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Is the size of a certain particle closer to \(10^{-3}\) centimeter than it is to \(10^{-2}\) centimeter?
We need to determine whether a particle's size is closer to \(10^{-3}\) centimeter or to \(10^{-2}\) centimeter.
To visualize this:
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.
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.
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 0.1Statement 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!
Let's test two carefully chosen values:
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.
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.
Using the same test values:
Both positions satisfy Statement 2, yet give different answers.
Statement 2 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices B and D.
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?
Let's verify our test values work under both conditions:
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.
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.