Can the instructors please help to solve this methodically. Thanks.

This is what I did:
n^2 < 1/100
Therefore:

-1/100 < n^2 < 1/100
-1/10 < n < 1/10
Taking the right side and cross multiplying, I get:
10 < 1/n

Taking the left side
-1/10 < n
1/10 > -n (multiplying both sides by -1)
1/n > -10 (cross multiplying)

Hence I get answer E

your mistake is in the last step here:



first of all, that's not what "cross multiplying" does; there's no way that "cross multiplying" would move n from the numerator (where it's located in the next-to-last step) into the denominator (where it's located at the end).
"cross multiplying" always eliminates all denominators.

second, and much more importantly, you can't perform “cross multiplication” on inequalities, anyway.
"cross multiplication" actually involves multiplication by the denominators of the fractions that are present -- so it won't work if one of those denominators happens to be negative. (your use of "cross multiplication" ignores the very important issue of whether you have to turn around the ">"/"<" sign.)

--

to get from your next-to-last to your last step, you have to perform two operations:

1/10 > -n (starting inequality)

MULTIPLY BY 10 --> this doesn't change the ">" sign
gives
1 > -10n

DIVIDE BY N --> in this instance, n is negative, so you have to turn the inequality sign around!
gives
1/n < -10

there you go.

by the way, you don't have to consider positive values of n at all, since the information in the problem's stem guarantees that n is negative.