater than or equal to 180 mL.
1. TRANSLATE the variable meanings
- Given information:
- \(\mathrm{x}\) = milliliters of 30% acid solution
- \(\mathrm{y}\) = milliliters of 60% acid solution
- \(\mathrm{(x, y) = (150, 200)}\)
- What this tells us:
- \(\mathrm{x = 150}\), so we have 150 mL of 30% solution
- \(\mathrm{y = 200}\), so we have 200 mL of 60% solution
2. TRANSLATE the inequality meaning
- The inequality \(\mathrm{0.3x + 0.6y \leq 180}\) represents:
- \(\mathrm{0.3x}\) = pure acid from the 30% solution
- \(\mathrm{0.6y}\) = pure acid from the 60% solution
- Total pure acid \(\mathrm{\leq 180}\) mL
3. SIMPLIFY by substituting our values
- Substitute \(\mathrm{x = 150}\) and \(\mathrm{y = 200}\):
\(\mathrm{0.3(150) + 0.6(200)}\)
\(\mathrm{= 45 + 120}\)
\(\mathrm{= 165}\)
- Since \(\mathrm{165 \leq 180}\) is true, the constraint is satisfied
4. TRANSLATE back to context
- This means: 150 mL of 30% solution and 200 mL of 60% solution mixed together will have pure acid content less than or equal to 180 mL
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students mix up which variable corresponds to which solution type
They see \(\mathrm{(x, y) = (150, 200)}\) and incorrectly think x represents the 60% solution and y represents the 30% solution. This happens because they don't carefully track the variable definitions given in the problem setup.
This may lead them to select Choice A (150 mL of 60% and 200 mL of 30%)
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly identify the solution volumes but misinterpret the inequality direction
After calculating \(\mathrm{0.3(150) + 0.6(200) = 165}\), they know 165 < 180 but get confused about whether this means the constraint is satisfied or not. They might think since 165 is less than 180, it should be "greater than or equal to" in the answer.
This may lead them to select Choice D (correct volumes but wrong inequality direction)
The Bottom Line:
This problem tests careful attention to variable definitions and the ability to translate mathematical notation into real-world context. Success requires methodically tracking which variable represents which solution throughout the entire problem.