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Director  Joined: 25 Apr 2012
Posts: 654
Location: India
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If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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47 00:00

Difficulty:   95% (hard)

Question Stats: 37% (02:10) correct 63% (01:51) wrong based on 354 sessions

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If $$x^{2} < x$$ and x is written as a terminating decimal, does x have a nonzero hundredths digit?

(1) 10x is not an integer.

(2) 100x is an integer.

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Math Expert V
Joined: 02 Sep 2009
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If $$x^{2} < x$$ and x is written as a terminating decimal, does x have a nonzero hundredths digit?

$$x^{2} < x$$ means that $$0<x<1$$.

(1) 10x is not an integer. This only means that x is NOT 0.1, 0.2, ...., 0.9. Now, if x=0.01, then the answer is YES but if x=0.001, then the answer is NO. Not sufficient.

(2) 100x is an integer. This means that x is 0.01, 0.02, ..., 0.10, ..., 0.99. The hundredths digit could be 0 as well as nonzero. Not sufficient.

(1)+(2) From (1) 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80 and 0.90 are excluded, thus from (2) only those decimals are left which have a nonzero hundredths digit: 0.01, 0.02, ..., 0.09, 0.11, ..., 0.19, ..., 0.21, ..., 0.99. Sufficient.

Hope it's clear.
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Manager  Joined: 03 Jul 2013
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Schools: ISB '17 (A), IIMC (A)
GMAT 1: 660 Q48 V32 Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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Hi. Suppose i choose two numbers 0.23 and 0.03 for both of them x^2<x

Both number satisfy both the statements:

x=0.23 10x=2.3, not an integer
x=0.23 100X=23 is an integer

x=0.03 10x=0.3 is not an integer
x=0.03 100x=3 is an integer

in case 1 100 digit is 2 and in case 2 100 digit is 0.

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Math Expert V
Joined: 02 Sep 2009
Posts: 59086
Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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3
Hi. Suppose i choose two numbers 0.23 and 0.03 for both of them x^2<x

Both number satisfy both the statements:

x=0.23 10x=2.3, not an integer
x=0.23 100X=23 is an integer

x=0.03 10x=0.3 is not an integer
x=0.03 100x=3 is an integer

in case 1 100 digit is 2 and in case 2 100 digit is 0.

In both cases the hundredths digit is 3: 0.23 and 0.03.

1234.567

1 - THOUSANDS
2 - HUNDREDS
3 - TENS
4 - UNITS
. - decimal point
5 - TENTHS
6 - HUNDREDTHS
7 - THOUSANDTHS

Hope it helps.
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Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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Hi Bunuel, could u elaborate the explanation for this one? As soon as I saw that "100x is an integer" , figures like .99, .45, .01 come to mind and thus 100x always has a zero hundredth digit.

example: if x=.abcd
100x is an integer then 100x= ab.cd
for ab.cd to be an integer, cd has to be zero. therefore, hundredth place of x is a zero integer.

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Math Expert V
Joined: 02 Sep 2009
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If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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shreyagmat wrote:
Hi Bunuel, could u elaborate the explanation for this one? As soon as I saw that "100x is an integer" , figures like .99, .45, .01 come to mind and thus 100x always has a zero hundredth digit.

example: if x=.abcd
100x is an integer then 100x= ab.cd
for ab.cd to be an integer, cd has to be zero. therefore, hundredth place of x is a zero integer.

How is hundredths digit of 0.99, 0.45, or 0.01 zeo?
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Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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say x is .45
then statement says that 100x is an integer ( of the form xx.0)
if thats true then x cant be .455 or .555, it has to be something like .45 or .10 ( since .45*100 = 45)

I dont even know where im wrong.
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Math Expert V
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Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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shreyagmat wrote:
say x is .45
then statement says that 100x is an integer ( of the form xx.0)
if thats true then x cant be .455 or .555, it has to be something like .45 or .10 ( since .45*100 = 45)

I dont even know where im wrong.

x CANNOT be 0.1 because (1) say that 10x is NOT an integer and if x=0.1, then 10x=1=integer. Anyway, what's your question?
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Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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I just realized the mistake i was making.
i was counting a unit digit after decimal point and thought that B is sufficient. But now i realized its not the case.

Here is what i understand now:
.abc . question : is b zero or nonzero?

statement 1: a.b is not integer.
x could be :: .001 or .01
b can be zero or non zero if c is non-zero (.011 , .001) ; but if c is zero then b has to be non zero in order for 10x to not be an integer
Insuff
statement 2: c is zero

(1+2)
when c is zero , b is non zero

Is my logic correct this time?
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If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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2
WoundedTiger wrote:
If $$x^{2} < x$$ and x is written as a terminating decimal, does x have a nonzero hundredths digit?

(1) 10x is not an integer.

(2) 100x is an integer.

Here's how I did it:

I represented x as x = .abc (I didn't go further because the question didn't ask for more than tens/hundredths digits).

1) 10x is NOT an integer. This means that a.bc is NOT an integer. Therefore, b and c cannot BOTH be zero; otherwise 10x would be an integer. Not sufficient.

2) 100x IS an integer. This means that ab.c is an integer, and therefore that c = 0. This just gives us x = .ab0 - no information about b. Not sufficient.

Together though, we know that c = 0. However, we also know that b and c cannot BOTH be zero. Therefore, b has to be nonzero.

Manager  Joined: 05 Jun 2014
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GMAT 1: 630 Q42 V35 Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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Bunuel plz explain where am i going wrong, if I select two numbers i.e 0.105 and 0.125.
Both when multiplied by 10 become 10.5 and 12.5 (non - integers)
both when multiplied by 100 become integers 105 and 125.
But in one the hundreth digit and 0 and the other one its 2..What am i missing??? Thanks in advance !
Math Expert V
Joined: 02 Sep 2009
Posts: 59086
Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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Bunuel plz explain where am i going wrong, if I select two numbers i.e 0.105 and 0.125.
Both when multiplied by 10 become 10.5 and 12.5 (non - integers)
both when multiplied by 100 become integers 105 and 125.
But in one the hundreth digit and 0 and the other one its 2..What am i missing??? Thanks in advance !

Check the red parts.

10x = 1.05 and 1.25, receptively.
100x = 10.5 and 12.5, receptively.
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Posts: 1382
Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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WoundedTiger wrote:
If $$x^{2} < x$$ and x is written as a terminating decimal, does x have a nonzero hundredths digit?

(1) 10x is not an integer.

(2) 100x is an integer.

x is a +ve fraction , in decimal form ... 0.abcd... , is b not 0

from 1

a.bcd... is not integer ... b or c or d could be any digit , i.e b = 0 but c is not or b a no zero digit and c,d = 0 ... insuff

from 2

ab.cd.. = ab thus c,d...etc = 0 ... no idea about b itself .. insuff

both

the decimal representation of x is in the form 0.ab and a.b is not integer thus b aint zero ... suff

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Re: If x^2 < x and x is written as a terminating decimal,does x  [#permalink]

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