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# If 0 < x < 1, is it possible to write x as a terminating dec

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If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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24 Jun 2013, 02:20
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If 0 < x < 1, is it possible to write x as a terminating decimal?

(1) 24x is an integer.

(2) 28x is an integer.
[Reveal] Spoiler: OA

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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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24 Jun 2013, 02:32
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If 0 < x < 1, is it possible to write x as a terminating decimal?

(1) 24x is an integer --> $$24x=m$$, where m an integer --> $$x=\frac{m}{24}=\frac{m}{2^3*3}$$, If m is a multiple of 3, then the answer is YES, else the answer is NO. Not sufficient.

(2) 28x is an integer --> $$28x=n$$, where n an integer --> $$x=\frac{n}{28}=\frac{n}{2^2*7}$$, If n is a multiple of 7, then the answer is YES, else the answer is NO. Not sufficient.

(1)+(2) $$x=\frac{m}{2^3*3}=\frac{n}{2^2*7}$$ --> $$\frac{m}{n}=\frac{2*3}{7}$$ --> m IS a multiple of 3 (as well as n IS multiple of 7). Sufficient.

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 $$2*5^2$$. 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 the 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.

Questions testing this concept:
does-the-decimal-equivalent-of-p-q-where-p-and-q-are-89566.html
any-decimal-that-has-only-a-finite-number-of-nonzero-digits-101964.html
if-a-b-c-d-and-e-are-integers-and-p-2-a3-b-and-q-2-c3-d5-e-is-p-q-a-terminating-decimal-125789.html
700-question-94641.html
is-r-s2-is-a-terminating-decimal-91360.html
pl-explain-89566.html
which-of-the-following-fractions-88937.html

Hope it helps.
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Re: If 0 < x < 1, is it possible to write x as a terminating [#permalink]

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27 Jun 2013, 22:37
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If 0 < x < 1, is it possible to write x as a terminating decimal?
(1) 24x is an integer.
(2) 28x is an integer.

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

Statement 1- If 24x is an integer than x can take the following values
1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 1/24
Some values of x can be reduced to a terminating decimal (1/2, 1/4, 1/8), while few can not be (1/3,1/6,1/12, 1/24)
Insufficient

Statement 2- If 28x is an integer than x can take the following values
1/2, 1/4, 1/7, 1/14, 1/28
Some values of x can be reduced to a terminating decimal (1/2, 1/4), while few can not be (1/7, 1/14, 1/28)
Insufficient

Statement 1& 2- If both 24x & 28x are integers than x can take the following values
1/2, 1/4
Both of these values of x can be reduced to a terminating decimal
Sufficient

Ans C.

Hope the explanation will help many.
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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28 Jul 2013, 11:18
Can someone explain me what is the meaning of terminating decimal.
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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28 Jul 2013, 11:25
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trafficspinners wrote:
Can someone explain me what is the meaning of terminating decimal.

A decimal number that has digits that do not go on forever.

Examples:

0.25 (it has two decimal digits)
0.123456789 (it has nine decimal digits)

In contrast a Recurring Decimal has digits that go on forever

Example of a Recurring Decimal: 1/3 = 0.333... (the 3 repeats forever)
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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20 Aug 2013, 04:16
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Rock750 wrote:
If 0 < x < 1, is it possible to write x as a terminating decimal?

(1) 24x is an integer.

(2) 28x is an integer.

I have a bit of difficulty in understanding the intended meaning of "is it possible" part of the question.

The answer can be yes and ofcourse no, but just that there is a possibility that the answer could be yes confuses me a bit. Had the question been framed like this " is x a terminating decimal?", then it would have been clearer. The use of the term "possible" makes it just a bit ambiguous.

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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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20 Aug 2013, 05:57
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agourav wrote:
Rock750 wrote:
If 0 < x < 1, is it possible to write x as a terminating decimal?

(1) 24x is an integer.

(2) 28x is an integer.

I have a bit of difficulty in understanding the intended meaning of "is it possible" part of the question.

The answer can be yes and ofcourse no, but just that there is a possibility that the answer could be yes confuses me a bit. Had the question been framed like this " is x a terminating decimal?", then it would have been clearer. The use of the term "possible" makes it just a bit ambiguous.

The question basically asks: if x is written as a decimal will it be a terminating decimal?

Hope it's clear.
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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06 Sep 2013, 11:58
since the question asks "is it possible", wouldn't the answer be D since .5 is a terminating decimal and 24*.5=12, and 28*.5=24?
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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07 Sep 2013, 05:30
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thebloke wrote:
since the question asks "is it possible", wouldn't the answer be D since .5 is a terminating decimal and 24*.5=12, and 28*.5=24?

