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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
[quote="Bunuel"]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. 3.507/10.02 is a shrinking number.

III. a/40 is a shrinking number.

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



The answer should be B as both I and III have denominators which have factors of only 2 and 5
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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
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Bunuel wrote:
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. 3.507/10.02 is a shrinking number.

III. a/40 is a shrinking number.

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

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
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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
Expert Reply
AndrewN wrote:
Bunuel wrote:
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. 3.507/10.02 is a shrinking number.

III. a/40 is a shrinking number.

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

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


Thank you AndrewN! Edited the question according to the source: GMAT Advanced Quant: 250+ Practice Problems & Online Resources (Manhattan).
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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
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.

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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
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
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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
MidhilaMohan wrote:
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


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.
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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
davro wrote:
MidhilaMohan wrote:
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


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.


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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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
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Re: A decimal is called a shrinking number if its value is between 0 and [#permalink]
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