A rectangle has length ell and width w. The inequality 2ell + 2w leq 27 gives the possible values of...
GMAT Algebra : (Alg) Questions
A rectangle has length \(\ell\) and width \(\mathrm{w}\). The inequality \(2\ell + 2\mathrm{w} \leq 27\) gives the possible values of \(\ell\) and \(\mathrm{w}\) for which the perimeter of this rectangle is less than or equal to 27. Which statement is the best interpretation of \((\ell, \mathrm{w}) = (8, 3)\) in this context?
If the rectangle has length 3 and width 8, its perimeter is \(\leq 27\).
If the rectangle has length 8 and width 3, its perimeter is \(\leq 27\).
If the rectangle has length 3 and width 8, its perimeter is \(\geq 27\).
If the rectangle has length 8 and width 3, its perimeter is \(\geq 27\).
1. TRANSLATE the ordered pair notation
- Given: \((ℓ, w) = (8, 3)\)
- This means: \(ℓ = 8\) and \(w = 3\)
- Since \(ℓ\) represents length and \(w\) represents width:
- Length = 8
- Width = 3
2. INFER what we need to check
- We have an inequality: \(2ℓ + 2w \leq 27\)
- This inequality tells us when the perimeter is \(\leq 27\)
- To interpret \((ℓ, w) = (8, 3)\), we need to test if these values satisfy the inequality
3. SIMPLIFY by substitution
- Substitute \(ℓ = 8\) and \(w = 3\) into \(2ℓ + 2w \leq 27\):
\(2(8) + 2(3) \leq 27\)
\(16 + 6 \leq 27\)
\(22 \leq 27\)
4. INFER the conclusion
- Since \(22 \leq 27\) is TRUE, the inequality is satisfied
- This means: If the rectangle has length 8 and width 3, its perimeter IS less than or equal to 27
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students mix up the order in the ordered pair notation.
They might think \((ℓ, w) = (8, 3)\) means length = 3 and width = 8, essentially reading the ordered pair backwards. This confusion about which position represents which variable is common when students haven't fully internalized ordered pair conventions.
This may lead them to select Choice A (If the rectangle has length 3 and width 8, its perimeter is less than or equal to 27).
Second Most Common Error:
Poor INFER reasoning about inequality direction: Students correctly substitute but then misinterpret what a true inequality means.
After getting \(22 \leq 27\) (which is true), they might think this means the perimeter is "greater than or equal to 27" instead of understanding that satisfying the inequality means the perimeter IS less than or equal to 27.
This may lead them to select Choice D (If the rectangle has length 8 and width 3, its perimeter is greater than or equal to 27).
The Bottom Line:
This problem tests whether students can correctly interpret mathematical notation (ordered pairs) and logical relationships (what it means when an inequality is satisfied). The key insight is that when specific values make an inequality true, those values represent a case where the stated condition holds.
If the rectangle has length 3 and width 8, its perimeter is \(\leq 27\).
If the rectangle has length 8 and width 3, its perimeter is \(\leq 27\).
If the rectangle has length 3 and width 8, its perimeter is \(\geq 27\).
If the rectangle has length 8 and width 3, its perimeter is \(\geq 27\).