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Which of the following is/are terminating decimal(s)?

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Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 06 Aug 2014, 09:08
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Which of the following is/are terminating decimal(s)?

I 299/(32^123)
II 189/(49^99)
III 127/(25^37)

A) I only
B) I and III
C) II and III
D) II only
E) I, II, III
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 06 Aug 2014, 10:24
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Terminating decimal is a such decimal that has only a finite number of non-zero digits. That's why if you will write such decimal as a fraction you will have as a denominator 10,100, 1000, etc. Since the fraction can be simplified, you can have in the denominator at the end some product of 2 or 5.

For example:
\(0.4=4/10=2/5\), 5 in the denominator
\(0.25=25/100=1/4\), 4=2*2 in the denominator.

But anyway, if you have terminating decimal you can't have anything other 2 or 5 in prime factorization of denominator. So we have a rule:

The fraction expressed as a decimal will be the terminating decimal if it can be presented as \(\frac{a}{{2^n\cdot 5^m}}\) where \(a\) is an integer, \(m=0,1,2,3..\). , and \(n=0,1,2,3...\) .

Or

if a fraction has in denominator any prime factor different from 2 and 5, such fraction will be infinite decimal.

Now, according to your problem:

I. Has only 2 as prime factor, since \(32=2^5\). Terminating
II. Has 7 as prime factor. Infinite
III. Has only 5 as prime factor. Terminating

The correct answer is B.


Hope this helps!:)
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 06 Aug 2014, 13:57
smyarga wrote:
Terminating decimal is a such decimal that has only a finite number of non-zero digits. That's why if you will write such decimal as a fraction you will have as a denominator 10,100, 1000, etc. Since the fraction can be simplified, you can have in the denominator at the end some product of 2 or 5.

For example:
\(0.4=4/10=2/5\), 5 in the denominator
\(0.25=25/100=1/4\), 4=2*2 in the denominator.

But anyway, if you have terminating decimal you can't have anything other 2 or 5 in prime factorization of denominator. So we have a rule:

The fraction expressed as a decimal will be the terminating decimal if it can be presented as \(\frac{a}{{2^n\cdot 5^m}}\) where \(a\) is an integer, \(m=0,1,2,3..\). , and \(n=0,1,2,3...\) .

Or

if a fraction has in denominator any prime factor different from 2 and 5, such fraction will be infinite decimal.

Now, according to your problem:

I. Has only 2 as prime factor, since \(32=2^5\). Terminating
II. Has 7 as prime factor. Infinite
III. Has only 5 as prime factor. Terminating

The correct answer is B.


Hope this helps!:)


Great explanation, i forgot the concept of terminating decimals.
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 06 Aug 2014, 14:08
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 06 Aug 2014, 23:36
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 07 Aug 2014, 00:41
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alphonsa wrote:


I like very much this source 700-800-level-quant-problem-collection-detailed-solutions-137388.html
You can find there problems divided by topics with detailed explanation.
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 07 Aug 2014, 18:46
1
alphonsa wrote:
Which of the following is/are terminating decimal(s)?

I 299/(32^123)
II 189/(49^99)
III 127/(25^37)

A) I only
B) I and III
C) II and III
D) II only
E) I, II, III
Please explain your method. Also if possible, please share the properties of terminating decimals, if any


I

\(\frac{299}{32^{123}} = \frac{299}{(2^5)^{123}} = \frac{299}{2^{(5*123)}}\)

\(= \frac{299}{2^{(5*123)}} * \frac{5^{(5*123)}}{5^{(5*123)}}\)

\(= \frac{299 * 5^{(5*123)}}{10^{(5*123)}}\) >> Power of 10 in denominator, this is a terminating decimal

II

\(\frac{189}{49^{99}} = \frac{7*27}{7^{198}} = \frac{27}{7^{197}}\) >> This is not a terminating decimal

III

\(\frac{127}{25^{37}} = \frac{127}{5^{74}} = \frac{127}{5^{74}} * \frac{2^{74}}{2^{74}}\)

\(= \frac{127 * 2^{74}}{10^{74}}\) >> Power of 10 in denominator, this is a terminating decimal

Answer = B
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 07 Aug 2014, 21:16
PareshGmat wrote:
alphonsa wrote:
Which of the following is/are terminating decimal(s)?

I 299/(32^123)
II 189/(49^99)
III 127/(25^37)

A) I only
B) I and III
C) II and III
D) II only
E) I, II, III
Please explain your method. Also if possible, please share the properties of terminating decimals, if any


I

\(\frac{299}{32^{123}} = \frac{299}{(2^5)^{123}} = \frac{299}{2^{(5*123)}}\)

\(= \frac{299}{2^{(5*123)}} * \frac{5^{(5*123)}}{5^{(5*123)}}\)

\(= \frac{299 * 5^{(5*123)}}{10^{(5*123)}}\) >> Power of 10 in denominator, this is a terminating decimal

II

\(\frac{189}{49^{99}} = \frac{7*27}{7^{198}} = \frac{27}{7^{197}}\) >> This is not a terminating decimal

III

\(\frac{127}{25^{37}} = \frac{127}{5^{74}} = \frac{127}{5^{74}} * \frac{2^{74}}{2^{74}}\)

\(= \frac{127 * 2^{74}}{10^{74}}\) >> Power of 10 in denominator, this is a terminating decimal

Answer = B


I could solve the problem by using rules of terminating number, but your solution is so interesting. Thanks for sharing!!! + 1 kudos
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 07 Aug 2014, 21:44
vad3tha wrote:
PareshGmat wrote:
alphonsa wrote:
Which of the following is/are terminating decimal(s)?

I 299/(32^123)
II 189/(49^99)
III 127/(25^37)

A) I only
B) I and III
C) II and III
D) II only
E) I, II, III
Please explain your method. Also if possible, please share the properties of terminating decimals, if any


I

\(\frac{299}{32^{123}} = \frac{299}{(2^5)^{123}} = \frac{299}{2^{(5*123)}}\)

\(= \frac{299}{2^{(5*123)}} * \frac{5^{(5*123)}}{5^{(5*123)}}\)

\(= \frac{299 * 5^{(5*123)}}{10^{(5*123)}}\) >> Power of 10 in denominator, this is a terminating decimal

II

\(\frac{189}{49^{99}} = \frac{7*27}{7^{198}} = \frac{27}{7^{197}}\) >> This is not a terminating decimal

III

\(\frac{127}{25^{37}} = \frac{127}{5^{74}} = \frac{127}{5^{74}} * \frac{2^{74}}{2^{74}}\)

\(= \frac{127 * 2^{74}}{10^{74}}\) >> Power of 10 in denominator, this is a terminating decimal

Answer = B


I could solve the problem by using rules of terminating number, but your solution is so interesting. Thanks for sharing!!! + 1 kudos


Thank you so much for calling the solution interesting :)

One thing to add. In such type of problems, just look out for powers of 2 and/or powers of 5 (because they only compose 10) in denominator

For any other number, its not possible (Subject to complete simplification of the term)
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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New post 07 Aug 2014, 23:11
The fraction will have terminating decimal if and only if the denominator of the fraction is of the form (2^n)(5^m).

If you look at the denominators:
1) (32^123) => (2^(5*123))(5^0) => Terminating Decimal
2) (49^99) => Can't be expressed as (2^n)(5^m) => Non terminating Decimal
3) (25^37) => (2^0)(5^(2*37)) => Terminating Decimal

Hence 1 and 3 are terminating decimals => Choice [B]
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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