A model estimates that whales from the genus Eschrichtius travel 72 to 77 miles in the ocean each day during...
GMAT Algebra : (Alg) Questions
A model estimates that whales from the genus Eschrichtius travel \(72\) to \(77\) miles in the ocean each day during their migration. Based on this model, which inequality represents the estimated total number of miles, \(\mathrm{x}\), a whale from the genus Eschrichtius could travel in \(16\) days of its migration?
1. TRANSLATE the problem information
- Given information:
- Daily distance range: 72 to 77 miles per day
- Time period: 16 days
- Need to find: total distance x over 16 days
- What this tells us: We have a range of daily values that need to be converted to a total value over multiple days
2. INFER the mathematical relationship
- To find total distance, we multiply daily distance by number of days
- Since we have a range of daily distances (72 to 77), we need to find the range of total distances
- Key insight: When you multiply every value in a range by the same number, you get a new range
3. Apply the multiplication to both bounds
- Minimum total distance: \(72 \times 16 = (72)(16)\) miles
- Maximum total distance: \(77 \times 16 = (77)(16)\) miles
4. Write the inequality
- The total distance x falls between these bounds: \((72)(16) \leq x \leq (77)(16)\)
Answer: B. \((72)(16) \leq x \leq (77)(16)\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what operation connects daily distance to total distance. Instead of recognizing that \(\mathrm{total} = \mathrm{daily} \times \mathrm{time}\), they think about adding days to daily distance.
This leads them to set up: \(72 + 16 \leq x \leq 77 + 16\)
This may lead them to select Choice A \((72 + 16 \leq x \leq 77 + 16)\)
Second Most Common Error:
Poor INFER reasoning: Students correctly identify that they need to use 16 somehow, but they misunderstand what x represents. They think x is the daily distance rather than the total distance, leading to backwards setup.
This causes them to write: \(72 \leq 16x \leq 77\) (treating x as daily distance)
This may lead them to select Choice D \((72 \leq 16x \leq 77)\)
The Bottom Line:
This problem tests whether students can correctly translate a real-world rate situation into mathematical language and recognize that operations on ranges require applying the operation to all bounds of the range.