What is the 25th digit to the right of the decimal point in

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...


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I dunno how anyone else would of got 0.5454 etc without using a calculator

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.

Answer: C.
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6/11=0.54545454 repeating, with 5 occuring at 1,3,5,7 etc. So the 25th digit will be 5.

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

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
Hence Answer C.

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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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....

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

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.

Answer C.
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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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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?

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

If we use LONG DIVISION and divide 6 by 11, we see a pattern emerge.

We get: 6/11 = 0.5454....
So, the 1st digit to the right of the decimal point is 5
The 2nd digit to the right of the decimal point is 4
The 3rd digit to the right of the decimal point is 5
The 4th digit to the right of the decimal point is 4
and so on...

Every ODD digit (1st, 3rd, 5th, etc) is 5
Every EVEN digit (2nd, 4th, 6th etc) is 4

So, the 25th digit will be 5

Answer: C

Cheers,
Brent

Here, first find out by basic manual division method, 6/11 will come out to be approx 0.545 (I determined upto 3 decimals), then I observed the pattern of repetition after every 2 decimals. I reconfirmed if it is recurring decimal by using the concept of prime factorization of denominator of the fractions.

As per the rule, if the denominator has only 2, and/or 5 as the prime factors, then such a fraction or decimal is terminating. If not, (like in this case), fraction is nonterminating. It has to be recurring (for non -recurring would mean irrational no). Thus, decimal must be 0.545454, etc. Now it is easy to observe the pattern at even and odd positions of the decimal and get the answer.

Alternatively, if you can remember some basic fractions byheart would also help like 1/9 ,1/7, 1/11, 1/15, and other std fractions,etc.

1/11 is 9.09090909...

So we can easily find out 6/11 from above value and identify the pattern. Although question is of 600 level, but I think it is a good question of more than 650 level.

You can just do long division and quickly see that the answer is 0.545454 repeating