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Any decimal that has only a finite number of nonzero digits

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

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New post 18 Dec 2012, 05:27
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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 decimals. If r and s are positive integers and the ratio r/s is expressed as a decimal, is r/s a terminating decimal?

(1) 90 < r < 100
(2) s = 4
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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New post 18 Dec 2012, 05:32
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THEORY:
Reduced fraction \(\frac{a}{b}\) (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only \(b\) (the 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.)

BACK TO THE QUESTION:
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 decimals. If r and s are positive integers and the ratio r/s is expressed as a decimal, is r/s a terminating decimal?

(1) 90 < r < 100. Nothing about the denominator. Not sufficient.

(2) s = 4. According to the above, any fraction r/4=r/2^2 when expressed as a decimal will be a terminating decimal. Sufficient.

Answer: B.

Questions testing this concept:
700-question-94641.html
is-r-s2-is-a-terminating-decimal-91360.html
pl-explain-89566.html
which-of-the-following-fractions-88937.html

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

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New post 18 Dec 2012, 05:33
A fraction r/s will only be a terminating decimal ONLY if it is of the form \(Numerator/ 2^m 5^n\), where n and m are non-negative.
Statement 1 gives the range of numerators, of which we are not concerned at all. Insufficient
Statement 2 gives the value of denominator which is of the form \(2^2\). Hence the fraction has to be a terminating decimal.
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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New post 18 Dec 2012, 05:33
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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New post 29 Jul 2013, 18:43
Question, I understand that a terminating decimal has to be of the form \(2^x5^x\) but four is only in the form of \(2^n\) to be a terminating decimal it can meet either of the requirements?

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

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New post 29 Jul 2013, 18:51
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hfbamafan wrote:
Question, I understand that a terminating decimal has to be of the form \(2^x5^x\) but four is only in the form of \(2^n\) to be a terminating decimal it can meet either of the requirements?

Thanks,
Hunter


for a fraction to be terminating two condition must satisfy:
1) numerator is an INTEGER.
2) denominator should be of form \(2^x 5^y\) \((x,y\)==>integers which also includes 0)

now in this question
denominator is \(2^2 5^0\)
hence it satisfies.

hope it helps
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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New post 03 Oct 2016, 10:33
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Walkabout wrote:
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 decimals. If r and s are positive integers and the ratio r/s is expressed as a decimal, is r/s a terminating decimal?

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


This problem is testing us on our knowledge of terminating decimals.

When solving this problem, we should remember that there is a special property of fractions that allows their decimal equivalents to terminate. In its most-reduced form, any fraction with a denominator whose prime factorization contains only 2s, 5s, or both produces decimals that terminate. A denominator with any other prime factors produces decimals that do not terminate. So to determine whether r/s is expressed as a terminating decimal, we need to determine whether the prime factorization of s contains only 2s, 5s, or both.

Statement One Alone:

90 < r < 100

Since statement one does not provide any information about s, we cannot determine whether r/s is expressed as a terminating decimal. If r = 91 and s = 1, then r/s is a terminating decimal. On the other hand, if r = 91 and s = 3, then r/s = 30.3333… and thus, r/s is not a terminating decimal. Statement one alone is not sufficient. We can eliminate answer choices A and D.

Statement Two Alone:

s = 4

Since we know that s = 4, we know that the prime factorization of s (2^2) only contains 2’s. Thus, r/s is expressed as a terminating decimal.

Answer: B
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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New post 12 Sep 2018, 15:32
Walkabout wrote:
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 decimals. If r and s are positive integers and the ratio r/s is expressed as a decimal, is r/s a terminating decimal?

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


\(r,s\,\, \geqslant 1\,\,\,{\text{ints}}\)

\(\frac{r}{s}\,\,\,\mathop = \limits^? \,\,\,\,{\text{terminating}}\)

\(\left( 1 \right)\,\,90 < r < 100\,\,\,\,\left\{ \begin{gathered}
\,\left( {r,s} \right) = \left( {95,5} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\left( {\frac{{95}}{5} = \operatorname{int} } \right) \hfill \\
\,\left( {r,s} \right) = \left( {91,3} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\left( {\frac{{91}}{3} = 30\frac{1}{3} = 30.333 \ldots } \right) \hfill \\
\end{gathered} \right.\)

\(\left( 2 \right)\,\,s = 4\)

\(\left( * \right)\,\,\,\,{\text{r/s}}\,\,\,{\text{division}}\,\,{\text{algorithm}}:\,\,\,\left\{ \begin{gathered}
\,r = qs + R\,\,\,\mathop = \limits^{s\, = \,4} \,\,\,4q + R \hfill \\
\,q\,\,\operatorname{int} \,\,\,,\,\,\,\,0\,\,\, \leqslant \,\,\,R\,\,\operatorname{int} \,\,\, \leqslant \,\,3\,\,\,\,\left( { = s - 1} \right) \hfill \\
\end{gathered} \right.\)

\(\frac{r}{s}\,\,\,\,\mathop = \limits^{\,\left( * \right)} \,\,\,\,\frac{{4q + R}}{4} = q + \frac{R}{4}\,\, = \,\,\operatorname{int} \,\, + \,\,\frac{R}{4}\,\,\,\,\,\,\,\)

\(\frac{R}{4} = \,\,\,\left\{ {\begin{array}{*{20}{c}}
{\,\,\frac{0}{4}} \\
{\,\,\frac{1}{4}} \\
{\,\,\frac{2}{4}} \\
{\,\,\frac{3}{4}}
\end{array}} \right.\begin{array}{*{20}{c}}
{\,\,{\text{if}}\,\,\,R = 0\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\left( {\frac{r}{4} = \operatorname{int} } \right)} \\
{\,\,{\text{if}}\,\,\,R = 1\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\left( {\frac{r}{4} = \operatorname{int} \,\, + \,\,0.25} \right)} \\
{\,\,{\text{if}}\,\,\,R = 2\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\left( {\frac{r}{4} = \operatorname{int} \,\, + \,\,0.5} \right)} \\
{\,\,{\text{if}}\,\,\,R = 3\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\left( {\frac{r}{4} = \operatorname{int} \,\, + \,\,0.75} \right)}
\end{array}\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Any decimal that has only a finite number of nonzero digits  [#permalink]

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New post Updated on: 17 Aug 2019, 09:50
Hi Bunuel,

Please clarify below one :

A denominator with any other prime factors apart from 2s, 3s produces decimals that do not terminate.

Is the above statement true in each case irrespective of numerator ?

Also I want to know If the denominator has prime factors as 2,3,5 For ex denominator is 30. Now irrespective of numerator will it be terminate as the denominator has prime factors 2 and 3 .

Thanks

Originally posted by a12bansal on 17 Aug 2019, 09:46.
Last edited by a12bansal on 17 Aug 2019, 09:50, edited 1 time in total.
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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New post 17 Aug 2019, 09:49
a12bansal wrote:
Hi Bunuel,

Please clarify below one :

A denominator with any other prime factors apart from 2s, 3s produces decimals that do not terminate.

Is the above statement true in each case irrespective of numerator ?

Thanks


If the denominator has only 2's or/and 5's the the fraction will terminate irrespective of the numerator (obviously provided the numerator is an integer).
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Re: Any decimal that has only a finite number of nonzero digits   [#permalink] 17 Aug 2019, 09:49
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