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Which of the following fractions has a decimal equivalent th

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Which of the following fractions has a decimal equivalent th [#permalink]

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Which of the following fractions has a decimal equivalent that is a terminating decimal?

A. 10/189
B. 15/196
C. 16/225
D. 25/144
E. 39/128

OPEN DISCUSSION OF THIS QUESTION IS HERE: which-of-the-following-fractions-has-a-decimal-equivalent-159322.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 01 Oct 2013, 07:37, edited 1 time in total.
Added the OA.
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Re: decimal equivalent [#permalink]

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New post 09 Jul 2009, 10:56
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The answer is E.

For a fraction to be a terminating decimal the denominator can contain only the prime factors 2 and/or 5 i.e. the denominator should be in the form \(2^a5^b\)

Looking at each denominator.

A. 189 has the prime factor 3 so the fraction is not a terminating decimal.
B. 196 has the prime factor 7 so the fraction is not a terminating decimal.
C. 225 has the prime factor 3 so the fraction is not a terminating decimal.
D. 144 has the prime factor 3 so the fraction is not a terminating decimal.
E. 128 has the prime factor 2 only and can be written in the form \(2^75^0\) which makes the fraction a terminating decimal.
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Re: decimal equivalent [#permalink]

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New post 09 Jul 2009, 10:58
Thanks a lot.
Nice explanation.
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Re: decimal equivalent [#permalink]

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New post 09 Jul 2009, 11:28
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One important point: you should look at numerator as well, as it can simplify denominator. For example, \(18/225\) is a terminating decimal: \(18/225=\frac{3*3*2}{3*3*5*5}=\frac{2}{5*5}\).
However, this problem doesn't try to trick you this way.
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Re: decimal equivalent [#permalink]

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Re: Which of the following fractions has a decimal equivalent th [#permalink]

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Which of the following fractions has a decimal equivalent that is a terminating decimal?
A. 10/189
B. 15/196
C. 16/225
D. 25/144
E. 39/128

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 \(b\) (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 \(2*5^2\). 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 the 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:

Only option E (when reduced to its lowest form) has the denominator of the form \(2^n5^m\): 39/128=39/2^7.

Answer: E.

Questions testing this concept:
does-the-decimal-equivalent-of-p-q-where-p-and-q-are-89566.html
any-decimal-that-has-only-a-finite-number-of-nonzero-digits-101964.html
if-a-b-c-d-and-e-are-integers-and-p-2-a3-b-and-q-2-c3-d5-e-is-p-q-a-terminating-decimal-125789.html
700-question-94641.html
is-r-s2-is-a-terminating-decimal-91360.html
pl-explain-89566.html
which-of-the-following-fractions-88937.html

OPEN DISCUSSION OF THIS QUESTION IS HERE: which-of-the-following-fractions-has-a-decimal-equivalent-159322.html
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Re: Which of the following fractions has a decimal equivalent th   [#permalink] 01 Oct 2013, 07:38
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