Last month, a moving company charged customers according to the equation 55H + 90J = 6{,}785, where H is the...
GMAT Algebra : (Alg) Questions
Last month, a moving company charged customers according to the equation \(55\mathrm{H} + 90\mathrm{J} = 6{,}785\), where \(\mathrm{H}\) is the total number of labor hours worked across all jobs, \(\mathrm{J}\) is the number of jobs completed, and \(6{,}785\) is the total amount, in dollars, charged that month. Which of the following is the best interpretation of \(55\) in this context?
The company charged $55 per job that month.
The company worked 55 labor hours that month.
The company charged $55 per hour of labor that month.
The company completed 55 jobs that month.
1. TRANSLATE the equation components
- Given equation: \(55\mathrm{H} + 90\mathrm{J} = 6{,}785\)
- Variable meanings:
- \(\mathrm{H}\) = total number of labor hours worked
- \(\mathrm{J}\) = number of jobs completed
- \(6{,}785\) = total amount charged in dollars
2. INFER what coefficients represent in context
- In equations modeling costs: coefficient × quantity = partial cost
- Since \(55\) multiplies \(\mathrm{H}\) (labor hours): \(55\) must be the cost per hour
- Since \(90\) multiplies \(\mathrm{J}\) (jobs): \(90\) must be the cost per job
- The equation structure is: \((\$/\text{hour})(\text{hours}) + (\$/\text{job})(\text{jobs}) = \text{total cost}\)
3. APPLY CONSTRAINTS to select the correct interpretation
- \(55\) is the coefficient of \(\mathrm{H}\) (hours), not the value of \(\mathrm{H}\)
- Therefore, \(55\) represents the rate charged per hour of labor
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse the coefficient \((55)\) with the variable value \((\mathrm{H})\)
They might think "55 appears in the equation, so the company worked 55 hours" or "completed 55 jobs," not recognizing that 55 is multiplied by the hours variable, making it a rate. This leads them to select Choice B (The company worked 55 labor hours) or Choice D (The company completed 55 jobs).
Second Most Common Error:
Poor INFER reasoning about equation structure: Students don't recognize that coefficients represent unit rates in context problems
They may randomly associate 55 with jobs instead of hours, leading them to select Choice A (The company charged $55 per job), not realizing that 90 is the coefficient associated with jobs, not 55.
The Bottom Line:
This problem tests whether students understand that in context equations, coefficients represent rates or prices per unit, while variables represent quantities. The key insight is recognizing what each coefficient is multiplied by.
The company charged $55 per job that month.
The company worked 55 labor hours that month.
The company charged $55 per hour of labor that month.
The company completed 55 jobs that month.