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non-repeating decimal

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by sachin_yadav » Mon Jun 10, 2013 10:22 am
I came across the following question on MGMAT

If the fraction d were converted into a decimal, would there be more than 3 nonzero digits to the right of the decimal point?
(1) The denominator of d is exactly 8 times the numerator of d.
(2) If d were converted into a decimal, d would be a non-repeating decimal.

OA is A

It was easy for me to spot the answer but got confused after reading about non-repeating decimal. https://www.mathwithlarry.com/lessons/lesson091.htm

non-repeating decimals:- These are decimal numbers that go on forever, but do not follow a pattern at all. An example might be 0.48376922026985321.... If there is no pattern, and no end in sight, it means that the number is a non-repeating decimal.

If I keep this in my mind, then statement 2 can be considered as the correct answer i.e d would be a non-repeating decimal (decimal numbers that go on forever, but do not follow a pattern at all).
So, there would there be more than 3 nonzero digits to the right of the decimal point.

Please help me in understanding this theory. I am a bit confused now.

Thanks & Regards
Sachin
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by aaggar7 » Tue Jun 11, 2013 12:48 am
A) alone is sufficient.

As let Numerator = x then denominator = 8x.

Hence,D=x/8x or 0.125.

B) As per B,decimals are non-repeating,which means the digits after decimal are unique and do not repeat in a pattern.But fails to comment on the number of digits.

Additionally,Repeating decimals are obtained when a number is divided by 9 or any multiple of 9 (9,99,999...)

for ex 2/9 = .2222..
Also known as non-terminating decimals.

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by Atekihcan » Tue Jun 11, 2013 3:33 am
aaggar7 wrote:Additionally,Repeating decimals are obtained when a number is divided by 9 or any multiple of 9 (9,99,999...)
That's a true statement but not an exhaustive one.

Repeating decimals are obtained when the denominator of a rational number is not entirely made up of 2 and/or 5.

So, any number when divided by 3, 7, 11, 13, ... etc or their multiples will always result in a repeating decimal.

For example,
  • 1/3 = 0.333333333....
    1/7 = 0.142857142857...
    1/11 = 0.09090909...
I've underlined the repeating part from which you can see that often the repeating part can be so large that apparently we may not identify the pattern.

To clarify things three type of decimals are
  • Terminating Decimal : Which do not go on forever. For example, 0.1 or 0.345 or 0.002005 etc. The denominator of the fractional form of these decimals will have only 2 and 5 as the prime factors.
    Repeating Decimals : Which goes on forever but there is a pattern. Already discussed the examples and how to identify them.
    Non-repeating Decimals : Which goes on forever but there is no pattern. These are basically the decimal representation of irrational numbers (whereas the previous two types are of rational numbers) like √2, √3, etc and π. For example, √2 = 1.4142135623... or π = 3.1415926535... goes on forever without any pattern.
There may be many more type of decimals depending upon how you define it, but these three are common types.


Now, to answer your query regarding statement 2, I think statement 2 is indeed sufficient because by definition a non-repeating decimal will have infinite number of non-zero digits to the right of the decimal point. If it has a finite number of non-zero digits to the right of the decimal point, it will become a terminating decimal.

I feel the problem means to say "If d were converted into a decimal, d would be a terminating decimal." as statement 2.

Hope that helps.
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by sachin_yadav » Wed Jun 12, 2013 9:09 am
Atekihcan wrote: Now, to answer your query regarding statement 2, I think statement 2 is indeed sufficient because by definition a non-repeating decimal will have infinite number of non-zero digits to the right of the decimal point. If it has a finite number of non-zero digits to the right of the decimal point, it will become a terminating decimal.

I feel the problem means to say "If d were converted into a decimal, d would be a terminating decimal." as statement 2.

Hope that helps.
Thanks Atekihcan.

I got your explanation. At first, i didn't read the language of the statement (2) attentively, but now it's clear. Thanks :D

Regards
Sachin
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by ravi_uppal2004 » Wed Jun 12, 2013 9:39 am
why is OA " A " then can someone explain ?
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by sachin_yadav » Wed Jun 12, 2013 1:09 pm
ravi_uppal2004 wrote:why is OA " A " then can someone explain ?
Try to understand the following or plug the values in the numerator and denominator you will come to know why OA is A:-

A) alone is sufficient.

As let Numerator = x then denominator = 8x.

Hence,D=x/8x or 0.125.

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by kul512 » Thu Jun 13, 2013 8:28 pm
Ravi_uppal...
I think Aaggar7's statement Now, to answer your query regarding statement 2, I think statement 2 is indeed sufficient has confused you. I think there is some error in this sentence. Sentence 2 is not sufficient because number can be 2.235 also, which is a non-repeating decimal but terminates at 3 decimal digits.[/quote]
Sometimes there is very fine line between right and wrong: perspective.
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