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# GMAT Diagnostic Test Question 8

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GMAT Diagnostic Test Question 8 [#permalink]  06 Jun 2009, 20:59
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GMAT Diagnostic Test Question 8
Field: number properties
Difficulty: 650

If $$m$$ and $$n$$ are positive integers, what is the smallest possible value of integer $$m$$ if $$\frac{m}{n}$$ = 0.3636363636...?

A. 3
B. 4
C. 7
D. 13
E. 22
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Last edited by Bunuel on 06 Oct 2013, 23:06, edited 2 times in total.
Updated
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Re: GMAT Diagnostic Test Question 8 [#permalink]  08 Jul 2009, 06:47
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Explanation:

We are dealing with a repeating decimal in this question. It's helpful to know that there's a way to write these kinds of decimals as a fraction. For example, the repeating decimal 0.444444444(4) may be written as $$\frac{4}{9}$$. So, $$\frac{5}{9}$$, $$\frac{7}{9}$$ and $$\frac{8}{9}$$ will all be repeating decimals. You might check it in your calculator. In order to make two decimal points repeat, you have to divide the two digit number by 99. For example, $$\frac{23}{99} = 0.232323232323(23)$$. Similarly, $$\frac{36}{99} = \frac{4}{11} = 0.36363636(36)$$. Now it's clear that the minimum value of $$m = 4$$.
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Last edited by bb on 28 Sep 2013, 11:12, edited 1 time in total.
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Re: GMAT Diagnostic Test Question 8 [#permalink]  07 Feb 2010, 13:08
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It was an easy question if you know the following concept.

let S = .363636 so on

i.e S = .36 + .0036 + .000036 till infinity.

This is basically and infinite GP series with first term as .36 and ratio as .01

Sum of infinite GP is = $$\frac{a}{1-r}$$

S = $$\frac{.36}{1-.01}$$
S = $$\frac{36}{99}$$
S = $$\frac{4}{11}$$

Thus minimum value will be 4 when n = 11

The above method is basically used to calculate the fraction from the repeating decimal using GP series.
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Re: GMAT Diagnostic Test Question 8 [#permalink]  15 Jul 2009, 14:08
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Alternate Solution:
 Rating:

In case you did not know the formula for the repeating decimal (most probably did not), there is another approach to solving this question - backsolving. This is not a typical backsolving question, however, since both of the variables are unknown and we have to make some assumptions to get to the solution.

1. Looking at the repeating decimal - 0.36.... - the ratio between m and n has to be slightly less than 1:3.
2. Let's run through the answer choices:
A. 3 - the number that's slightly less than 3*3 is 8. $$\frac{3}{8} = 0.375$$. Does not work
B. 4 - the number that's slightly less than 3*4 is 11. $$\frac{4}{11} = 0.3636$$. Works!
C. 7
D. 13
E. 22

We could continue going through answer choices C, D, and E, but the question asks us for the smallest possible value of m, and 4 is the smallest of the ones that work (even if multiple do) so there is no value to check others.

-
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Re: GMAT Diagnostic Test Question 8 [#permalink]  17 Feb 2010, 01:00
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This is a easier way......

consider m/n = x.

x= .36(bar)
100x = 36.36(bar)

100x-x = 36.

i.e., 99x=36

x=4/11.
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Re: GMAT Diagnostic Test Question 8 [#permalink]  18 Jul 2009, 13:42
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actually 7 devided by 11 is 0.63636363
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Re: GMAT Diagnostic Test Question 8 [#permalink]  05 May 2010, 02:08
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Expert's post
dzyubam wrote:
I'm not sure if there is a simple way to obtain 0.99999999(9). $$\frac{9}{9}$$ should result in 0.9999(9), but $$\frac{9}{9}$$ = 1.
thanatoz wrote:
So is there a way to get 0.99999999~?

Note: it's not tested on GMAT.

$$0.(9)=1$$ - long known mathematical equality.

Proof: $$1=3*\frac{1}{3}=3*0.(3)=0.(9)$$.
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Re: GMAT Diagnostic Test Question 8 [#permalink]  03 Aug 2009, 04:20
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bb wrote:
GMAT Diagnostic Test Question 8
Field: number properties
Difficulty: 700
 Rating:

If $$m$$ and $$n$$ are integers, what is the smallest possible value of integer $$m$$ if $$\frac{m}{n}$$ = 0.3636363636...?

A. 3
B. 4
C. 7
D. 13
E. 22

In the original question (which I downloaded from download/file.php?id=9039) , it doesn't say that n is an integer... that mean that m and n could be 3 and 33/4 respectivelly. In that case A would be the correct choice.
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Re: GMAT Diagnostic Test Question 8 [#permalink]  17 Jul 2009, 09:32
Hi again,

the answer choices in the PDF are different from those of the post:

A. 7
B. 13
C. 19
D. 23
E. 29

Thanks for the test =)
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Founder
Affiliations: AS - Gold, HH-Diamond
Joined: 04 Dec 2002
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GMAT 1: 750 Q49 V42
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Kudos [?]: 14054 [0], given: 3876

Re: GMAT Diagnostic Test Question 8 [#permalink]  17 Jul 2009, 10:54
Expert's post
saruba wrote:
Hi again,

the answer choices in the PDF are different from those of the post:

A. 7
B. 13
C. 19
D. 23
E. 29

Thanks for the test =)

Thanks! Version control is starting to take its toll on me....
Fixed - new version of document was uploaded.
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Re: GMAT Diagnostic Test Question 8 [#permalink]  18 Jul 2009, 11:48
So is there a way to calculate the answer for the 0.636363636 version of the question!

