A decimal is called a shrinking number if its value is between 0 and

Question

A decimal is called a “shrinking number” if its value is between 0 and 1 and each digit to the right of the decimal is not less than the digit to its immediate right. For instance, 0.86553221 is a shrinking number. If x is a shrinking number, which of the following must be true?

I. 9x/10 is a shrinking number.

II. (x + 9)/10 is a shrinking number.

III. x/10 is a shrinking number.

(A) I only
(B) II only
(C) I and II only
(D) III only
(E) I, II, and III

chetan2u · Accepted Answer

So, shrinking form is of type: 0.yyyyyy or 0.yy(y-1)(y-2)(y-2)......, that is the numbers after decimal are never increasing as we move to further right

I.  \frac{9x}{10}  is a shrinking number.
Multiplication of x by 0.9 will surely lead to change in the layout of digits after decimals.
For example: x = 0.21, so  0.21*\frac{9}{10}=0.21*0.9=0.189

II.  \frac{(x + 9)}{10}  is a shrinking number.
 Addition of 0.9 to x/10 will add a 9 to immediate right of decimal followed by the existing sequence of x, thereby resulting in a shrinking number 
 \frac{(x + 9)}{10} 
Say x = 0.998761, then x+9=9.998761
 \frac{(x + 9)}{10}=0.9998761

III. x/10 is a shrinking number.
Always NO, as we will add a 0 to immediate right of decimal followed by the existing sequence of x.
Say x = 0.998761, then  \frac{x }{10}=0.09998761 
Only possibility when the number is 0.0.

B

Bunuel · Answer

Check more  Special Numbers and Sequences  questions.

AndrewN · Answer

This question is a mess, as written. Statement II appears to be missing an  x , while Statement III has a mysterious  a  where we would expect  x  to be. Unless the Manhattan Prep question-writers just tossed in a random variable to catch some people off guard, Statement III almost certainly looks different in the original question. Furthermore, looking at Statement II, we can disprove the purely arithmetic fraction right away. I multiplied both halves of the fraction by 1000 to move the decimal, and then I worked out the long division.

\frac{3507}{10020} = 0.35

Of course, 0.35 is NOT a shrinking number, per the definition provided in the problem. If  x  were in there somewhere, we would be looking at a different proposition.

Finally, Statement I does NOT have to be true. We can consider nothing more than the first few digits of the given decimal, 0.865, to disprove it:

\frac{9 * 0.865}{10} = 7.785

After two minutes or so, I chose (B), having disproven anything with Statement I in it—answer choices (A), (C), and (E)—and Statement III seemed spurious.

Bunuel , would you please review the source material and correct the question accordingly? Many thanks.

- Andrew

Bunuel · Answer

Thank you  AndrewN ! Edited the question according to the source: GMAT Advanced Quant: 250+ Practice Problems & Online Resources (Manhattan).

Capakki · Answer

Lets take x= 0.332

X/10 = 0.0332 - Not satisfying the condition. So, III is ruled out.

Now we are left with A,B &C.

9x/10: 
0.332 * 9 = 2.998
2.998/10 = 0.2998 - VOILATES the condition.

Hence, option C and A are ruled out. B is our answer.

(X+9)/10: here any digit after 9 will either be equal or be less than 9. 
9+0.332 = 9.332 
9.332/10 = 0.9332 - satisfies the condition.

Posted from my mobile device

MidhilaMohan · Answer

Assume the number to be 0.21 and try and to do suggested calculations
I. 9x/10 is a shrinking number.
=> 9*0.21/10= 1.89/10 = 0.189 hence not true

II. (x + 9)/10 is a shrinking number.
=> (0.21+9)/10 = 0.921-> shrinking decimal; understand that this situation will ALWAYS result in a shrinking decimal because the condition states that the decimal digit to the right of the decimal should not be LESS than its immediate right-> TRUE

III. x/10 is a shrinking number.
0.21/10= 0.021-> hence not true

II only

davro · Answer

I think "x" in the first statement means a digit, not a number to multiply by 9. So, the two-digit number "9x" satisfies the rule of a shrinking number.

MidhilaMohan · Answer

Now "x" is defined as a shrinking number and not a digit. A shrinking number is defined as a decimal number that is between 0 and 1. In the first statement, 9x is not appending x to 9 but multiplying 9 to x. Hope this clears the doubt.

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A decimal is called a shrinking number if its value is between 0 and

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