Rectangle A has length 3x + 4 and width x - 2, where x is a real number such that...
GMAT Advanced Math : (Adv_Math) Questions
Rectangle A has length \(3\mathrm{x} + 4\) and width \(\mathrm{x} - 2\), where x is a real number such that both dimensions are positive. Rectangle B has length \(2\mathrm{x} + 5\) and width \(\mathrm{x} + 1\). Which of the following expressions is equivalent to \(\mathrm{P_A} - \mathrm{P_B}\), where \(\mathrm{P_A}\) and \(\mathrm{P_B}\) are the perimeters of rectangles A and B, respectively?
1. TRANSLATE the problem information
- Given information:
- Rectangle A: length = \(\mathrm{3x + 4}\), width = \(\mathrm{x - 2}\)
- Rectangle B: length = \(\mathrm{2x + 5}\), width = \(\mathrm{x + 1}\)
- Need to find: \(\mathrm{P_A - P_B}\) (difference in perimeters)
2. INFER the approach needed
- Since we need perimeters, we'll use the rectangle perimeter formula
- We'll calculate each perimeter separately, then subtract
- Key insight: We must be extra careful with signs during subtraction
3. SIMPLIFY to find \(\mathrm{P_A}\) (Rectangle A's perimeter)
- \(\mathrm{P_A = 2(length + width) = 2[(3x + 4) + (x - 2)]}\)
- Combine terms inside brackets: \(\mathrm{P_A = 2[3x + 4 + x - 2] = 2[4x + 2]}\)
- Distribute: \(\mathrm{P_A = 8x + 4}\)
4. SIMPLIFY to find \(\mathrm{P_B}\) (Rectangle B's perimeter)
- \(\mathrm{P_B = 2(length + width) = 2[(2x + 5) + (x + 1)]}\)
- Combine terms inside brackets: \(\mathrm{P_B = 2[2x + 5 + x + 1] = 2[3x + 6]}\)
- Distribute: \(\mathrm{P_B = 6x + 12}\)
5. SIMPLIFY the final subtraction \(\mathrm{P_A - P_B}\)
- \(\mathrm{P_A - P_B = (8x + 4) - (6x + 12)}\)
- Distribute the negative sign carefully: \(\mathrm{= 8x + 4 - 6x - 12}\)
- Combine like terms: \(\mathrm{= (8x - 6x) + (4 - 12) = 2x - 8}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Poor SIMPLIFY execution: Sign error during the final subtraction step
Students correctly find \(\mathrm{P_A = 8x + 4}\) and \(\mathrm{P_B = 6x + 12}\), but when calculating \(\mathrm{P_A - P_B = (8x + 4) - (6x + 12)}\), they incorrectly handle the negative sign distribution. They might write: \(\mathrm{8x + 4 - 6x + 12 = 2x + 16}\), forgetting that subtracting \(\mathrm{(6x + 12)}\) means subtracting both terms.
This leads them to select Choice D \(\mathrm{(2x + 16)}\)
Second Most Common Error:
Weak TRANSLATE reasoning: Incorrectly applying the perimeter formula
Some students might add the dimensions incorrectly or forget to multiply by 2 in the perimeter formula. They might calculate something like \(\mathrm{P_A = (3x + 4) + (x - 2) = 4x + 2}\) without the factor of 2, leading to incorrect perimeter values and ultimately an answer that doesn't match any choice.
This leads to confusion and guessing among the available options.
The Bottom Line:
This problem tests careful algebraic manipulation more than conceptual understanding. The perimeter formula is straightforward, but students must methodically handle each step of the algebra, especially the critical subtraction step where sign errors are most likely to occur.