The table shows the number of hours worked and the total pay in dollars for a job. There is a...
GMAT Algebra : (Alg) Questions
The table shows the number of hours worked and the total pay in dollars for a job. There is a linear relationship between hours and pay. Which of the following equations represents this relationship, where \(\mathrm{x}\) is hours and \(\mathrm{y}\) is pay?
| \(\mathrm{x}\) | \(\mathrm{y}\) |
|---|---|
| 0 | 10 |
| 1 | 25 |
| 2 | 40 |
1. TRANSLATE the table information
- Given points from table:
- \((0, 10)\): 0 hours worked, $10 pay
- \((1, 25)\): 1 hour worked, $25 pay
- \((2, 40)\): 2 hours worked, $40 pay
- Need to find: equation \(\mathrm{y = mx + b}\) where x is hours and y is pay
2. INFER the y-intercept directly
- When \(\mathrm{x = 0}\), \(\mathrm{y = 10}\)
- This means the y-intercept \(\mathrm{b = 10}\)
- The equation starts as: \(\mathrm{y = mx + 10}\)
3. INFER strategy for finding slope
- Need slope m using any two points
- Use points \((0, 10)\) and \((1, 25)\) with slope formula
4. SIMPLIFY the slope calculation
- \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
\(= \frac{25 - 10}{1 - 0}\)
\(= \frac{15}{1}\)
\(= 15\) - Complete equation: \(\mathrm{y = 15x + 10}\)
5. SIMPLIFY verification with remaining point
- Check with \((2, 40)\):
\(\mathrm{y = 15(2) + 10}\)
\(= 30 + 10\)
\(= 40\) ✓ - Matches the table data
Answer: C (\(\mathrm{y = 15x + 10}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that \(\mathrm{x = 0}\) gives the y-intercept directly. Instead, they try to use the slope formula with all three points separately or get confused about which number represents what in the linear equation.
They might calculate slope correctly as 15, but then incorrectly think the y-intercept is 25 (from the second data point), leading them to select Choice B (\(\mathrm{y = 15x + 25}\)).
Second Most Common Error:
Poor SIMPLIFY execution: Students set up the slope formula correctly but make arithmetic errors, calculating the slope as 10 instead of 15 (possibly by using (25-10)/(1-0) but somehow getting 10).
This leads them to select Choice A (\(\mathrm{y = 10x + 10}\)), which has the correct y-intercept but wrong slope.
The Bottom Line:
This problem tests whether students can systematically extract information from a data table and translate it into the standard linear equation form. The key insight is recognizing that the point where \(\mathrm{x = 0}\) immediately gives you the y-intercept, making the problem much simpler than it initially appears.