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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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09 Jul 2009, 09:29
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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, 06:37, edited 1 time in total.
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Re: decimal equivalent [#permalink]

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09 Jul 2009, 09:56
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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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09 Jul 2009, 09:58
Thanks a lot.
Nice explanation.
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Re: decimal equivalent [#permalink]

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09 Jul 2009, 10:28
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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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01 Oct 2013, 06:34
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Re: Which of the following fractions has a decimal equivalent th [#permalink]

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01 Oct 2013, 06:38
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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.

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
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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, 06:38
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# Which of the following fractions has a decimal equivalent th

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