Is the size of a certain particle closer to 10^(-3) centimeter than it is to 10^(-2) centimeter? The size of...

Understanding the Question

We need to determine whether a particle's size is closer to \(10^{-3}\) centimeter or to \(10^{-2}\) centimeter.

To visualize this:

• \(10^{-3}\) cm = 0.001 cm
• \(10^{-2}\) cm = 0.01 cm

So we're asking: is the particle closer to 0.001 cm or to 0.01 cm?

What "sufficient" means here: Since this is a yes/no question, we need to definitively answer either YES (it's closer to \(10^{-3}\)) or NO (it's closer to \(10^{-2}\)). If we can determine this with certainty, the statement(s) are sufficient.

Key Insight

Picture this on a number line. The particle will be closer to \(10^{-3}\) if it's in the "left half" of the interval between these two values, and closer to \(10^{-2}\) if it's in the "right half." We can solve this without calculating exact boundaries—we just need to know which side our particle falls on.

Analyzing Statement 1

Statement 1: The particle is closer to \(10^{-4}\) centimeter than it is to \(10^{-1}\) centimeter.

Let's visualize this on our number line:

10^{-4} -------- 10^{-3} -------- 10^{-2} -------- 10^{-1}
0.0001          0.001            0.01             0.1

Statement 1 tells us the particle is somewhere in the left portion between \(10^{-4}\) and \(10^{-1}\). But this is a HUGE range—it spans three orders of magnitude!

Testing Strategic Positions

Let's test two carefully chosen values:

• Particle at 0.0005: This is between \(10^{-4}\) and \(10^{-3}\), so it's definitely closer to \(10^{-4}\) than to \(10^{-1}\) ✓. And it's closer to \(10^{-3}\) than to \(10^{-2}\). Answer: YES
• Particle at 0.005: This is between \(10^{-3}\) and \(10^{-2}\), still much closer to \(10^{-4}\) than to \(10^{-1}\) ✓. But now it's closer to \(10^{-2}\) than to \(10^{-3}\). Answer: NO

Both positions satisfy Statement 1, but give different answers to our question.

Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The particle is closer to \(10^{-3}\) centimeter than it is to \(10^{-1}\) centimeter.

This tells us the particle is in the left portion between \(10^{-3}\) and \(10^{-1}\). Again, this is a large range—spanning two orders of magnitude.

Testing Strategic Positions

Using the same test values:

• Particle at 0.0005: This is less than \(10^{-3}\), so definitely closer to \(10^{-3}\) than to \(10^{-1}\) ✓. And it's closer to \(10^{-3}\) than to \(10^{-2}\). Answer: YES
• Particle at 0.005: This is between \(10^{-3}\) and \(10^{-2}\), still much closer to \(10^{-3}\) than to \(10^{-1}\) ✓. But it's closer to \(10^{-2}\) than to \(10^{-3}\). Answer: NO

Both positions satisfy Statement 2, yet give different answers.

Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices B and D.

Combining Statements

Now let's see what happens when we use both statements together.

From Statement 1: The particle is closer to \(10^{-4}\) than to \(10^{-1}\)
From Statement 2: The particle is closer to \(10^{-3}\) than to \(10^{-1}\)

Both statements place the particle in "left portions" of overlapping intervals. The key question: does their intersection narrow down the range enough?

Testing Our Positions with Both Constraints

Let's verify our test values work under both conditions:

• Particle at 0.0005:
  • Closer to \(10^{-4}\) than \(10^{-1}\)? YES ✓
  • Closer to \(10^{-3}\) than \(10^{-1}\)? YES ✓
  • Closer to \(10^{-3}\) than \(10^{-2}\)? YES
• Particle at 0.005:
  • Closer to \(10^{-4}\) than \(10^{-1}\)? YES ✓
  • Closer to \(10^{-3}\) than \(10^{-1}\)? YES ✓
  • Closer to \(10^{-3}\) than \(10^{-2}\)? NO

Both positions satisfy BOTH statements, yet they give different answers to our question!

The statements together are NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice C.

The Answer: E

Even when we combine both statements, we still can't determine whether the particle is closer to \(10^{-3}\) or \(10^{-2}\) centimeter. The constraints from both statements are too broad to pinpoint the particle's position precisely enough.

Answer Choice E: The statements together are not sufficient.