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Math Expert V
Joined: 02 Sep 2009
Posts: 59012
What is the 25th digit to the right of the decimal point in  [#permalink]

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17 00:00

Difficulty:   5% (low)

Question Stats: 86% (00:53) correct 14% (01:01) wrong based on 1849 sessions

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What is the 25th digit to the right of the decimal point in the decimal form of 6/11?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Practice Questions
Question: 20
Page: 154
Difficulty: 600

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Math Expert V
Joined: 02 Sep 2009
Posts: 59012
Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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5
SOLUTION

What is the 25th digit to the right of the decimal point in the decimal form of 6/11?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

$$\frac{6}{11}=0.5454...$$.

Every odd-numbered digit, to the right of the decimal point, is 5, thus 25th (odd) digit will also be 5.

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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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11
1
an easy way to find 6/11 in decimal terms

expand it by 9

(6*9)/(11*9) = 54/99

note that 54/99 is the fractional representation of circulating (recurring) decimal 0,545454...

___

I dunno how anyone else would of got 0.5454 etc without using a calculator
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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1
6/11=0.54545454 repeating, with 5 occuring at 1,3,5,7 etc. So the 25th digit will be 5.
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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2
C) 5
Same solution as above, but in case the repeating digits are more numerous then see:
1) Number of repeating digits (in this case 2 -> .5454..)
2) Divide the # of the digit you're supposed to view (25th digit) by # of repeats = (25/2)th digit
3) If no remainder, then its 4, if there's a Remainder = 1 then it has to be the next digit or 5, as in this case
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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3
1
6/11 results in a Non-terminating decimal equal to .545454......54
Every even place (2nd,4th,6th,,,,,,,,,24th) digit is 4.
Every Odd place (1,3,5.....25) digit is 5.
Thus thus the odd place digit is 5
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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1
But is there anyway to remember how to calculate 6/11 or 7/11 for that matter?I can't possibly remember all fractions...
Math Expert V
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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Markster wrote:
But is there anyway to remember how to calculate 6/11 or 7/11 for that matter?I can't possibly remember all fractions...

Try to simply divide 6 by 11 to get 0.5454...

Else notice that 6/11=54/99=0.5454... Check Converting Decimals to Fractions here: math-number-theory-88376.html
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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Bunuel wrote:
What is the 25th digit to the right of the decimal point in the decimal form of 6/11?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Practice Questions
Question: 20
Page: 154
Difficulty: 600

multiply 9 on both 6 & 11 then u will get 54/99...

whatever 2 digit number divided 99 will have the same digits repeat... say here 0.5454....

25th digit will be 5....
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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Just want to be clear here. Are we expected to know non terminating fractions or expected to do the math of 6/11? I could do, but I felt it would be math intensive and wanted to avoid that. The rule that seemed relevant from the number theory link is that terminating fractions have a denominator that fits the 2^m*5^n formula. What's the best way to approach this, and what math is unnecessary? Please advise.
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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I divided 6/11 and got to .545, after that I realized the pattern would repeat itself. As there's just two different digits the even decimals will be 5 and the odds will be 4. Thus 5 is the answer and C as the answer choice.
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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1
Hi braveally,

The GMAT cannot realistically expect you to work through 25 digits one at a time, so this question MUST have a pattern to it.

There are certain pieces of information that you're expected to have memorized (formulas, math and grammar rules, etc.) and other types of information that would be beneficial to have memorized (fraction-to-decimal conversions, commonly tested 'patterns', etc.). It shouldn't be difficult to memorize the decimal equivalent of a set of fractions, BUT if you don't want to, then that's fine - the work will just take a little longer.

So the question in this case becomes "how quickly can you mathematically determine that 6/11 = .5454 repeating?" You were concerned that the work would be time-sensitive, so you didn't attempt it. But I'm curious about how quickly you COULD have done it. Grab a timer and find out. I bet it takes 10-15 seconds (at most) until you have the proof that you need.

GMAT assassins aren't born, they're made,
Rich
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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Bunuel wrote:
What is the 25th digit to the right of the decimal point in the decimal form of 6/11?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

To solve this question, we first have to use some long division. This long division allows us to get 6/11 in decimal form, which is 0.545454… where “54” is repeating.