You misinterpret the question. The question asks: if x is written as a decimal will it be a terminating decimal? Thus the correct answer is C, not D.
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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05 Jan 2014, 09:48
we know that x is a proper positive fraction. we need to check whether x has powers of 5 or 2 in the denominator or not.

1. 24(x)=INT ---> $$x=Int/24$$ if our integer is 3 then x is can be written as a terminating decimal otherwise x will be a non-terminating decimal

2. 28(x)=INT ----> same story here if our int is 7 then x can be written as a terminating decimal, otherwise x will be a non-terminating decimal

1+2 $$Int/3(2^3)=Int/7(2^2)$$ -----> 7(4)Int=8(3)Int the expression has to be equal on both sides thus on the right hand side we need a 7 and on the right hand side we need a 3 and a two. We now know that our integer a terminating decimal because we can get rid of both 7 and 3 in the denominator.

C.

Hope it helps.
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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03 Mar 2015, 03:34
fameatop wrote:
If 0 < x < 1, is it possible to write x as a terminating decimal?
(1) 24x is an integer.
(2) 28x is an integer.

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

Statement 1- If 24x is an integer than x can take the following values
1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 1/24
Some values of x can be reduced to a terminating decimal (1/2, 1/4, 1/8), while few can not be (1/3,1/6,1/12, 1/24)
Insufficient

Statement 2- If 28x is an integer than x can take the following values
1/2, 1/4, 1/7, 1/14, 1/28
Some values of x can be reduced to a terminating decimal (1/2, 1/4), while few can not be (1/7, 1/14, 1/28)
Insufficient

Statement 1& 2- If both 24x & 28x are integers than x can take the following values
1/2, 1/4
Both of these values of x can be reduced to a terminating decimal
Sufficient

Ans C.

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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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13 May 2015, 15:46
thebloke wrote:
since the question asks "is it possible", wouldn't the answer be D since .5 is a terminating decimal and 24*.5=12, and 28*.5=24?

i agree with this. The way it is phrased, it should be D. I understand how the answer C is achieved, but I don't think this question is worded well....

The fact that a terminating decimal is possible should be enough. The only way, in my mind that C is correct is if the question asks "is X a terminating decimal?".
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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20 May 2016, 11:45
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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20 May 2016, 13:35
Rock750 wrote:
If 0 < x < 1, is it possible to write x as a terminating decimal?

(1) 24x is an integer.

(2) 28x is an integer.

Given information :- 0 < x < 1 (x is a fraction between 0 and 1)

Question asked:- x as a terminating decimal?

x can only be written as terminating decimal when the denominator can be written in the form of 2^n or 5^n

(1) 24x is an integer

x can be any factor of 24
1, 2, 4, 3, 6, 8, 12, 24

So x may or may not be a terminating decimal.

(2) 28x is an integer

x can be any factor of 28
1, 2, 4, 7, 14, 28

So x may or may not be a terminating decimal

Combining both statements, only 4 is common between these two. Hence, combining the statements we get one digit that could be the denominator- 4.

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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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21 May 2016, 03:48
Bunuel
I am sorry
I can not get the combining statements.
We do not need to prove that n is a multiple of 7 ?
another question regarding the denominator, is it enough to be 2s or 5s to terminate X as decimal or 2s * 5s is a must.
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Re: If 0 < x < 1, is it possible to write x as a terminating dec [#permalink]

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21 May 2016, 03:59
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hatemnag wrote:
Bunuel
I am sorry
I can not get the combining statements.
We do not need to prove that n is a multiple of 7 ?
another question regarding the denominator, is it enough to be 2s or 5s to terminate X as decimal or 2s * 5s is a must.

From (1) we have that $$x=\frac{m}{24}=\frac{m}{2^3*3}$$. If m is a multiple of 3, then 3 in the denominator will be reduced and x will be a terminating decimal.

Similarly, from (2) we have that $$x=\frac{n}{28}=\frac{n}{2^2*7}$$. If n is a multiple of 7, then 7 in the denominator will be reduced and x will be a terminating decimal.

The answer to your other question is yes, if a fraction has only 2's or 5's in the denominator it'll terminate.

Check Terminating and Recurring Decimals Problems in our Special Questions Directory.

Hope it helps.
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Re: If 0 < x < 1, is it possible to write x as a terminating dec   [#permalink] 21 May 2016, 03:59
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