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Any decimal that has only a finite number of nonzero digits  [#permalink]

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Question Stats: 83% (00:47) correct 17% (01:37) wrong based on 272 sessions

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Any decimal that has only a finite number of nonzero digits is a terminating decimal. for example, 24, 0.82, and 5.096 are three terminating numbers. If r and s are positive integers and the ratio is r/s is expressed as a decimal, is r/s a terminating decimal?

(1) 90<r< 100
(2) s = 4

Originally posted by tarn151 on 20 Aug 2010, 13:18.
Last edited by Bunuel on 12 Apr 2012, 14:28, edited 1 time in total.
Edited the question
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Any decimal that has only a finite number of nonzero digits  [#permalink]

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tarn151 wrote:
OK, so the question I have below has a very similar post on this forum (as well as other's I've checked) but it appears that my question is slightly different. With data point #2, every other forum post I've checked has s=4 for this question. In this case, the answer is B as any integer divided by 4 results in a terminating decimal. HOWEVER, in my OG11, I have s=4B (with no other reference to B anywhere in the question, as you see below). Please take a look, and let me know your thoughts.

any decimal that has only a finite number of nonzero digits is a terminating decimal. for example, 24, 0.82, and 5.096 are three terminating numbers. If r and s are positive integers and the ratio is r/s is expressed as a decimal, is r/s a terminating decimal?
1. 90<r< 100
2. s = 4B

answer is still choice B in my version! I don't see how this is possible as once you introduce the other variable, s could really be an infinite number of positive integers

It must be a typo. The answer to the question if statement (2) says $$s=4B$$ would be E, as we have no info about B.

As for the terminating decimals:

Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example $$\frac{x}{2^n5^m}$$, (where x, n and m are integers) will always be terminating decimal.

(We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction $$\frac{6}{15}$$ has 3 as prime in denominator and we need to know if it can be reduced.)

Questions testing this concept:
http://gmatclub.com/forum/700-question- ... %20decimal
http://gmatclub.com/forum/is-r-s2-is-a- ... %20decimal
http://gmatclub.com/forum/pl-explain-89 ... %20decimal
http://gmatclub.com/forum/which-of-the- ... %20decimal

Hope it helps.
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Any decimal that has only a finite number of nonzero digits  [#permalink]

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tarn151 wrote:
OK, so the question I have below has a very similar post on this forum (as well as other's I've checked) but it appears that my question is slightly different. With data point #2, every other forum post I've checked has s=4 for this question. In this case, the answer is B as any integer divided by 4 results in a terminating decimal. HOWEVER, in my OG11, I have s=4B (with no other reference to B anywhere in the question, as you see below). Please take a look, and let me know your thoughts.

any decimal that has only a finite number of nonzero digits is a terminating decimal. for example, 24, 0.82, and 5.096 are three terminating numbers. If r and s are positive integers and the ratio is r/s is expressed as a decimal, is r/s a terminating decimal?
1. 90<r< 100
2. s = 4B

answer is still choice B in my version! I don't see how this is possible as once you introduce the other variable, s could really be an infinite number of positive integers

My understanding is as follows.

For (r/s) to be a terminating decimal, its denominator 's' should contain powers of 2 and/or 5 in it. Hence any denominator which can be expressed as (2^x * 5^ y), where x, y could be any positive integers.

In the two statements
a) 90 < r < 100 --- This does not give us any clue about the denominator. Hence insufficient.

b) s= 4B. Now (r/s) becomes (r/(4s)) and the denominator is of the form (2^2 * 5 ^ 0 * s) and hence the division would result in a terminating decimal.

Hence statement is sufficient.

 I don't see how this is possible as once you introduce the other variable, s could really be an infinite number of positive integers

Even after introducing a new variable B, s could be a infinite number of positive integers but still it would be of the form (2^ x * 5 ^y) and hence for all those infinite combination the fraction would be a terminating decimal.

Hope my explanation helps.
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Re: OG 11 Data Sufficiency 107  [#permalink]

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Just to clarify your point s=4B, so it is NOT of the form 2^x 5^y, but will contain these factors for sure. If I understand right all it takes for a terminating decimal is to have the denominator contain 2 and/or 5 as factors..
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Thanks Bunuel for the explanation.
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Re: OG 11 Data Sufficiency 107  [#permalink]

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Great explanation. Thanks Bunuel
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Re: PT #13 DS 25  [#permalink]

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Interesting question...

Here is my take:

what determines the terminating/terminating nature of a decimal is the denominator

(1) provides numerator info only so insufficient

(2) any integer divided by 4 will have terminating decimal so i think this is sufficient

please provide explanation if you disagree with my logic
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Bunuel, can you edit the original post and correct the s=4B?
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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BN1989 wrote:
Bunuel, can you edit the original post and correct the s=4B?

Original post is edited.
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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1
IMO B,
the decimal is terminating or not depends on the DENOMINATOR
1)90 < r < 100
but we need to know what is s.
for example any number that when divided by 3 gives remainder will be non terminating.. ex-91/3, 94/3 etc

2)s=4
for any number which is divided by 4, remainder is 0,1,2or 3
for all of them you get terminating decimals
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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any decimal that has only a finite number of nonzero digits is a terminating decimal. for example, 24, 0.82, and 5.096 are three terminating numbers. If r and s are positive integers and the ratio is r/s is expressed as a decimal, is r/s a terminating decimal?
1. 90<r< 100
2. s = 4B

There is a mistake in the text. It says that any decimal number has only a finite number of nonzero digits, but this is not true. For this to be true, It must say: any decimal number has only a finite number of digits.
Do you agree?
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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Andrake26 wrote:

There is a mistake in the text. It says that any decimal number has only a finite number of nonzero digits, but this is not true. For this to be true, It must say: any decimal number has only a finite number of digits.
Do you agree?

The reason it says "non zero digits" specifically is because theoretically every decimal has infinite trailing 0s at the right of the decimal after the last non zero digit.

15.6 = 15.6000000000000000...
15.0903 = 15.0903000000000000000000...
15 = 15.0000000000000000000...

But all of 15.6, 15.0903 and 15 are terminating.
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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