GMAT Set 8 Q23

[Abhijit K]

In the decimal representation of x, where 0 < x < 1, is the tenths digit of x nonzero?
(1) 16x is an integer.
(2) 8x is an integer.

Approach please....

[GMATGuruNY]

In the decimal representation of x, where 0 < x < 1, is the tenths digit of x nonzero?
(1) 16x is an integer.
(2) 8x is an integer.

Since 1/10 = 0.1, any positive fraction LESS THAN 1/10 will be LESS THAN 0.1 and thus will have a tenths digit of 0.
To illustrate:
1/11 = 0.090909...
1/20 = 0.05
1/100 = 0.01.

By extension, any positive fraction greater than or equal to 1/10 will have a NONZERO tenths digit:
1/10 = 0.1
1/5 = 0.2
1/2 = 0.5.

Question stem, rephrased:
Is 1/10 ≤ x < 1?

Statement 1: 16x is an integer
If x = 1/16, then x < 1/10.
If x = 15/16, then 1/10 ≤ x < 1.
INSUFFICIENT.

Statement 2: 8x is an integer
Here, the least possible value for x is 1/8, which is greater than 1/10.
Thus, 1/10 ≤ x < 1.
SUFFICIENT.

The correct answer is B.

[Brent@GMATPrepNow]

In the decimal representation of x, where 0 < x < 1, is the tenths digit of x nonzero?

(1) 16x is an integer.
(2) 8x is an integer.

Target question:Is the tenths digit of x nonzero?

This is a good candidate for rephrasing the target question.

Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

First recognize that the tenths digit of x will equal ZERO, if x is LESS THAN 0.1
For example, if x = 0.04, the tenths digit is 0
So, the tenths digit of x will be NONZERO if x > 0.1
In other words, the tenths digit of x will be NONZERO if x > 1/10
Since we're already told that x < 1, we can REPHRASE the target question...
REPHRASED target question: Is 1/10 < x < 1?

Statement 1: 16x is an integer.
Since we're told that 0 < x (i.e., x is positive), we know that 16x is positive
So, let's say that 16x = k, where k is some positive integer
Solve for x to get: x = k/16 (where k is some positive integer)
There are several values of k that satisfy statement 1. Here are two:
Case a: k = 8, in which case x = 8/16 = 1/2, which means it IS the case that 1/10 < x < 1
Case b: k = 1, in which case x = 1/16, which means it is NOT the case that 1/10 < x < 1
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 8x is an integer.
Let's say that 8x = j, where j is some positive integer
Solve for x to get: x = j/8 (where j is some positive integer)
Since j is a positive integer, then j = 1 or 2 or 3 or ....
In all of these cases, j/8 will be GREATER than 1/10
In other words, x must be GREATER than 1/10
Since we're also told that x < 1, we can be certain that 1/10 < x < 1
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

[NandishSS]

In the decimal representation of x, where 0 < x < 1, is the tenths digit of x nonzero?
(1) 16x is an integer.
(2) 8x is an integer.

Since 1/10 = 0.1, any positive fraction LESS THAN 1/10 will be LESS THAN 0.1 and thus will have a tenths digit of 0.
To illustrate:
1/11 = 0.090909...
1/20 = 0.05
1/100 = 0.01.

By extension, any positive fraction greater than or equal to 1/10 will have a NONZERO tenths digit:
1/10 = 0.1
1/5 = 0.2
1/2 = 0.5.

Question stem, rephrased:
Is 1/10 ≤ x < 1?

Statement 1: 16x is an integer
If x = 1/16, then x < 1/10.
If x = 15/16, then 1/10 ≤ x < 1.
INSUFFICIENT.

Statement 2: 8x is an integer
Here, the least possible value for x is 1/8, which is greater than 1/10.
Thus, 1/10 ≤ x < 1.
SUFFICIENT.

The correct answer is B.

Hey Guru,

Just wanted to go in depth.I was wondering here what are they trying to test?

Terminating Decimals? or anything else?

Thanks
Nandish