1. 99x is an integer
This means the denominator of x is a factor of 99
So x can be \(\frac{a}{99}\), where a is an integer from 1 to 98, That is x is any of \(\frac{1}{99}, \frac{2}{99}, \frac{3}{99},...... \frac{98}{99}, \)
If x is \(\frac{1}{99}....till....\frac{9}{99}\), the tenth digit is 0 => \(\frac{1}{99}=0.010101...\)
If x is \(\frac{10}{99}\), the tenth digit is 1, as \(\frac{10*1}{99}=10*0.010101...=0.10101...\)
Insufficient

2. 81x is an integer
This means the denominator of x has to be a factor of 81
So x can be \(\frac{a}{81}\), where a is an integer from 1 to 80, That is x is any of \(\frac{1}{81}, \frac{2}{81},\frac{3}{81},......\frac{80}{81}, \)
If x is \(\frac{1}{81}....till ....\frac{8}{81}\), the tenth digit is 0 => \(\frac{1}{81}=0.012345...\)
If x is \(\frac{27}{81}=\frac{1}{3}\), the tenth digit is 3, as \(\frac{1}{3}=0.3333...\)
Insufficient

Combined
x has to have a denominator equal to or less than the HCF(99,81) or 9
Thus the value of x can be 1/9, 2/9, 1/3, 2/3, 8/9 etc.
If the denominator is 3 or 9 and even if we take the least value of numerator as 1, the least value of fraction is 1/9=0.10101...
Thus,the tens digit will never be 0

C

Hi chetan
I always learn from your robust solution but this one is confusing a little bit. I don't know if it's because of the formatting not appearing well or my misreading it.

Pls check out my solution, see how i got C and help me point out the error in my process bcos i don't understand how i concluded differently that the tenth digit of x must always be non zero while you got that it must always be zero.

THIS IS A YES/NO DS QUESTION. is the info in (1) and (2) SUFFICIENT to answer no or yes with 100% certainty? that is the question.

For (1), x could be 1/3 or 1/9 --- then answer is NO, tenth digit of x is not zero
x could be either 1/11, 1/33, or 1/99 ---- then answer is YES tenth digit of x is zero
(1) is NOT SUFFICIENT

For (2) x could be 1/3 or 1/9..... we reply NO tenth digit of x is not zero
x could be 1/27 or 1/81 ... we reply YES tenth digit of x is zero
(2) is NOT SUFFICIENT

Combining (1) and (2) we see the intersection between (1) and (2) is 1/3,1/9 which belongs to a "NO, tenth digit of x is not zero". BOTH ARE SUFFICIENT

Kindly advice

Hi

Sorry as there were a lot of formatting issues. Edited the solution.
You are correct in your solution except there are other values too that x can take. Example : 2/9,5/9,2/3 etc
We consider the least value which is 1/9=0.10101..

Kudos to you

chetan2u

I complicated the solution by taking LCM and not HCF. Can you please explain why we need to take HCF here?


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