The table shows four values of x and their corresponding values of \(\mathrm{f(x)}\).x\(\mathrm{f(x)}\)302235774013245187There is a linear relationshi...
GMAT Algebra : (Alg) Questions
The table shows four values of \(\mathrm{x}\) and their corresponding values of \(\mathrm{f(x)}\).
| \(\mathrm{x}\) | \(\mathrm{f(x)}\) |
|---|---|
| 30 | 22 |
| 35 | 77 |
| 40 | 132 |
| 45 | 187 |
There is a linear relationship between \(\mathrm{x}\) and \(\mathrm{f(x)}\) that is defined by the equation \(\mathrm{f(x) = m(x - 28)}\), where \(\mathrm{m}\) is a constant.
What is the value of \(\mathrm{m}\)?
1. TRANSLATE the problem information
- Given information:
- Data points: \((30,22)\), \((35,77)\), \((40,132)\), \((45,187)\)
- Linear equation form: \(\mathrm{f(x) = m(x - 28)}\)
- Need to find: constant m
2. INFER the solution strategy
- Since we have the equation form and data points, we can substitute any point to solve for m
- Strategy: Pick one data point, substitute into equation, solve for m
- Smart approach: Verify answer with a second point
3. TRANSLATE and substitute the first data point
- Using point (30, 22): when x = 30, f(x) = 22
- Substitute: \(22 = \mathrm{m}(30 - 28)\)
- This gives us: \(22 = \mathrm{m}(2)\)
4. SIMPLIFY to solve for m
- \(22 = 2\mathrm{m}\)
- \(\mathrm{m} = 22 ÷ 2 = 11\)
5. INFER that verification is wise, then TRANSLATE second point
- Using point (35, 77): when x = 35, f(x) = 77
- Substitute: \(77 = \mathrm{m}(35 - 28)\)
- This gives us: \(77 = 7\mathrm{m}\)
- SIMPLIFY: \(\mathrm{m} = 77 ÷ 7 = 11\) ✓
Answer: D) 11
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misread the equation form \(\mathrm{f(x) = m(x - 28)}\) and try to force it into slope-intercept form \(\mathrm{y = mx + b}\), leading them to think they need to find both slope and y-intercept separately. They may calculate the slope correctly as 11, but then get confused about how to use the (x - 28) format.
This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Poor SIMPLIFY execution: Students substitute correctly but make arithmetic errors when dividing. For example, with \(22 = 2\mathrm{m}\), they might incorrectly calculate \(\mathrm{m} = 22 ÷ 2 = 12\) instead of 11, or with \(77 = 7\mathrm{m}\), they might get confused and calculate 77 ÷ 7 incorrectly.
This may lead them to select Choice A (2) or cause other calculation-based errors.
The Bottom Line:
This problem tests whether students can work with linear functions in non-standard form. The key insight is recognizing that the given form \(\mathrm{f(x) = m(x - 28)}\) directly gives you everything needed - you just substitute and solve, rather than trying to convert to familiar slope-intercept form first.