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

(1) The size of the particle is closer to \(10^{-4}\) centimeter than it is to \(10^{-1}\) centimeter.

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

A)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)EACH statement ALONE is sufficient.
E)Statements (1) and (2) TOGETHER are NOT sufficient.

Here is a different visual representation that might help:

Which of the two values 0.01 and 0.001, is x more close to?

(1) x is more close to 0.0001 than to 0.1.
(2) x is more close to 0.001 than to 0.1.



Attachment:
Number line.png
Number line.png [ 10.07 KiB | Viewed 15252 times ]
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XimeSol
Is the size of a certain particle closer to \(10^{-3}\) centimeter than it is to \(10^{-2}\) centimeter?

(1) The size of the particle is closer to \(10^{-4}\) centimeter than it is to \(10^{-1}\) centimeter.

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

A)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)EACH statement ALONE is sufficient.
E)Statements (1) and (2) TOGETHER are NOT sufficient.
I have trouble dealing with questions such as these. So I eliminated the negative powers as much as I could.

Question: Is a number x closer to 10^-3 than it is to 10^-2?
Multiplying the equation with 10^3

Is a new number (1000x) closer to 1 than it is to 10? We can call the new number y just to simplify it further. Our new problem statement: is y closer to 1 than it is to 10?

Since we multiplied the question statement by 10^3, we will multiply the statements as well by the same number.
Statement 1: y is closer to 0.1 than it is to 100.
y could be 0.2, and the answer to the problem statement would be YES in that case.
y could be 10, and the answer would be NO.
NOT SUFFICIENT

Statement 2: y is closer to 1 than it is to 100.
y = 0.2, YES.
y = 10, NO.
NOT SUFFICIENT

Both (1) and (2) together: NOT SUFFICIENT.

Answer: E
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XimeSol
Is the size of a certain particle closer to \(10^{-3}\) centimeter than it is to \(10^{-2}\) centimeter?

(1) The size of the particle is closer to \(10^{-4}\) centimeter than it is to \(10^{-1}\) centimeter.

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

A)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)EACH statement ALONE is sufficient.
E)Statements (1) and (2) TOGETHER are NOT sufficient.

Draw it on the number line: 0, 1/1000, 1/100 and 1/10 (the point 1/10,000 can be assumed to be right next to 0)
(or you could multiply everything by 10,000 and look at relative placements of 1, 10, 100 and 1000. It won't change the question at all)

Attachment:
Screenshot 2025-05-06 at 11.55.47 AM.png
Screenshot 2025-05-06 at 11.55.47 AM.png [ 32.86 KiB | Viewed 3588 times ]

(1) The size of the particle is closer to 10^-4 than 10^-1

The green line is the mid point of 1/10000 and 1/10. We know that the particle is to the left of the green line. Is it closer to 1/100 or 1/1000? We cannot say. It could be anywhere to the left of the green mid line. It could even lie on top of 1/100 or on top of 1/1000.
Not sufficient alone.


(2) The size of the particle is closer to 10^-3 than 10^-1

Once again, the mid point will be very close to the green line shown in the diagram above. The particle is to the left of the green line. Is it closer to 1/100 or 1/1000? We cannot say. It could be anywhere to the left of the green mid line. It could even lie on top of 1/100 or on top of 1/1000.
Not sufficient alone.

Almost same possible region is given by both statements. Hence both together are also not sufficient.

Answer (E)

Here is a useful discussion on Number Line: https://youtu.be/3gxVx3Y9xJA
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For anyone thinking of solving it mathematically, calculate mid point of each of the statements and you will see that the range defined would capture the whole range of the question stem.

Basically, question is asking us whether the value of x is on the left of the midpt or right of the midpt of the range given in the question.

And both the statements give a range which doesn't help us identify that. And both together also gives us a similar range overlapping the range of the question stem, hence again it could be to left of mid pt or right. Hence E)

This one took me sometime, but the number line approach is much more intuitive.
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