A small business tracks its profit using two different accounting methods. Method A gives the profit equation 9x + 14y...
GMAT Algebra : (Alg) Questions
A small business tracks its profit using two different accounting methods. Method A gives the profit equation \(\mathrm{9x + 14y = 11}\), where \(\mathrm{x}\) represents thousands of units sold and \(\mathrm{y}\) represents price per unit in dollars. Method B, using different cost assumptions, gives the profit equation \(\mathrm{-27x + ry = \frac{1}{5}}\), where \(\mathrm{r}\) is a constant related to the overhead rate.
If these two methods can never give the same profit values (the system has no solution), what is the value of \(\mathrm{r}\)?
Answer Format: Fill-in-the-blank
1. TRANSLATE the problem requirements
- Given information:
- Method A: \(\mathrm{9x + 14y = 11}\)
- Method B: \(\mathrm{-27x + ry = \frac{1}{5}}\)
- "Can never give same profit values" = system has no solution
2. INFER what "no solution" means mathematically
- A system has no solution when the lines are parallel but distinct
- Parallel lines have equal slopes but different y-intercepts
- Strategy: Find slopes of both equations and set them equal
3. INFER the slope of each equation
- For standard form \(\mathrm{Ax + By = C}\), the slope is \(\mathrm{-\frac{A}{B}}\)
- Method A: \(\mathrm{9x + 14y = 11}\) → \(\mathrm{slope_1 = -\frac{9}{14}}\)
- Method B: \(\mathrm{-27x + ry = \frac{1}{5}}\) → \(\mathrm{slope_2 = -\frac{(-27)}{r} = \frac{27}{r}}\)
4. SIMPLIFY by setting slopes equal
- For parallel lines: \(\mathrm{slope_1 = slope_2}\)
- \(\mathrm{-\frac{9}{14} = \frac{27}{r}}\)
- Cross multiply: \(\mathrm{-9r = 27 \times 14}\)
- \(\mathrm{-9r = 378}\)
- \(\mathrm{r = -\frac{378}{9} = -42}\)
5. INFER verification that lines are distinct
- Check coefficient ratios: \(\mathrm{\frac{A_2}{A_1} = \frac{-27}{9} = -3}\), \(\mathrm{\frac{B_2}{B_1} = \frac{-42}{14} = -3}\)
- Since \(\mathrm{\frac{A_2}{A_1} = \frac{B_2}{B_1}}\), lines are parallel ✓
- Check: \(\mathrm{\frac{C_2}{C_1} = \frac{\frac{1}{5}}{11} = \frac{1}{55} \neq -3}\)
- Since \(\mathrm{\frac{C_2}{C_1} \neq \frac{A_2}{A_1}}\), lines are distinct ✓
Answer: -42
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't connect "can never give same profit values" to the mathematical condition of parallel but distinct lines. Instead, they might try to solve the system directly or assume they need to find where the equations intersect. This leads to confusion about what the problem is actually asking for, causing them to get stuck and guess randomly.
Second Most Common Error:
Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{-\frac{9}{14} = \frac{27}{r}}\) but make arithmetic errors in cross multiplication, such as getting the sign wrong (\(\mathrm{-9r = -378}\) instead of \(\mathrm{-9r = 378}\)) or miscalculating \(\mathrm{27 \times 14}\). This could lead them to answer choices like 42 instead of -42.
The Bottom Line:
This problem requires recognizing that business language about "never giving same values" translates to a specific mathematical condition about parallel lines, then executing the slope comparison correctly.