Valentina bought two containers of beads. In the first container 30% of the beads are red, and in the second...
GMAT Algebra : (Alg) Questions
Valentina bought two containers of beads. In the first container \(30\%\) of the beads are red, and in the second container \(70\%\) of the beads are red. Together, the containers have at least \(400\) red beads. Which inequality shows this relationship, where \(\mathrm{x}\) is the total number of beads in the first container and \(\mathrm{y}\) is the total number of beads in the second container?
\(0.3\mathrm{x} + 0.7\mathrm{y} \geq 400\)
\(0.7\mathrm{x} + 0.3\mathrm{y} \leq 400\)
\(\frac{\mathrm{x}}{3} + \frac{\mathrm{y}}{7} \leq 400\)
\(30\mathrm{x} + 70\mathrm{y} \geq 400\)
1. TRANSLATE the given information into mathematical expressions
- Given information:
- Container 1: x total beads, \(30\%\) are red
- Container 2: y total beads, \(70\%\) are red
- Combined containers have at least 400 red beads
- What this tells us:
- Red beads in container 1 = \(30\%\) of x = \(0.3\mathrm{x}\)
- Red beads in container 2 = \(70\%\) of y = \(0.7\mathrm{y}\)
2. INFER how to combine the information
- Since we want the total red beads from both containers, we add them:
Total red beads = \(0.3\mathrm{x} + 0.7\mathrm{y}\)
- The phrase "at least 400" means the total is greater than or equal to 400
3. TRANSLATE "at least" into the correct inequality symbol
- "At least 400" means \(\geq 400\)
- Therefore: \(0.3\mathrm{x} + 0.7\mathrm{y} \geq 400\)
Answer: A. \(0.3\mathrm{x} + 0.7\mathrm{y} \geq 400\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skills: Students mix up the percentages or misinterpret "at least"
Some students switch the percentages, thinking container 1 has \(70\%\) red beads and container 2 has \(30\%\) red beads, leading to \(0.7\mathrm{x} + 0.3\mathrm{y}\). Combined with misreading "at least" as "at most," they arrive at \(0.7\mathrm{x} + 0.3\mathrm{y} \leq 400\).
This may lead them to select Choice B (\(0.7\mathrm{x} + 0.3\mathrm{y} \leq 400\)).
Second Most Common Error:
Conceptual confusion about percentage representation: Students don't properly convert percentages to decimals for mathematical operations
Instead of converting \(30\%\) to \(0.3\), they might think "\(30\%\) of x" means dividing x by 30, leading to expressions like \(\mathrm{x}/3 + \mathrm{y}/7\). When combined with the "at least" misinterpretation, this results in \(\mathrm{x}/3 + \mathrm{y}/7 \leq 400\).
This may lead them to select Choice C (\(\mathrm{x}/3 + \mathrm{y}/7 \leq 400\)).
The Bottom Line:
This problem tests whether students can accurately translate percentages into decimal form and correctly interpret inequality language. The key insight is that percentages must be converted to decimals when setting up algebraic expressions, and "at least" always means "greater than or equal to."
\(0.3\mathrm{x} + 0.7\mathrm{y} \geq 400\)
\(0.7\mathrm{x} + 0.3\mathrm{y} \leq 400\)
\(\frac{\mathrm{x}}{3} + \frac{\mathrm{y}}{7} \leq 400\)
\(30\mathrm{x} + 70\mathrm{y} \geq 400\)