A function A gives the area, in square units, of a rectangle whose width is x units and whose length...
GMAT Advanced Math : (Adv_Math) Questions
A function \(\mathrm{A}\) gives the area, in square units, of a rectangle whose width is \(\mathrm{x}\) units and whose length is \(\mathrm{7}\) less than three times the width. Which equation defines \(\mathrm{A}\)?
1. TRANSLATE the problem information
- Given information:
- Width of rectangle = \(\mathrm{x}\) units
- Length = "7 less than three times the width"
- TRANSLATE the length description:
- Three times the width = \(\mathrm{3x}\)
- 7 less than three times the width = \(\mathrm{3x - 7}\)
2. INFER the solution approach
- We need to find a function \(\mathrm{A(x)}\) for the area
- Since this is a rectangle: Area = width × length
- Therefore: \(\mathrm{A(x) = x \times (3x - 7)}\)
3. SIMPLIFY the expression
- \(\mathrm{A(x) = x(3x - 7)}\)
- Using distributive property: \(\mathrm{A(x) = 3x^2 - 7x}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Misinterpreting "7 less than three times the width" as \(\mathrm{(x - 7)}\) instead of \(\mathrm{(3x - 7)}\). Students often read this phrase as "x minus 7" rather than carefully parsing that 7 is being subtracted from the entire expression "three times the width."
This leads to \(\mathrm{A(x) = x(x - 7) = x^2 - 7x}\), but this doesn't match any answer choice exactly, causing confusion and potentially guessing.
Second Most Common Error:
Poor INFER reasoning: Not recognizing that area requires multiplication of dimensions. Some students try to combine the width and length by adding: \(\mathrm{A(x) = x + (3x - 7) = 4x - 7}\), or they partially multiply: \(\mathrm{A(x) = x + 3x - 7 = x^2 + 3x - 7}\).
This may lead them to select Choice A (\(\mathrm{A(x) = x^2 + 3x - 7}\)).
Third Most Common Error:
Inadequate SIMPLIFY execution: Correctly setting up \(\mathrm{A(x) = x(3x - 7)}\) but making distribution errors, such as \(\mathrm{A(x) = 3x^2 - 7}\) (forgetting to multiply the -7 by x).
This may lead them to select Choice B (\(\mathrm{A(x) = 3x^2 - 7}\)).
The Bottom Line:
This problem tests your ability to carefully translate verbal descriptions into algebra and systematically apply the rectangle area formula. The key challenge is parsing the phrase "7 less than three times the width" accurately.