What is the 99th digit after the decimal point in the

Question

What is the 99th digit after the decimal point in the decimal expansion of 2/9 + 3/11?

A. 1
 B. 2
 C. 4
 D. 7
 E. 9

Bunuel · Answer

\frac{2}{9} + \frac{3}{11}=\frac{22}{99} + \frac{27}{99}=\frac{49}{99}=0.494949... . So, 99th (odd placed after the decimal) digit will be 4. Answer: C. For more on converting a recurring decimal to fraction check this: math-number-theory-88376.html Hope it helps.

Carcass · Answer

I have not added OA because I hoped in a more long discussion.

Thanks

PareshGmat · Answer

\frac{2}{9} = 0.222222222...........2

\frac{3}{11}= 0.272727.............. 27

Addition (Does not have carries)

\frac{2}{9} + \frac{3}{11} = 0.4949494949....... 49

Odd's have 4

99th would have 4

Answer = C

0ld · Answer

\frac{2}{9}  = 0.22222.... ==> 99th digit is  2

\frac{3}{11}  = 0.27272727.... ==> Every odd digit is 2. So, 99th digit will be 2.

2 +2 = 4 ( Ans C)

rhine29388 · Answer

\frac{2}{9}  +  \frac{3}{11}  =  \frac{49}{99}

\frac{49}{99}  = 0.49494949..............

so the digits are getting repeated in order of two digits 49

so the 99 th digit is 4

correct answer - C

hero_with_1000_faces · Answer

HI what do you mean by 99th digit could you explain, I understand the theory of repeating no.'s but what do you mean by 99th Digit ?

Bunuel · Answer

We need 99th digit  after the decimal point  of 0  .  494949...

The first digit is 4, the second digit is 9, the third digit is 4 again, the fourth digit is 9 again, ... 99th (odd placed after the decimal) digit will be 4.

JeffTargetTestPrep · Answer

2/9 + 3/11 = 22/99 + 27/99 = 49/99 = 0.494949….

SInce each odd-positioned digit to the right of the decimal point has a value of 4, the 99th digit is 4.

Alternate Solution:

We see that 2/9 = 0.222… and 3/11 = 0.272727…

If we add the two decimal expressions, we see that we obtain 0.494949… Since all the odd-positioned digit to the right of the decimal point has a value of of 4, the 99th digit is 4.

Answer: C

analystpgh · Answer

I was able to get to 49/99, but do you just need to memorize that anything divided by 99 is itself repeated? ~~And why is the 9 the odd place? It looks to me that the first (odd place) after after the decimal is the 4 and the 9 would be the 2nd place so even?~~ Edit: realize that 9 is the even place and 4 is the odd

Rhitib · Answer

This is an aeasy sum
2/9+3/11=49/99

Now break 99 into int's prime factorization 99=3*3*11

According to the cyclicity rule of numbers, the number with the highest cyclicity or higher in value will determine the last digit, here 11 and has a cyclicity 2.

49/11= 4.454545... goes on. Here, 98th term will 5, then plus 1= 99th term will be 4.

Answer is C

egmat · Answer

Hi analystpgh,

Good news on both counts. First, your edit is spot on: the block is  49 , so the 1st digit is  4 , the 2nd is  9 , the 3rd is  4  again, and so on. Odd positions are  4 , even positions are  9 , and since  99  is odd, the  99 th digit is  4 . Nothing more to fix there.

Now the real question:  you don't need to memorize the /99 rule  - you can derive it in seconds, which is safer than trusting memory.

Why x/99 repeats the block

Take your result and let  d = 0.494949... . The block is two digits long, so multiply by  100  to shift it one full block:

-  100 d = 49.494949...
- d = 0.494949...
- Subtract:  99 d =  49 , so d =  49 / 99 .

Run that backward and you see  why  49 / 99  = 0.494949...: dividing by  99  is exactly what reproduces the two-digit block. The  100  you multiplied by is what forces a  two-digit  repeat - and  100  pairs with the  99 .

The pattern behind it

The number of 9s in the denominator tells you the block length:

-  7 / 9  = 0.7777... - one  9 , so a  1-digit  block.
-  7 / 99  = 0.070707... - two  9 s, so a  2-digit  block (write  7  as "07").
-  7 / 999  = 0.007007... - three  9 s, so a  3-digit  block.

So  49 / 99  gives the  2 -digit block  49  repeating - no memorizing required, just the shift-and-subtract idea. Once you trust that, the only remaining job is matching the position (odd/even) to the right digit, which you already nailed.

Answer: C

What is the 99th digit after the decimal point in the

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