A table of values for a linear function g is shown below: t 5 10 15 20 \(\mathrm{g(t)}\) 45 75...
GMAT Algebra : (Alg) Questions
A table of values for a linear function \(\mathrm{g}\) is shown below:
| \(\mathrm{t}\) | 5 | 10 | 15 | 20 |
|---|---|---|---|---|
| \(\mathrm{g(t)}\) | 45 | 75 | 105 | 135 |
The function is defined by the equation \(\mathrm{g(t) = 6t + k}\), where \(\mathrm{k}\) is a constant. What is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem information
- Given information:
- Table showing t and g(t) values: \(\mathrm{(5,45)}\), \(\mathrm{(10,75)}\), \(\mathrm{(15,105)}\), \(\mathrm{(20,135)}\)
- Equation form: \(\mathrm{g(t) = 6t + k}\)
- Need to find: constant k
2. INFER the solution strategy
- Key insight: Since we know the equation form and have specific input-output pairs, we can substitute any table point into the equation to solve for k
- Strategy: Pick any point from the table and substitute into \(\mathrm{g(t) = 6t + k}\)
3. SIMPLIFY by substitution and solving
- Using the first point \(\mathrm{(5, 45)}\):
- Substitute: \(45 = 6(5) + \mathrm{k}\)
- Simplify: \(45 = 30 + \mathrm{k}\)
- Solve: \(\mathrm{k} = 45 - 30 = 15\)
4. INFER the need to verify (good practice)
- Using second point \(\mathrm{(10, 75)}\) to check:
- \(75 = 6(10) + \mathrm{k} = 60 + \mathrm{k}\)
- \(\mathrm{k} = 75 - 60 = 15\) ✓
Answer: C) 15
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize they can directly substitute table values into the given equation. Instead, they might try to calculate the slope from the table (getting \(30/5 = 6\)) and then get confused about what to do next, not realizing the slope is already given as 6 in the equation.
This leads to confusion and potential guessing among the answer choices.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(45 = 6(5) + \mathrm{k}\) but make arithmetic errors, such as:
- Calculating \(6(5) = 25\) instead of 30, leading to \(\mathrm{k} = 20\)
- Solving \(45 = 30 + \mathrm{k}\) as \(\mathrm{k} = 45 + 30 = 75\) instead of \(\mathrm{k} = 45 - 30 = 15\)
This may lead them to select Choice E (75) or create confusion that results in guessing.
The Bottom Line:
This problem tests whether students understand that having an equation form with one unknown and specific coordinate pairs gives them everything needed to solve directly through substitution. The key breakthrough is recognizing this substitution strategy rather than trying to derive the equation from scratch.