Question:A rectangular garden has area A = 24x^3y - 18x^2y^2 square meters, where x and y are positive real numbers...
GMAT Advanced Math : (Adv_Math) Questions
A rectangular garden has area \(\mathrm{A = 24x^3y - 18x^2y^2}\) square meters, where x and y are positive real numbers representing lengths. The length of the garden is \(\mathrm{6x^2y}\) meters. Which expression gives the width, in meters?
- \(\mathrm{4x - 3y}\)
- \(\mathrm{4x + 3y}\)
- \(\mathrm{6x - 3y}\)
- \(\mathrm{4xy - 3}\)
- \(\mathrm{4x - 3y^2}\)
1. TRANSLATE the problem information
- Given information:
- Area = \(24\mathrm{x}^3\mathrm{y} - 18\mathrm{x}^2\mathrm{y}^2\) square meters
- Length = \(6\mathrm{x}^2\mathrm{y}\) meters
- Find: Width in meters
- What this tells us: We need to use the rectangle area relationship to find the missing dimension.
2. INFER the approach
- Since \(\mathrm{Area} = \mathrm{Length} \times \mathrm{Width}\) for rectangles, we can solve for width by rearranging: \(\mathrm{Width} = \mathrm{Area} \div \mathrm{Length}\)
- This means we need to divide the area expression by the length expression
3. SIMPLIFY through polynomial division
- Set up the division: \(\mathrm{Width} = (24\mathrm{x}^3\mathrm{y} - 18\mathrm{x}^2\mathrm{y}^2) \div (6\mathrm{x}^2\mathrm{y})\)
- Divide each term in the area by the length:
- First term: \(24\mathrm{x}^3\mathrm{y} \div 6\mathrm{x}^2\mathrm{y} = 4\mathrm{x}\)
- Second term: \(18\mathrm{x}^2\mathrm{y}^2 \div 6\mathrm{x}^2\mathrm{y} = 3\mathrm{y}\)
- Combine: \(\mathrm{Width} = 4\mathrm{x} - 3\mathrm{y}\)
Answer: (A) \(4\mathrm{x} - 3\mathrm{y}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make mistakes with exponent rules during division. They might incorrectly think \(\mathrm{x}^3 \div \mathrm{x}^2 = \mathrm{x}\) or forget that \(\mathrm{y}^2 \div \mathrm{y} = \mathrm{y}\), leading to answers like \(4\mathrm{xy} - 3\) or other incorrect expressions.
This may lead them to select Choice (D) \((4\mathrm{xy} - 3)\) or cause confusion that results in guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students might not recognize that they need to divide area by length to find width. Instead, they might try to factor the area expression directly or attempt other algebraic manipulations without understanding the geometric relationship.
This leads to confusion about how to proceed and often results in random answer selection.
The Bottom Line:
This problem tests whether students can connect geometric formulas with algebraic operations. The key insight is recognizing that finding a missing dimension requires division, not just algebraic manipulation of the given expressions.