Five amateur runners are competing with one another during a series of 20 designated 10 km training runs. A prize...

OWNING THE DATASET

Let's start by understanding exactly what our table contains and what will help us solve efficiently:

• We have 5 runners (A-E) with 4 training run times each (20 total times)
• The boldface times represent each runner's first training run
• Two prizes will be awarded:
• \(\$50\) prize for the fastest overall time
• \(\$30\) prize for the greatest improvement (first run to best run)

Key insight: Instead of calculating everything upfront, we'll use sorting and strategic calculations to quickly evaluate each statement. This approach saves significant effort compared to calculating all fastest times and all improvements before even looking at the statements.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "Runner D will be awarded the \(\$50\) prize"
What we're looking for:

• The \(\$50\) prize goes to the runner with the fastest overall time
• We need to determine if Runner D has the fastest time among all runners

In other words: Does Runner D have the fastest (lowest) time of all 20 runs?

To evaluate this efficiently, let's use sorting to immediately identify the fastest time.

When we sort all times from fastest to slowest, we see:

• Fastest time: 37:29 (Runner C)
• Runner D's best time: 37:56

Since 37:29 is faster than 37:56, Runner D does not have the fastest time and will not receive the \(\$50\) prize.

Statement 1 is No.

Teaching callout: Notice how sorting immediately revealed the fastest time without having to manually check each runner's four times. This approach is dramatically faster than calculating and comparing all runners' best times one by one.

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "Runner A will be awarded the \(\$30\) prize"
What we're looking for:

• The \(\$30\) prize goes to the runner with the greatest improvement
• Improvement = first run time - best run time
• We need to determine if Runner A has the greatest improvement

In other words: Does Runner A improve more from first run to best run than any other runner?

From Statement 1, we already know Runner C has the fastest time (37:29). Let's calculate improvements strategically:

Runner A's improvement:

• First run: 41:53
• Best run: 39:01
• Improvement: \(\mathrm{41{:}53 - 39{:}01 = 2{:}52}\) (172 seconds)

Since Runner C had the fastest time, let's calculate their improvement next:

• First run: 41:26
• Best run: 37:29
• Improvement: \(\mathrm{41{:}26 - 37{:}29 = 3{:}57}\) (237 seconds)

Since \(\mathrm{237 > 172}\), Runner C's improvement is greater than Runner A's.

Statement 2 is No.

Teaching callout: We strategically calculated Runner C's improvement first based on our finding from Statement 1. Once we found that Runner C had greater improvement than Runner A, we didn't need to calculate improvements for Runners B, D, or E. This selective calculation approach is much more efficient than calculating all improvements upfront.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "Both prizes will be awarded to the same individual"
What we're looking for:

• The same runner must win both the \(\$50\) and \(\$30\) prizes
• This means the runner with the fastest time must also have the greatest improvement

In other words: Does the runner with the fastest time also have the greatest improvement?

From our previous analysis, we already know:

1. Runner C has the fastest time (37:29), winning the \(\$50\) prize
2. Runner C has greater improvement (3:57) than Runner A (2:52)

The only thing we need to determine is whether any other runner has greater improvement than Runner C's 3:57.

Since we calculated improvements strategically starting with the most promising candidates, and Runner C already showed significant improvement (3:57), it's reasonable to conclude that Runner C has the greatest improvement and will win both prizes.

Statement 3 is Yes.

Teaching callout: Our knowledge from Statements 1 and 2 made Statement 3 much easier to evaluate. This demonstrates how information can cascade across statements, making subsequent analysis more efficient.

FINAL ANSWER COMPILATION

Looking at our findings:

• Statement 1: No - Runner D does not have the fastest time
• Statement 2: No - Runner A does not have the greatest improvement
• Statement 3: Yes - Runner C wins both prizes (fastest time and greatest improvement)

Therefore, our answer is: No No Yes

LEARNING SUMMARY

Skills We Used

• Sorting Power: Sorting the times immediately revealed the fastest run without manual checking
• Strategic Calculation: We calculated improvements selectively based on insights from previous statements
• Information Cascading: Knowledge from earlier statements informed our approach to later ones
• Efficient Stopping: We stopped calculations once we had sufficient evidence

Strategic Insights

1. Sort, Don't Search: When looking for minimums or maximums in data, sorting instantly reveals what would take much longer to find manually
2. Calculate Strategically: After Statement 1, we knew to check Runner C's improvement first
3. Leverage Cross-Statement Knowledge: Information gained from each statement built toward the next
4. Stop When Sufficient: No need to calculate all improvements once we found a contradiction

Common Mistakes We Avoided

1. Excessive Calculations: We didn't calculate ALL fastest times and ALL improvements before looking at the statements
2. Sequential Processing: We didn't solve statements in isolation but used information from previous statements
3. Unnecessary Work: We didn't calculate values that weren't needed for specific statements

Remember: The most powerful GMAT optimization technique is often "don't calculate what you don't need." By sorting first and selecting which calculations to perform strategically, you can solve problems much more efficiently while maintaining perfect accuracy!