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

Question

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

smyarga · Accepted Answer

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!:)

smyarga · Answer

Try to solve these problems (in this order): which-of-the-following-fractions-has-a-decimal-equivalent-159322.html which-of-the-following-is-a-terminating-decimal-when-139050.html which-of-the-following-has-a-decimal-equivalent-that-is-a-161892.html non-terminating-decimal-81024.html a-terminating-decimal-is-defined-as-a-decimal-that-has-a-94641.html is-r-s-2-a-terminating-decimal-91360.html if-a-b-c-d-and-e-are-integers-and-p-2-a3-b-and-q-2-c3-125789.html another-terminating-decimal-125790.html

smyarga · Answer

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.

PareshGmat · Answer

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

oss198 · Answer

Great explanation, i forgot the concept of terminating decimals.

alphonsa · Answer

Thank you so much :-D

I ve done the problems mentioned in all the links :-D

Is there a particular post for number properties question of difficulty 600- 700 and 700+

vad3tha · Answer

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

PareshGmat · Answer

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)

meghaswaroop · Answer

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

bumpbot · Answer

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

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