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Math Expert V
Joined: 02 Sep 2009
Posts: 57155
D01-08  [#permalink]

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18 00:00

Difficulty:   45% (medium)

Question Stats: 60% (01:15) correct 40% (01:48) wrong based on 141 sessions

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$$m$$ 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

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 57155
Re D01-08  [#permalink]

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8
Official Solution:

$$m$$ 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

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 $$m = 4$$.

Alternate Solution:

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.

Answer: B
_________________
Intern  Joined: 01 Jun 2014
Posts: 44
Concentration: General Management, International Business
GMAT 1: 660 Q48 V34 Re: D01-08  [#permalink]

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1
When you look at the question , it does strike that 0.3636.. is 4 * 0.0909.. . I think from there on it becomes very simple.
Intern  Joined: 27 Nov 2014
Posts: 44
D01-08  [#permalink]

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2
1
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
------------------------------------------------------------------------------------------------------------------------------------------------

it is easy to memorize that
n will be 11 because if we choose any other value smaller than 11 it will not give 2 repeating non terminating values in return.

for eg. 1/3 = .3333...(1 repeating decimal)
1/7 = .142857 142857. ... (6 repeating decimals)
1/9 = .11111... (1 repeating decimals)
1/11= .090909 (2 repeating decimals) --> this is wat we willl pick

Now look at the ans choices
a. 3 upon dividing with 11 will not yield .363636... out
b. 4 upon dividing with 11 will yield .363636 ... correct

Hence B ans!

Regards
SG
Intern  Joined: 11 Aug 2012
Posts: 2
Re: D01-08  [#permalink]

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18
5
Bunuel wrote:
Official Solution:

$$m$$ 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

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 $$m = 4$$.

Answer: B

Alternatively,
$$\frac{m}{n}$$ = 0.363636... - Eq 1
100$$\frac{m}{n}$$ = 36.363636... - Eq 2
(Multiplied by 100 as we have 2 repeating decimals, in case of 3 repeating decimals we'd multiply by 1000 and so on)

Subtract equation1 from 2
99$$\frac{m}{n}$$ = 36
$$\frac{m}{n}$$ = $$\frac{4}{11}$$
Smallest possible value of m is 4.

It can be used in the similar way for any number of repeating decimals.
Senior Manager  B
Joined: 23 Sep 2015
Posts: 371
Location: France
GMAT 1: 690 Q47 V38 GMAT 2: 700 Q48 V38 WE: Real Estate (Mutual Funds and Brokerage)
Re: D01-08  [#permalink]

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It is easy to spot the answer once we know that 1/11 yields 0,0909
_________________
Manager  Joined: 11 Oct 2013
Posts: 102
Concentration: Marketing, General Management
GMAT 1: 600 Q41 V31 Re D01-08  [#permalink]

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I think this is a high-quality question and I agree with explanation.
_________________
Its not over..
Manager  Joined: 11 Oct 2013
Posts: 102
Concentration: Marketing, General Management
GMAT 1: 600 Q41 V31 Re: D01-08  [#permalink]

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james2329 wrote:
Bunuel wrote:
Official Solution:

$$m$$ 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

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 $$m = 4$$.

Answer: B

Alternatively,
$$\frac{m}{n}$$ = 0.363636... - Eq 1
100$$\frac{m}{n}$$ = 36.363636... - Eq 2
(Multiplied by 100 as we have 2 repeating decimals, in case of 3 repeating decimals we'd multiply by 1000 and so on)

Subtract equation1 from 2
99$$\frac{m}{n}$$ = 36
$$\frac{m}{n}$$ = $$\frac{4}{11}$$
Smallest possible value of m is 4.

It can be used in the similar way for any number of repeating decimals.

This is exactly how Bunuel's formula is derived.
_________________
Its not over..
Intern  Joined: 16 Sep 2015
Posts: 3
Re: D01-08  [#permalink]

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1
1. Looking at the repeating decimal - 0.36.... - the ratio between m and n has to be slightly less than 1:3.

Could you please explain how did we arrive at this when we know 0.36> 0.33 how is it that the ratio is slightly less than 1:3.

Could you please clarify this point.

Thanks Senior Manager  Joined: 31 Mar 2016
Posts: 375
Location: India
Concentration: Operations, Finance
GMAT 1: 670 Q48 V34 GPA: 3.8
WE: Operations (Commercial Banking)
Re D01-08  [#permalink]

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I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 11 Oct 2017
Posts: 11
Re: D01-08  [#permalink]

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1
all 'repeating decimals" can be written as a fraction with a denominator that is made up of only 9s i.e. 9, 99, 999. below is an explanation how.
but before that, an interesting fact to know is that when the repeating digits are 0 and 1, then the nominator is ALWAYS 1 and the denominator is ALWAYS made up of 9s.
i.e. 0.11111.... = 1/9
0.010101.... = 1/99
0.001001001..... = 1/999 i.e
note that for every additional 9 in the denominator in addition to the first 9, a 0 is added before and after the 1s. except the first 9. (see above)
now, knowing this simple fact, you can tackle all such questions.

0.363636.... = 0.36 + 0.0036 + 0.000036 +......
= 36 (0.01 + 0.0001 + 0.000001 + .....)
= 36 (0.010101......) = 36 (1/99) = 36/99 = 4/11
Manager  S
Joined: 26 Feb 2018
Posts: 51
Location: India
GMAT 1: 640 Q45 V34 GPA: 3.9
WE: Web Development (Computer Software)
Re D01-08  [#permalink]

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I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 12 Nov 2017
Posts: 18
Re: D01-08  [#permalink]

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The most simple way to solve it:
0.36363636 here 36 repeats.
For such a number to convert to fraction :
We know 0.36 = 36/100, but since 36 repeats deduct 1 from the denominator. In this case 100-1= 99
So, this fraction is 36/99.
Now we need the smallest possible value of integer m.
Therefore, simplify 36/99 = 4/11
So the answer is m=4.
Intern  B
Joined: 25 Apr 2018
Posts: 12
Location: India
Concentration: Strategy, Finance
GMAT 1: 600 Q40 V33 Re: D01-08  [#permalink]

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x=0.3636.. (1)
10x=3.6363
100x=36.3636 (2)

Subtracting (2) & (1) we get,

99x=36
x=4/11
therefore m/n=4/11

Smallest value of m is 4.
_________________
Gmat aspirant
Target score 750 ! Re: D01-08   [#permalink] 03 Aug 2019, 09:38
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