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# Is \frac{r}{s^2} a terminating decimal? 1. s = 225 2. r = 81

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Retired Moderator
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Is \frac{r}{s^2} a terminating decimal? 1. s = 225 2. r = 81 [#permalink]  06 Nov 2008, 11:46
Is $$\frac{r}{s^2}$$ a terminating decimal?

1. s = 225
2. r = 81

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient
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Re: Terminating Decimal [#permalink]  06 Nov 2008, 12:15
C.
When r = 81, if s = 2 will lead to terminating decimal. If s=21 will lead to non terminating.
When s = 225, if r = 81 will lead to terminating decimal. If r=1 will lead to non terminating.

When s=225, r=81 then
r/s^2 = 9*9/(25*25*9*9) = 0.04*0.04 a terminating decimal.
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Re: Terminating Decimal [#permalink]  06 Nov 2008, 20:03
1
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Is $$\frac{r}{s^2}$$ a terminating decimal?

1. s = 225
2. r = 81

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient

terminating decimal is one in which the denominator has only 2's and 5's

s=225. means 5 ^ 4 X 3 ^ 4 Insuff because we dont know what numerator can be

r = 81 we dont know what the denominator is

use both, we get 5 ^ 4 and 2 ^ 0 in the denominator. Yes it is terminating.
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Re: Terminating Decimal [#permalink]  07 Nov 2008, 06:23
When you say "it only has 2's and 5's in the denominator"

Does that mean prime factors of 2's and 5's?

Let's say the denominator is 50.

Prime factors are 5*5*2, therefore it is terminating?

Also, what does this mean?

s=225. means 5 ^ 4 X 3 ^ 4 Insuff because we dont know what numerator can be

icandy wrote:
Is $$\frac{r}{s^2}$$ a terminating decimal?

1. s = 225
2. r = 81

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient

terminating decimal is one in which the denominator has only 2's and 5's

s=225. means 5 ^ 4 X 3 ^ 4 Insuff because we dont know what numerator can be

r = 81 we dont know what the denominator is

use both, we get 5 ^ 4 and 2 ^ 0 in the denominator. Yes it is terminating.
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Re: Terminating Decimal [#permalink]  07 Nov 2008, 10:57
1
KUDOS
When you say "it only has 2's and 5's in the denominator"

Does that mean prime factors of 2's and 5's?

Let's say the denominator is 50.

Prime factors are 5*5*2, therefore it is terminating?

Also, what does this mean?

s=225. means 5 ^ 4 X 3 ^ 4 Insuff because we dont know what numerator can be

To make it simpler, a fraction will be a terminating decimal when the decimal point terminates (e.g. 1/3 cannot be a terminating decimal as it will be 0.3333333333333 without any termination. However, 9/3 is a terminating decimal as it is 3.0).

With this, in order to find out whether a fraction is a terminating decimal, we need to know the values of numerator as well as denominator. Additionally, if the denominator contains only 2's (e.g. 2, 2^2, 2^10, etc.), only 5's or a combination of 2's and 5's, then the fraction will always be a terminating decimal irrespective of the numerator.

However, from stmt1, we know that denominator = 5^2 * 3^2 and the presence of 3^2 adds to the complexity. We do not know whether the numerator is a multiple of 3^2. If it is, the fraction is a terminating decimal. Else, it is not.

Similarly, from stmt2 only, we do not know what denominator is.

Combining both, we know that numerator is a multiple of 3^2 and additionally, denominator contains only 5^2. Hence, the fraction will be a terminating decimal.
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Re: Terminating Decimal [#permalink]  07 Nov 2008, 11:10
+1 for the both of you. Awesome explanations!
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Re: Terminating Decimal [#permalink]  07 Nov 2008, 13:22
icandy wrote:
Is $$\frac{r}{s^2}$$ a terminating decimal?

1. s = 225
2. r = 81

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient

terminating decimal is one in which the denominator has only 2's and 5's

s=225. means 5 ^ 4 X 3 ^ 4 Insuff because we dont know what numerator can be

r = 81 we dont know what the denominator is

use both, we get 5 ^ 4 and 2 ^ 0 in the denominator. Yes it is terminating.

I think you made a mistake here. 225 = 5^2 x 3^2 (instead of 5^4 x 3^4).
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Re: Terminating Decimal [#permalink]  08 Nov 2008, 08:44
scthakur wrote:
When you say "it only has 2's and 5's in the denominator"

Does that mean prime factors of 2's and 5's?

Let's say the denominator is 50.

Prime factors are 5*5*2, therefore it is terminating?

Also, what does this mean?

s=225. means 5 ^ 4 X 3 ^ 4 Insuff because we dont know what numerator can be

To make it simpler, a fraction will be a terminating decimal when the decimal point terminates (e.g. 1/3 cannot be a terminating decimal as it will be 0.3333333333333 without any termination. However, 9/3 is a terminating decimal as it is 3.0).

With this, in order to find out whether a fraction is a terminating decimal, we need to know the values of numerator as well as denominator. Additionally, if the denominator contains only 2's (e.g. 2, 2^2, 2^10, etc.), only 5's or a combination of 2's and 5's, then the fraction will always be a terminating decimal irrespective of the numerator.

However, from stmt1, we know that denominator = 5^2 * 3^2 and the presence of 3^2 adds to the complexity. We do not know whether the numerator is a multiple of 3^2. If it is, the fraction is a terminating decimal. Else, it is not.

Similarly, from stmt2 only, we do not know what denominator is.

Combining both, we know that numerator is a multiple of 3^2 and additionally, denominator contains only 5^2. Hence, the fraction will be a terminating decimal.

scthakur : how is 81/225^2 = 81/50625 = .0016 a terminating decimal ?

As far as I know, a terminating decimal is one, with numerator as an integer and denominator must be an interger that can be expressed as 2^x * 5^y, where x and y are non-negative integers, that means they can't be zero.

Even though the answer to the above DS problem is C, isn't the answer to the question Is r/s^2, a clear No ?
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Re: Terminating Decimal [#permalink]  08 Nov 2008, 09:02
nganle08 wrote:

I think you made a mistake here. 225 = 5^2 x 3^2 (instead of 5^4 x 3^4).

The denominator is S^2 and NOT S. I made a mistake in writing the den as S.

225 ^2 = 15 ^4 = 5 ^4 X 3 ^4 ; numerator = 3 ^4
Re: Terminating Decimal   [#permalink] 08 Nov 2008, 09:02
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