We can see that the 1st, 3rd, 5th (and so on) digit to the right of the decimal point is a 5 and that the 2nd, 4th, 6th (and so on) digit to the right of the decimal point is a 4. In other words, each odd-positioned digit is a 5, and each even-positioned digit is a 4.

Because we are being asked about the 25th digit to the right of the decimal point and we see that 25 is odd, we know that the 25th digit is a 5.

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Manager  Joined: 20 Mar 2015
Posts: 54
Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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Aximili85 wrote:
C) 5
Same solution as above, but in case the repeating digits are more numerous then see:
1) Number of repeating digits (in this case 2 -> .5454..)
2) Divide the # of the digit you're supposed to view (25th digit) by # of repeats = (25/2)th digit
3) If no remainder, then its 4, if there's a Remainder = 1 then it has to be the next digit or 5, as in this case

If remainder is 2 then, 2+1?
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GRE 1: Q169 V154 Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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4
2
Another way of solving this one=>
6/11=> multiply 9 to both he numerator and denominator => 54/99
hence the decimal expansion would be 0.54545454545454
Clearly the 57th digit is 5=> Smash that C

Why did we write 6/11 as 54/99 => if denominator and numerator have same number of digits and if the denominator can be written as 10^n-1 ie 9,99,999,9999 etc then we need not carry any division process . The result would be given by the numerator

E.g => 69/99 = 0.69696969...
3/9=0.33333...
158/999=0.158158158...
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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6/11=0.545454545454.....................

We have to only look at the digit at odd's place because 25 is odd.
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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EMPOWERgmatRichC wrote:
Hi braveally,

The GMAT cannot realistically expect you to work through 25 digits one at a time, so this question MUST have a pattern to it.

There are certain pieces of information that you're expected to have memorized (formulas, math and grammar rules, etc.) and other types of information that would be beneficial to have memorized (fraction-to-decimal conversions, commonly tested 'patterns', etc.). It shouldn't be difficult to memorize the decimal equivalent of a set of fractions, BUT if you don't want to, then that's fine - the work will just take a little longer.

So the question in this case becomes "how quickly can you mathematically determine that 6/11 = .5454 repeating?" You were concerned that the work would be time-sensitive, so you didn't attempt it. But I'm curious about how quickly you COULD have done it. Grab a timer and find out. I bet it takes 10-15 seconds (at most) until you have the proof that you need.

GMAT assassins aren't born, they're made,
Rich

Hi Rich,

Are you recommending that 6/11 is useful to memorize directly?
Of course we can't memorize everything, but is this a common/repeated fraction that would be useful to know directly?
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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1
Hi omegan3,

I actually recommend to all of my Clients that they memorize the fractions 1/2, 1/3, 1/4..... 1/11. That's just 10 fractions - and I'll bet that you know at least half of them already. Once you know what 1/11 is, it's not very hard to figure out what 6/11 is.

GMAT assassins aren't born, they're made,
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Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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EMPOWERgmatRichC wrote:
Hi omegan3,

I actually recommend to all of my Clients that they memorize the fractions 1/2, 1/3, 1/4..... 1/11. That's just 10 fractions - and I'll bet that you know at least half of them already. Once you know what 1/11 is, it's not very hard to figure out what 6/11 is.

GMAT assassins aren't born, they're made,
Rich

Thank you, Rich, for sharing your knowledge.
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GMAT 1: 560 Q39 V28 GMAT 2: 670 Q48 V34 Re: What is the 25th digit to the right of the decimal point in  [#permalink]

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Easy way to remember 1/11 till 10/11 is:

1/11 --> The repeating pattern is 10-1=09 so 0.09090909
2/11 --> The repeating pattern is 20-2=18 so 0.18181818
3/11 --> The repeating pattern is 30-3=27 so 0.27272727
...
10/11 --> The repeating pattern is 100-10=90 so 0.90909090
_________________ Re: What is the 25th digit to the right of the decimal point in   [#permalink] 31 Aug 2018, 02:01

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