For each of 5 airlines (Airlines 1 through 5), the table shows the percent of flights offered by that airline...

OWNING THE DATASET

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:

1. Ranking Structure: The airlines are ranked by total flights (\(\mathrm{T_1} > \mathrm{T_2} > \mathrm{T_3} > \mathrm{T_4} > \mathrm{T_5}}\)), meaning Airline 1 has the most total flights and Airline 5 has the least.
2. Percentage Representation: All values are percentages of each airline's own flights, not absolute numbers. This means we need to consider both the percentage AND the relative size of each airline when comparing absolute numbers.

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!

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "Airline 5 had the least flights delayed by more than 60 minutes."
What we're looking for:

• Compare absolute number (not percentage) of flights delayed >60 minutes across all airlines
• Determine if Airline 5 has the smallest absolute number in this category

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:

• Airline 5 has the lowest percentage (1.2%) in this category
• Airline 5 also has the lowest total flights (\(\mathrm{T_5}\)) overall

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:

• For any airline: \(\text{(% of flights delayed >60 min)} \times \text{(Total flights)} = \text{Absolute number of flights delayed >60 min}\)
• When both factors on the left are smallest for Airline 5, their product must be smallest

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.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "Airline 3 did NOT have the least total delayed flights."
What we're looking for:

• Compare the absolute number of total delayed flights across all airlines
• Determine if some airline other than Airline 3 has the least total delays

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:

• Airline 4 has the lowest percentage of total delays (20.3%)
• Airline 3 has 23.0% total delays

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:

• The difference between 23.0% and 20.3% is about 2.7 percentage points
• This represents roughly a 13% advantage for Airline 4 \((2.7\% \div 20.3\% \approx 13\%)\)

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.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "Airline 2 had the greatest number of flights delayed by 1 to 15 minutes."
What we're looking for:

• Compare absolute number of flights delayed 1-15 minutes across all airlines
• Determine if Airline 2 has the largest absolute number in this category

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:

• Airline 2 has the highest percentage (9.2%)
• Airline 1 has the second highest percentage (8.5%)

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.

FINAL ANSWER COMPILATION

Let's summarize our findings for each statement:

• Statement 1: Need not be true - We cannot conclusively determine that Airline 2 had the greatest number of flights delayed by 1-15 minutes.
• Statement 2: Must be true - Airline 5 definitely had the least flights delayed by more than 60 minutes.
• Statement 3: Must be true - Airline 3 definitely did NOT have the least total delayed flights.

Therefore, our answer is: B and C are true.

LEARNING SUMMARY

Skills We Used

1. Pattern Recognition: Identifying "double extreme" situations where both percentage and total flights align to give a definitive answer.
2. Visual Estimation: Using rough percentage differences rather than exact calculations for comparative questions.
3. Strategic Statement Ordering: Tackling the clearest statement first to build momentum.

Strategic Insights

1. Sort Immediately: Sorting reveals patterns that eliminate the need for calculations. This was especially powerful for Statement 2.
2. Double Extremes = Instant Answers: When both factors align (lowest % AND lowest total), the conclusion is immediate.
3. Small Percentage Advantages Can Be Offset: As we saw in Statement 1, a small percentage advantage (8%) can easily be offset by a difference in total volume.

Common Mistakes We Avoided

1. Unnecessary Calculations: We didn't need to calculate the exact number of flights for each airline.
2. Ignoring Pre-existing Order: We leveraged the given ranking of airlines by total flights.
3. Fixed Statement Order: By tackling Statement 2 first, we built confidence and gained insights that helped with other statements.

For Future Table Analysis Questions

Always look for:

• Pre-existing ordering in the dataset
• "Double extreme" cases for quick answers
• Opportunities to use visual estimation rather than exact calculation
• Ways to attack statements in order of clarity/certainty, not the given sequence

This approach transforms table analysis from methodical calculation to pattern recognition, making your solving process both more efficient and more accurate!