In a school experiment, students timed each other as they went various distances using various gaits, or ways of moving...

OWNING THE DATASET

Let's start by understanding the dataset with a focus on patterns and relationships that will make our solving more efficient.

This table shows the speed (in m/s) of four different movement gaits at two different distances (5m and 10m):

Gait	5m Speed (m/s)	10m Speed (m/s)
Hopping	\(\mathrm{2.0}\)	\(\mathrm{3.1}\)
Walking backward	\(\mathrm{1.2}\)	\(\mathrm{1.6}\)
Speed walking	\(\mathrm{2.4}\)	\(\mathrm{2.6}\)
Walking forward	\(\mathrm{1.6}\)	\(\mathrm{1.4}\)

Key insight: The most important patterns to notice are:

1. Starting speeds at 5m (from fastest to slowest):
   • Speed walking (\(\mathrm{2.4}\) m/s)
   • Hopping (\(\mathrm{2.0}\) m/s)
   • Walking forward (\(\mathrm{1.6}\) m/s)
   • Walking backward (\(\mathrm{1.2}\) m/s)
2. Rate of change as distance increases from 5m to 10m:
   • Hopping: \(\mathrm{+1.1}\) m/s (dramatic increase)
   • Walking backward: \(\mathrm{+0.4}\) m/s (moderate increase)
   • Speed walking: \(\mathrm{+0.2}\) m/s (slight increase)
   • Walking forward: \(\mathrm{-0.2}\) m/s (decreases with distance)

These patterns will help us make predictions about speeds at distances not shown in the table (3m and 15m) without needing to calculate complete equations!

ANALYZING STATEMENT 1

Statement 1 Translation:

Original: "At a distance of 3m, hopping is faster than speed walking."

What we're looking for:

• The speed of hopping at 3m
• The speed of speed walking at 3m
• Whether hopping speed \(\mathrm{>}\) speed walking speed at 3m

In other words: Is hopping faster than speed walking when the distance is shorter (3m)?

Let's use pattern recognition instead of calculating exact equations:

1. At 5m, speed walking (\(\mathrm{2.4}\) m/s) is faster than hopping (\(\mathrm{2.0}\) m/s)
2. As distance decreases from 5m to 3m, we need to understand how each gait's speed changes:
   • Hopping has the steepest increase with distance (\(\mathrm{+1.1}\) m/s per 5m)
   • Speed walking increases less dramatically (\(\mathrm{+0.2}\) m/s per 5m)
3. This means when distance decreases:
   • Hopping speed will decrease more dramatically (steeper negative slope)
   • Speed walking speed will decrease less dramatically (gentler negative slope)
4. Since hopping already starts behind speed walking at 5m AND will lose speed more rapidly as we go back to 3m, speed walking will maintain its advantage and be even faster at 3m.

Statement 1 is No

Teaching note: Notice how we didn't need to calculate exact speeds! By understanding the relative rates of change, we could determine which gait would be faster at 3m. This pattern recognition is much faster than calculating linear equations.

ANALYZING STATEMENT 2

Statement 2 Translation:

Original: "At a distance of 15m, speed walking is faster than hopping."

What we're looking for:

• The speed of speed walking at 15m
• The speed of hopping at 15m
• Whether speed walking speed \(\mathrm{>}\) hopping speed at 15m

In other words: Does speed walking overtake hopping when we extend to a longer distance (15m)?

Again, let's use our pattern recognition:

1. At 10m, hopping (\(\mathrm{3.1}\) m/s) is already faster than speed walking (\(\mathrm{2.6}\) m/s)
2. As distance increases from 10m to 15m:
   • Hopping increases very rapidly (\(\mathrm{+1.1}\) m/s per 5m)
   • Speed walking increases slowly (\(\mathrm{+0.2}\) m/s per 5m)
3. This means when extending to 15m:
   • Hopping would gain approximately \(\mathrm{1.1}\) m/s more
   • Speed walking would gain approximately \(\mathrm{0.2}\) m/s more
4. The gap between them would widen even further in hopping's favor, not close.

