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

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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?

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.

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

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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. +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, 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?

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

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29 Jul 2013, 18:51

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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23 Aug 2014, 10:42

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

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08 Sep 2015, 04:40

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

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30 Sep 2016, 01:36

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

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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
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03 Oct 2016, 10:33

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