Let's summarize the rules first:
- If a passenger checks luggage, it must be checked by 3:00 PM.
- If a passenger reaches the gate after 3:30 PM, they must have checked in online before 3:30 PM.
- No one can board after 3:50 PM.
We are told Mateo and Zahra reached the gate at different times.
We need to determine whether Mateo reached the gate later than Zahra.
Statement (1):
Zahra was allowed to board but Mateo was not.
Mateo could have been denied boarding for many reasons:
- Arrived after 3:50 PM.
- Arrived after 3:30 PM without checking in online.
- Failed to check luggage by 3:00 PM.
We cannot determine who reached the gate later.
Not sufficient.
Statement (2):
Only one traveler checked luggage, and they did so at 2:45 PM.
Only one traveler checked in online before 3:30 PM.
Checking luggage at 2:45 PM satisfies the luggage rule.
However, we still do not know who checked luggage who checked in online, or when either person reached the gate.
Not sufficient.
Statements (1) and (2) together:
From (1), Zahra boarded and Mateo did not.
From (2):
- One traveler checked luggage (properly at 2:45 PM).
- One traveler checked in online before 3:30 PM.
Since Zahra boarded, she satisfied all applicable rules.
Mateo did not board but we still don't know why.
For example:
Case 1:
- Zahra reaches the gate at 3:20 PM.
- Mateo reaches the gate at 3:40 PM without online check-in.
-----> Mateo arrived later.
Case 2:
- Mateo reaches the gate at 3:20 PM but failed to check luggage by 3:00 PM.
- Zahra reaches the gate at 3:40 PM after checking in online.
----> Mateo arrived earlier.
Both satisfy the statements yet give different answers.
Therefore, even together the statements are not sufficient.
Answer: E