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For each of 5 airlines (Airlines 1 through 5), the table shows the percent of flights offered by that airline last year that were delayed by certain ranges of time to the nearest minute and the total percent of flights offered by that airline last year that were delayed. The airlines are numbered from greatest total number of flights offered last year (Airline 1) to least total number of flights offered last year (Airline 5).
| Airline | 1 to 15 | 16 to 30 | 31 to 45 | 46 to 60 | More than 60 | Total delays |
|---|---|---|---|---|---|---|
| 4 | 6.3% | 4.8% | 5% | 1.7% | 2.5% | 20.3% |
| 3 | 7.5% | 7.1% | 4.5% | 2.2% | 1.7% | 23% |
| 1 | 8.5% | 6.5% | 5.8% | 2.1% | 1.6% | 24.5% |
| 5 | 8.8% | 5.9% | 7.1% | 1.9% | 1.2% | 24.9% |
| 2 | 9.2% | 6.9% | 4.9% | 2.4% | 2.8% | 26.2% |
For each statement, determine whether it must be true or need not be true based on the information provided.
Airline 2 had the greatest number of flights last year that were delayed by 1 to 15 minutes, to the nearest minute.
Airline 5 had the least number of flights last year that were delayed by more than 60 minutes, to the nearest minute.
Airline 3 did NOT have the least number of total delayed flights last year.
Let's start by understanding what we're working with. The table shows flight delay data for 5 airlines, broken down by delay duration categories expressed as percentages of each airline's total flights.
Two key insights immediately jump out:
For example, if Airline 1 has 8.5% of flights delayed by 1-15 minutes and Airline 2 has 9.2%, we can't immediately conclude which has more flights in this category because their total flight counts differ.
These insights will be crucial for efficiently analyzing the statements without unnecessary calculations!
Statement 2 Translation:
Original: "Airline 5 had the least flights delayed by more than 60 minutes."
What we're looking for:
In other words: Does Airline 5 have fewer total flights in the >60 minute delay category than any other airline?
Let's approach this statement first because it offers the clearest path to a quick answer.
The Efficient Approach:
First, let's sort the airlines by the "More than 60 min" column. This immediately shows us:
This creates a "double extreme" situation - Airline 5 has both the lowest percentage AND the lowest base number of total flights. When both factors are at their minimum, the absolute number must also be at its minimum.
Mathematically, this makes sense:
No calculations needed! When the extremes align like this, we can immediately determine that Airline 5 must have the least absolute number of flights delayed by more than 60 minutes.
Statement 2 Must be true.
Statement 3 Translation:
Original: "Airline 3 did NOT have the least total delayed flights."
What we're looking for:
In other words: Is there at least one airline with fewer total delayed flights than Airline 3?
Now let's look at the "Total delays" column to analyze this statement efficiently.
The Efficient Approach:
Sorting by the "Total delays" column reveals:
The key question: Could Airline 4 have fewer absolute delays than Airline 3?
We know that \(\mathrm{T_3} > \mathrm{T_4}\) (Airline 3 has more total flights than Airline 4). But is this difference in total flights enough to offset the percentage difference?
Let's do a quick visual estimation:
For Airline 3 to have fewer absolute delays than Airline 4, Airline 3's total flights would need to be at least 13% lower than Airline 4's. But we know the opposite is true - Airline 3 has MORE total flights than Airline 4.
Therefore, Airline 4 must have fewer total delayed flights than Airline 3, meaning Airline 3 cannot have the least total delayed flights.
Statement 3 Must be true.
Statement 1 Translation:
Original: "Airline 2 had the greatest number of flights delayed by 1 to 15 minutes."
What we're looking for:
In other words: Does Airline 2 have more flights in the 1-15 minute delay category than any other airline?
The Efficient Approach:
Let's sort by the "1 to 15 min" column:
The difference between these percentages is small - only 0.7 percentage points, or about an 8% advantage for Airline 2 \((0.7\% \div 8.5\% \approx 8\%)\).
Here's the crucial insight: We know that \(\mathrm{T_1} > \mathrm{T_2}\) (Airline 1 has more total flights than Airline 2). If Airline 1's total flights are more than 8% higher than Airline 2's, then Airline 1 could actually have more absolute flights in the 1-15 minute delay category despite having a lower percentage.
Since the airlines are ranked by total flights, it's quite reasonable that Airline 1 has at least 8% more flights than Airline 2. In fact, given industry patterns, the difference in size between the largest airline and the second largest is often substantial.
Therefore, we cannot conclusively determine that Airline 2 had the greatest number of flights delayed by 1-15 minutes.
Statement 1 Need not be true.
Let's summarize our findings for each statement:
Therefore, our answer is: B and C are true.
Always look for:
This approach transforms table analysis from methodical calculation to pattern recognition, making your solving process both more efficient and more accurate!
Airline 2 had the greatest number of flights last year that were delayed by 1 to 15 minutes, to the nearest minute.
Airline 5 had the least number of flights last year that were delayed by more than 60 minutes, to the nearest minute.
Airline 3 did NOT have the least number of total delayed flights last year.