Question:The table shows four values of x and their corresponding values of \(\mathrm{p(x)}\). There is a relationship between x and...
GMAT Algebra : (Alg) Questions
The table shows four values of \(\mathrm{x}\) and their corresponding values of \(\mathrm{p(x)}\). There is a relationship between \(\mathrm{x}\) and \(\mathrm{p(x)}\) that is defined by the equation \(\mathrm{p(x) = \frac{k}{x} + 7}\), where \(\mathrm{k}\) is a constant. What is the value of \(\mathrm{k}\)?
| \(\mathrm{x}\) | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| \(\mathrm{p(x)}\) | 27 | 22 | 19 | 17 |
- 20
- 27
- 60
- 81
- 102
1. TRANSLATE the problem information
- Given information:
- Table of x and p(x) values: \(\(3,27\)\), \(\(4,22\)\), \(\(5,19\)\), \(\(6,17\)\)
- Equation form: \(\mathrm{p(x) = \frac{k}{x} + 7}\)
- Need to find constant k
- What this tells us: Any coordinate pair from the table must satisfy the equation
2. INFER the solution strategy
- Since any point on the table satisfies the equation, we can substitute any coordinate pair
- Choose the simplest point to work with - let's use \(\(3, 27\)\)
- Substitute \(\mathrm{x = 3}\) and \(\mathrm{p(x) = 27}\) into \(\mathrm{p(x) = \frac{k}{x} + 7}\)
3. SIMPLIFY the algebraic equation
- Start with: \(\mathrm{27 = \frac{k}{3} + 7}\)
- Subtract 7 from both sides: \(\mathrm{27 - 7 = \frac{k}{3}}\)
- Simplify: \(\mathrm{20 = \frac{k}{3}}\)
- Multiply both sides by 3: \(\mathrm{k = 60}\)
4. Verify the solution
- Test with another point, say \(\(4, 22\)\):
- \(\mathrm{p(4) = \frac{60}{4} + 7 = 15 + 7 = 22}\) ✓
Answer: C) 60
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make arithmetic mistakes when isolating k, particularly when working with fractions or negative numbers.
For example, they might incorrectly solve \(\mathrm{27 = \frac{k}{3} + 7}\) by writing \(\mathrm{27 + 7 = \frac{k}{3}}\) instead of \(\mathrm{27 - 7 = \frac{k}{3}}\), leading to \(\mathrm{34 = \frac{k}{3}}\) and \(\mathrm{k = 102}\). This may lead them to select Choice E (102).
Second Most Common Error:
Poor TRANSLATE reasoning: Students might not recognize that they can use any point from the table, or they might substitute the values incorrectly into the equation.
Some students might switch x and p(x) values, substituting \(\mathrm{3 = \frac{k}{27} + 7}\) instead of \(\mathrm{27 = \frac{k}{3} + 7}\). This leads to completely different arithmetic and confusion about which answer choice to select.
The Bottom Line:
This problem tests whether students can systematically substitute known values into an equation and solve for an unknown constant - a fundamental skill in working with functions that have parameters.