If n denotes a number to the left of 0 on the number line such that the square of n is less than 1/100, then the reciprocal of n must be

A. less than -10
B. between -1 and -1/10
C. between -1/10 and 0
D. between 0 and -1/10
E. greater than 10





Why the answer is (A), not (E). Thank you.

Tough one to follow. I feel like my mind just got off a roller coaster after that one.

"If n denotes a number to the left of 0 on the number line" = n is negative

the square of n is less than 1/100. so what must the square of n look like? 1/101, 1/1000 2/3000 etc...

what kind of negative numbers squared will give us that result? n could be -1/11, -1/100 etc

n is negative so n < -1/10...i think

reciprocal of n = 1/n therefore 1/n < -10

Phew...my mind is still in loops.

You may want to wait for a MGMAT instructor to take this one, before you run with my explanation. I got the correct answer, but feel a little unsure

Just to start, it can't be E because the problem tells us n is negative. The reciprocal of a number maintains its original sign - so the reciprocal of -2 is -1/2 and the reciprocal of 1/3 is 3. So if n starts as a negative number, then the reciprocal also must be negative.

It seems that you may have gotten yourself to the idea of a number bigger than 10 or smaller than -10 (as opposed to between -10 and 10) and just messed up the sign somewhere along the way - so go back and check through your work. If you have questions on how to get the specific value regardless on the pos / neg (that is, why it isn't B, C, or D), come back and let us know.

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

Thank you Stacey: My calculation was if n^2 < 1/(10^2), then -1/10 < n < 1/10. Therefore, the reciprocal of n is less than -10 AND greater than 10. Now, I understand why it's not (E)....n is negative and its reciprocal must be negative.

No problem!

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

Given that n < 0 ...A
Also, n^2 < 1/100
i.e. |n| < 1/10
i.e. -1/10 < n < 1/10
using A from above,
-1/10 < n < 0
taking reciprocal, (applies since both are on same side of 0)
-10 > n [editor: this should be 1/n]
i.e. less than -10

why is the reciprocal taken on one side only ?

-1/10 < n

=> taking reciprocal will give us ::

-10 > 1 /n ??

please let me know what I am missing here ?

you are correct. i've edited that post.

Why does the sign of the inequality switch when you are taking a reciprocal?

i.e.
-1/10 < n
-10 > 1/n ?

Isn't the reciprocal the number you multiply by to get you to 1? I wasn't able to find the section on this in the GMAT prep books. Thanks.

I have the same confusion at to why does the inequality flip when taking the reciprocal. Can someone pls. explain?

Dividing or multiplying an inequality by a negative number will always flip the direction of the inequality. It can be best understood with an example

5 > -3

Now, multiplying both sides by a negative number, say -1 gives
LHS = -5
RHS = 3

So, -5 ? 3 ; we need to flip the inequality ie -5 < 3, for it to be true

Guys,

we are all clear upto this point -1/10 < n

There are two ways to find the reciprocal of n from this point onwards.

1.) since we know that n is a negative number , which makes both the sides of inequality -1/10 < n. Negative ! Taking reciprocal on both the sides of the inequality makes the inquality flip ie -10 > 1/n OR 1/n is less than -10.

This analogous to 1< 2 < 3.... .Now, taking reciprocal of all the elements in the inequality gives us-------> 1 > 1/2 >1/3. This manipulation is also valid for negative numbers, will leave that for you guys to test . Some food for thought ;)

Second approach from this point onwards -1/10 < n

Multiply both the sides of the given inequality by -10 we get ,

1 > -10 n ( notice that inequality sign flips since we are multiplying by -10 )

multiply 1 > -10 n once again by 1/n
1/n < -10 ( notice that the inequality sign flips again since n is a negative value making 1/n a negative quantity)

GMAT math is so much fun ... hehehe !

I like geetesht's response. Good work!

_________________
Ben Ku
Instructor
ManhattanGMAT

Your appreciation is my inspiration Ben , Thank you !

You're welcome!

_________________
Ben Ku
Instructor
ManhattanGMAT