Is x/y a terminating decimal?

Actually \(4=2^2*5^0\), thus \(\frac{3}{4}=0.75\) is a terminating decimal. If the denominator has only 2-s and/or 5-s then the fraction always will be a terminating decimal (in this case it also doesn't matter whether the fraction is reduced or not).


Is x/y a terminating decimal?

(1) x is a multiple of 2. Not sufficient since no info about y.
(2) y is a multiple of 3. Not sufficient since no info about x.

(1)+(2) If \(x=2\) and \(y=3\), then \(\frac{x}{y}=\frac{2}{3}=0.666...\) which is NOT a terminating decimal, but if \(x=6\) and \(y=12\), then \(\frac{x}{y}=0.5\) which is a terminating decimal. Not sufficient.

Answer: E.

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 \(2*5^3\). 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 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.)

Hope it helps.

I did not know (or notice) the red part albeit I read the theory in the gmatclub math book

Thanks

CONCEPT: A ration x/y will be Terminating when in its least form the denominator has NO PRIME FACTOR OTHER THAN 2 and 5

Question: Is x/y a terminating decimal?

Statement 1: x is a multiple of 2
No information about y hence
NOT SUFFICIENT

Statement 2: y is a multiple of 3
No information about y hence
NOT SUFFICIENT

Combining the two statements
using both statements as well we have no idea what will be the least form of x/y hence
NOT SUFFICIENT

Answer: Option E

­2K1/3K2 is the form. 2/3 is any way an non-terminating decimal. So the values of K1 and K2 are going to say whether x/y is terminating or not. So we can't say with clarity whether x/y is terminationg or not. Hence E

So by looking at this I automatically thought if denominator is a factor of 3 there will not be a terminating decimal 1/3, 2/3, 4/3/5/3 etc are not terminating. But I dug deeper in the question.
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(1) so if we do not know what Y is we cannot determine this answer so insufficient​

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( 3) We dont know what X is so we cannot determine the answer.

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( 1+2) if if X is 2/3--> .666, however what if X = 6 and Y = 3, then it will be 2 a terminating decimal, still insufficient

E