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

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Current Student
Joined: 22 Jul 2014
Posts: 123
Concentration: General Management, Finance
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Which of the following is/are terminating decimal(s)?  [#permalink]

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06 Aug 2014, 09:08
1
3
00:00

Difficulty:

35% (medium)

Question Stats:

67% (01:10) correct 33% (01:26) wrong based on 193 sessions

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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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06 Aug 2014, 10:24
10
6
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.

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

Hope this helps!:)
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##### General Discussion
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Joined: 18 Jul 2013
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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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.

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

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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06 Aug 2014, 14:08
3
Current Student
Joined: 22 Jul 2014
Posts: 123
Concentration: General Management, Finance
GMAT 1: 670 Q48 V34
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Re: Which of the following is/are terminating decimal(s)?  [#permalink]

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

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07 Aug 2014, 00:41
1
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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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

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

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

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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

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

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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07 Aug 2014, 21:44
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

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

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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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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11 Nov 2017, 11:17
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Re: Which of the following is/are terminating decimal(s)? &nbs [#permalink] 11 Nov 2017, 11:17
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