Cause I now get it about 0.363636 but am still stuck how to get the answer to 0.6363636 version

Founder
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Re: GMAT Diagnostic Test Question 8 [#permalink]  18 Jul 2009, 12:59
Expert's post
So is there a way to calculate the answer for the 0.636363636 version of the question!

Cause I now get it about 0.363636 but am still stuck how to get the answer to 0.6363636 version

0.63 version was a typo and will not work. The problem should read 0.36. I have updated the download file.
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Founder
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Re: GMAT Diagnostic Test Question 8 [#permalink]  18 Jul 2009, 14:02
Expert's post
actually 7 devided by 11 is 0.63636363

You did not need my help afterall! For some reason I assumed 63 was a prime and not divisible

Using the rule Dzyubam outlined - $$\frac{63}{99} = \frac{7}{11}$$
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Re: GMAT Diagnostic Test Question 8 [#permalink]  03 Aug 2009, 04:38
Thanks, +1. I've updated the PDF.
DFG5150 wrote:
In the original question (which I downloaded from download/file.php?id=9039) , it doesn't say that n is an integer... that mean that m and n could be 3 and 33/4 respectivelly. In that case A would be the correct choice.

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Re: GMAT Diagnostic Test Question 8 [#permalink]  03 Oct 2009, 07:29
hi, my solution is backsolving with picking numbers:

First of all, since the division is <|1|, then |m|<|n|

I started with 3 being divided by many numbers:
3/4 = 0,7xxx (not the answer)
3/5=0,6xxx (uh oh)
3/7 =0,4xxx (nope)

Do you see a pattern here? It is kind of logical, but just to be sure you noticed that: the greater the dividend, the smaller is the result of the division. Going on:

3/8 =0,37x (almost!)
3/9 = 3 we "crossed the line", move on to the next alternative

(x means that I stopped the division, even though I knew I could continue dividing)

since 4/8 = 0.5, let's start from here

4/9 = 0,4x (no)
4/10 = 0,4 (no)
4/11 = 0,3636363636363636363636363636363636363636363636
(ok, I did not the division until shown above :D)

It took some time... number properties always takes more than 2 minutes each for me... I try to overcome doing faster the others...
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Re: GMAT Diagnostic Test Question 8 [#permalink]  19 Dec 2009, 12:31
I totally started this wrong. I thought this is a reminder question. It's actually a fraction question.
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Re: GMAT Diagnostic Test Question 8 [#permalink]  14 Jan 2010, 19:48
diogoguitarrista wrote:
hi, my solution is backsolving with picking numbers:

First of all, since the division is <|1|, then |m|<|n|

I started with 3 being divided by many numbers:
3/4 = 0,7xxx (not the answer)
3/5=0,6xxx (uh oh)
3/7 =0,4xxx (nope)

Do you see a pattern here? It is kind of logical, but just to be sure you noticed that: the greater the dividend, the smaller is the result of the division. Going on:

3/8 =0,37x (almost!)
3/9 = 3 we "crossed the line", move on to the next alternative

(x means that I stopped the division, even though I knew I could continue dividing)

since 4/8 = 0.5, let's start from here

4/9 = 0,4x (no)
4/10 = 0,4 (no)
4/11 = 0,3636363636363636363636363636363636363636363636
(ok, I did not the division until shown above :D)

It took some time... number properties always takes more than 2 minutes each for me... I try to overcome doing faster the others...

Thanks for sharing your process. I just looked at fraction/decimal/% equivalents last night and was pissed that I didn't remember this one! Nonetheless, I appreciate the lesson learned about repeating decimals. I'm definitely adding it to my list of hints/tricks.
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Re: GMAT Diagnostic Test Question 8 [#permalink]  17 Jan 2010, 05:08
if you divide any no by 11(except multiple of 11 ) you will get repeating decimal.

so just divide answers by 11.

3/11=0.2727272727

4/11=0.363636363636
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Re: GMAT Diagnostic Test Question 8 [#permalink]  24 Jan 2010, 10:04
quite tough
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Re: GMAT Diagnostic Test Question 8 [#permalink]  16 Feb 2010, 19:24
dzyubam wrote:
Explanation:
 Rating:

We are dealing with a repeating decimal in this question. It's helpful to know that there's a way to write these kinds of decimals as a fraction. For example, the repeating decimal 0.444444444(4) may be written as $$\frac{4}{9}$$. So, $$\frac{5}{9}$$, $$\frac{7}{9}$$ and $$\frac{8}{9}$$ will all be repeating decimals. You might check it in your calculator. In order to make two decimal points repeat, you have to divide the two digit number by 99. For example, $$\frac{23}{99} = 0.232323232323(23)$$. Similarly, $$\frac{36}{99} = \frac{4}{11} = 0.36363636(36)$$. Now it's clear that the minimum value of $$m = 4$$.

is this a rule or something that any divisor when divided by 99 or a factor of it (9 or 11) will end up giving a repeating decimal? Was this rule taught in high school...I don't remember. Where can we brush up these special rules. Any book to practice from?

thanks
Re: GMAT Diagnostic Test Question 8   [#permalink] 16 Feb 2010, 19:24

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# GMAT Diagnostic Test Question 8

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