A shipping company charges different rates based on package weight. Packages weighing less than 8 pounds are charged the 'light'...
GMAT Algebra : (Alg) Questions
A shipping company charges different rates based on package weight. Packages weighing less than 8 pounds are charged the 'light' rate, and packages weighing more than 18 pounds are charged the 'heavy' rate. Packages weighing exactly 8 pounds through exactly 18 pounds are charged the 'standard' rate. Which inequality represents the weight \(\mathrm{w}\), in pounds, of a package that is charged the standard rate?
1. TRANSLATE the rate conditions into mathematical language
- Given information:
- Light rate: 'less than 8 pounds' → \(\mathrm{w \lt 8}\)
- Heavy rate: 'more than 18 pounds' → \(\mathrm{w \gt 18}\)
- Standard rate: 'exactly 8 pounds through exactly 18 pounds'
- The key phrase here is 'exactly...through exactly...' which tells us both endpoints are included.
2. INFER what the standard rate condition means mathematically
- 'Exactly 8 pounds through exactly 18 pounds' means:
- The package CAN weigh exactly 8 pounds (so we need ≥, not >)
- The package CAN weigh exactly 18 pounds (so we need ≤, not <)
- The package must satisfy BOTH conditions simultaneously
3. TRANSLATE the standard rate into compound inequality
- Lower boundary: \(\mathrm{w \geq 8}\) (includes exactly 8 pounds)
- Upper boundary: \(\mathrm{w \leq 18}\) (includes exactly 18 pounds)
- Combined: \(\mathrm{8 \leq w \leq 18}\)
Answer: C (\(\mathrm{8 \leq w \leq 18}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret 'exactly...through exactly...' as excluding the endpoints, thinking it means 'between but not including.'
This leads them to write \(\mathrm{8 \lt w \lt 18}\), but since this isn't an answer choice, they get confused and may guess or incorrectly select Choice D (\(\mathrm{w \geq 8}\)) thinking it's 'close enough.'
Second Most Common Error:
Incomplete TRANSLATE reasoning: Students correctly identify that \(\mathrm{w \geq 8}\) but forget that standard rate also has an upper limit.
They focus only on 'at least 8 pounds' and select Choice D (\(\mathrm{w \geq 8}\)), missing that packages over 18 pounds get the heavy rate, not standard rate.
The Bottom Line:
This problem tests whether students can accurately translate inclusive language ('exactly...through exactly...') into proper mathematical notation and remember to capture both boundary conditions in a compound inequality.