A chemist studying the impact of salt on a process mixes kilograms of a low-salt mixture, which is salt by...
GMAT Algebra : (Alg) Questions
A chemist studying the impact of salt on a process mixes kilograms of a low-salt mixture, which is salt by weight, with kilograms of a high-salt mixture, which is salt by weight, to create kilograms of a mixture that is salt by weight. Which equation represents this situation?
1. TRANSLATE the problem information
- Given information:
- x kg of low-salt mixture (2% salt by weight)
- y kg of high-salt mixture (96% salt by weight)
- Final mixture: 24 kg (4% salt by weight)
- Convert percentages to decimals: \(2\% = 0.02\), \(96\% = 0.96\), \(4\% = 0.04\)
2. INFER the conservation principle
- In mixture problems, the pure substance (salt) is conserved
- Total salt going in = Total salt coming out
- We need to find the salt contribution from each mixture
3. TRANSLATE each salt amount
- Salt from low-salt mixture = (amount) × (concentration) = \(0.02\mathrm{x}\) kg
- Salt from high-salt mixture = (amount) × (concentration) = \(0.96\mathrm{y}\) kg
- Salt in final mixture = (amount) × (concentration) = \(0.04 \times 24\) kg
4. Set up the conservation equation
- Salt in = Salt out
- \(0.02\mathrm{x} + 0.96\mathrm{y} = 0.04 \times 24\)
- This gives us: \(0.02\mathrm{x} + 0.96\mathrm{y} = (0.04)(24)\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Poor TRANSLATE reasoning: Students confuse which percentage goes with which variable, thinking the low-salt mixture has the high percentage (96%) and vice versa.
They might reason: "x is the first variable, so it gets the first percentage mentioned" or mix up "low-salt" with "high percentage." This leads them to write \(0.96\mathrm{x} + 0.02\mathrm{y} = (0.04)(24)\), which would make them select Choice A.
Second Most Common Error:
Weak INFER skill: Students don't recognize that the right side should represent the salt in the final mixture, not just the total amount of final mixture.
They might think "we have 24 kg total" and write \(0.02\mathrm{x} + 0.96\mathrm{y} = 24\), leading them to select Choice D or Choice C depending on their percentage assignment.
The Bottom Line:
This problem tests your ability to systematically translate mixture language into mathematics while keeping track of what each percentage represents. The key insight is recognizing that we're tracking the pure substance (salt), not just the total amounts.