Statement 2 is No

Teaching note: Here's where understanding the slopes (rates of change) is powerful. Since hopping's speed increases more than 5 times faster than speed walking's with distance, there's no way speed walking could overtake hopping at a greater distance once hopping is already ahead.

ANALYZING STATEMENT 3

Statement 3 Translation:

Original: "At a distance of 15m, walking backward is faster than walking forward at 3m."

What we're looking for:

• The speed of walking backward at 15m
• The speed of walking forward at 3m
• Whether walking backward at 15m \(\mathrm{>}\) walking forward at 3m

In other words: When comparing these two different gaits at two different distances, which is faster?

This comparison requires more careful analysis since we're comparing different gaits at different distances:

1. First, let's establish what we know:
   • Walking backward at 10m = \(\mathrm{1.6}\) m/s (and increases with distance at \(\mathrm{+0.4}\) m/s per 5m)
   • Walking forward at 5m = \(\mathrm{1.6}\) m/s (and decreases with distance at \(\mathrm{-0.2}\) m/s per 5m)
2. For walking backward at 15m:
   • Starting with \(\mathrm{1.6}\) m/s at 10m
   • Moving from 10m to 15m would add approximately \(\mathrm{+0.4}\) m/s
   • Therefore, walking backward at 15m would be \(\mathrm{> 1.6}\) m/s (approximately \(\mathrm{2.0}\) m/s)
3. For walking forward at 3m:
   • Starting with \(\mathrm{1.6}\) m/s at 5m
   • Moving from 5m to 3m would increase the speed (since this gait moves faster at shorter distances)
   • The increase would be approximately \(\mathrm{+0.2}\) m/s per 5m, so for 2m decrease, it would be less than \(\mathrm{+0.2}\) m/s
   • Therefore, walking forward at 3m would be \(\mathrm{< 1.8}\) m/s
4. Comparing the bounds:
   • Walking backward at 15m: \(\mathrm{> 1.6}\) m/s (approximately \(\mathrm{2.0}\) m/s)
   • Walking forward at 3m: \(\mathrm{< 1.8}\) m/s

Statement 3 is Yes

Teaching note: We used bounded thinking here rather than exact calculations. By establishing that walking backward at 15m must be greater than \(\mathrm{1.6}\) m/s and walking forward at 3m must be less than \(\mathrm{1.8}\) m/s, we can be confident in our answer even without calculating exact values.

FINAL ANSWER COMPILATION

Based on our analysis:

• Statement 1: No
• Statement 2: No
• Statement 3: Yes

The answer pattern is: No, No, Yes

LEARNING SUMMARY

Skills We Used:

1. Pattern Recognition: We identified the rate of change for each gait without calculating formal equations
2. Bounded Thinking: For Statement 3, we established value bounds rather than calculating exact speeds
3. Relative Reasoning: We focused on comparing values rather than calculating precise numbers

Strategic Insights:

1. Look for patterns first, calculate only when necessary
   • The growth rates (slopes) were visible directly from the table
   • Understanding which gaits increased/decreased with distance was more important than exact equations
2. Use bounds and limits rather than exact calculations
   • For Statement 3, knowing both values were on opposite sides of certain thresholds was sufficient
3. Consider the direction of extrapolation when using trends
   • Going from 5m to 3m reverses the advantage of steeper slopes
   • Going from 10m to 15m amplifies the advantage of steeper slopes

Common Mistakes We Avoided:

1. Calculating full linear equations for each gait (\(\mathrm{y = mx + b}\)) would have been time-consuming and unnecessary
2. Making exact predictions for all speeds when only relative comparisons were needed
3. Ignoring the pattern of changes which would have led to much more calculation

Remember that in GMAT table analysis, patterns and relationships often matter more than exact calculations. By first understanding how the data behaves, we can solve complex problems with minimal calculation!