A car rental company charges a base fee plus an additional amount per mile driven. The table shows the total...
GMAT Algebra : (Alg) Questions
A car rental company charges a base fee plus an additional amount per mile driven. The table shows the total cost \(\mathrm{C}\), in dollars, for driving \(\mathrm{m}\) miles. Which equation represents this relationship?
| Miles driven | Total cost (dollars) |
|---|---|
| 50 | 85 |
| 80 | 115 |
| 110 | 145 |
1. TRANSLATE the problem information
- Given information:
- Table showing miles driven (m) vs total cost (C)
- Need to find equation relating C and m
- Context: base fee plus cost per mile (linear relationship)
2. INFER the mathematical approach
- This is a linear relationship of the form \(\mathrm{C = mx + b}\)
- Need to find: slope (m) = cost per mile, y-intercept (b) = base fee
- Strategy: Use any two points to calculate slope, then find y-intercept
3. SIMPLIFY to find the slope
- Choose two points from table: \(\mathrm{(50, 85)}\) and \(\mathrm{(80, 115)}\)
- Apply slope formula:
\(\mathrm{slope = \frac{115-85}{80-50}}\)
\(\mathrm{= \frac{30}{30}}\)
\(\mathrm{= 1}\) - The cost per mile is $1
4. SIMPLIFY to find the y-intercept (base fee)
- Use point-slope relationship with \(\mathrm{(50, 85)}\):
- \(\mathrm{85 = 1(50) + b}\)
- \(\mathrm{85 = 50 + b}\)
- \(\mathrm{b = 35}\)
- The base fee is $35
5. Write the final equation and verify
- Equation: \(\mathrm{C = m + 35}\)
- Check: When \(\mathrm{m = 110}\), \(\mathrm{C = 110 + 35 = 145}\) ✓
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize they need to find both slope and y-intercept systematically. They might try to work backwards from answer choices or make assumptions about the base fee without calculating it properly.
This often leads them to select Choice B (C = m + 50) by incorrectly assuming the base fee equals the cost at 50 miles, or they might select Choice C or D by focusing only on the per-mile rate without accounting for the base fee structure.
Second Most Common Error:
Poor SIMPLIFY execution: Students make arithmetic errors when calculating the slope (getting 30 instead of 1) or when solving for the y-intercept (algebraic manipulation errors).
These calculation mistakes can lead to selecting any of the incorrect choices, particularly if they don't verify their answer against all data points in the table.
The Bottom Line:
This problem requires students to understand that real-world linear relationships often have both a rate component (slope) and a starting value (y-intercept), and that systematic calculation using the slope formula and point-slope form is more reliable than trying to spot patterns or guess from answer choices.