A certain apprentice has enrolled in 85 hours of training courses. The equation 10x + 15y = 85 represents this...
GMAT Algebra : (Alg) Questions
A certain apprentice has enrolled in 85 hours of training courses. The equation \(10\mathrm{x} + 15\mathrm{y} = 85\) represents this situation, where x is the number of on-site training courses and y is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course?
1. TRANSLATE the equation information
- Given equation: \(10\mathrm{x} + 15\mathrm{y} = 85\)
- Where:
- \(\mathrm{x}\) = number of on-site training courses
- \(\mathrm{y}\) = number of online training courses
- \(85\) = total hours of training
2. INFER what the coefficients represent
- In the equation \(10\mathrm{x} + 15\mathrm{y} = 85\):
- \(10\mathrm{x}\) must represent total hours from on-site courses
- \(15\mathrm{y}\) must represent total hours from online courses
- Since \(\mathrm{x}\) is the number of on-site courses, \(10\) must be the hours per on-site course
- Since \(\mathrm{y}\) is the number of online courses, \(15\) must be the hours per online course
3. Calculate the difference
- Hours per online course: \(15\)
- Hours per on-site course: \(10\)
- Difference: \(15 - 10 = 5\)
Answer: 5
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what the coefficients \(10\) and \(15\) represent in the equation context.
Instead of recognizing that \(10\) and \(15\) are hours per individual course, students might think these numbers represent total hours for each course type, or they might focus on the variables \(\mathrm{x}\) and \(\mathrm{y}\) without understanding that the coefficients carry meaning. This leads to confusion about what to subtract and they may guess randomly.
Second Most Common Error:
Poor INFER reasoning: Students correctly identify that \(10\) and \(15\) represent hours per course but subtract in the wrong order.
They calculate \(10 - 15 = -5\) and either select this as a negative answer (if available) or become confused about whether the answer can be negative. This may lead them to guess or select an incorrect positive value.
The Bottom Line:
The key challenge is recognizing that in contextual linear equations, coefficients often represent rates or amounts per unit, not just mathematical multipliers. Students need to connect the equation structure to the real-world meaning.