The difference between Larry's and John's heights is twice the difference between Larry's and Ken's heights. If Larry is the...

Understanding the Question

We need to find the average height of three men: Larry, John, and Ken.

Given Information

• Larry is the tallest of the three men
• The difference between Larry's and John's heights equals twice the difference between Larry's and Ken's heights

What We Need to Determine

Let's translate this mathematically to see what's really happening:

• If we call the heights L, J, and K respectively
• We're told: \(\(\mathrm{L} - \mathrm{J}\) = 2\(\mathrm{L} - \mathrm{K}\)\)
• We need: \(\(\mathrm{L} + \mathrm{J} + \mathrm{K}\) ÷ 3\)

Key Insight

Here's where it gets interesting. Let's simplify that equation:

• \(\mathrm{L} - \mathrm{J} = 2\mathrm{L} - 2\mathrm{K}\)
• \(-\mathrm{J} = \mathrm{L} - 2\mathrm{K}\)
• \(\mathrm{J} = 2\mathrm{K} - \mathrm{L}\)

This tells us something remarkable: Ken's height is exactly at the midpoint between John's and Larry's heights!

Think about it visually: If Larry is 4 units above John, then Larry is 2 units above Ken. That places Ken exactly halfway between them:
- John ← (2 units) → Ken ← (2 units) → Larry

For this question to be sufficient, we need to determine one specific value for the average height.

Analyzing Statement 1

Statement 1: Ken's height is 180 centimeters.

What Statement 1 Tells Us

We now know \(\mathrm{K} = 180 \text{ cm}\). Since Ken is at the midpoint between John and Larry, his height is the average of their two heights.

The Beautiful Insight

When one person is at the midpoint of two others, that middle person's height is the average of all three!

Why? Because the deviations cancel out perfectly:

• Larry is some distance above Ken
• John is the same distance below Ken
• So the average must be Ken's height: 180 cm

Conclusion

Statement 1 is sufficient - we can determine the average is exactly 180 cm.

[STOP - Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: Larry's height is 190 centimeters.

What Statement 2 Provides

We know \(\mathrm{L} = 190 \text{ cm}\), and from our relationship \(\mathrm{J} = 2\mathrm{K} - \mathrm{L}\), we get:
\(\mathrm{J} = 2\mathrm{K} - 190\)

Testing Different Scenarios

Let's see if different values of K lead to different averages:

Scenario 1: If Ken is 185 cm

• John = \(2(185) - 190 = 180 \text{ cm}\)
• Average = \((190 + 180 + 185) ÷ 3 = 185 \text{ cm}\)
• Heights in order: John (180), Ken (185), Larry (190) ✓

Scenario 2: If Ken is 180 cm

• John = \(2(180) - 190 = 170 \text{ cm}\)
• Average = \((190 + 170 + 180) ÷ 3 = 180 \text{ cm}\)
• Heights in order: John (170), Ken (180), Larry (190) ✓

Conclusion

Different values of Ken's height lead to different averages. Without knowing Ken's actual height, we cannot determine a unique average.

Statement 2 is NOT sufficient.

This eliminates choice D.

The Answer: A

Only Statement 1 is sufficient to determine the average height, while Statement 2 alone is not.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."