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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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18 Dec 2012, 04:27
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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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18 Dec 2012, 04:32
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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.

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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29 Jul 2013, 17:51
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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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18 Dec 2012, 04:33
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Carcass rightly said, you are a machine Bunuel.
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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14 Oct 2014, 22:27
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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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03 Oct 2016, 09:33
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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

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.

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

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18 Dec 2012, 04:33
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.
+1B
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Re: Any decimal that has only a finite number of nonzero digits  [#permalink]

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29 Jul 2013, 17:43
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?

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

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14 Oct 2014, 15:01
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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12 Sep 2018, 14:32
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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Updated on: 17 Aug 2019, 08:50
Hi Bunuel,

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, 08:46.
Last edited by a12bansal on 17 Aug 2019, 08:50, 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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17 Aug 2019, 08:49
a12bansal wrote:
Hi Bunuel,

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]

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02 Apr 2020, 08:21
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Re: Any decimal that has only a finite number of nonzero digits   [#permalink] 02 Apr 2020, 08:21