If n denotes a number to the left of 0 on the number line such that

Question

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

CrushGMAT · Answer

A video explanation can be found here:
https://www.youtube.com/watch?v=lG5iq9xcsws

First note that if n^2 < 1/100, the range of values of n would be

-1/10 < n < 1/10

But that range is further restricted – since we’re told n is to the left of 0, then

-1/10 < n < 0

(Note that C is a trap answer)

Thinking about reciprocals, we move from negative fractions to negative whole numbers. (The correct answer may become easier to visualize if you draw out the number line.)

Answer A

gmexamtaker1 · Answer

Gladiator59 chetan2u Bunuel VeritasKarishma
Hello could you provide your assistance regarding my question? It has to do with inequalities.

lets say we have -1<10n (n<0) when we divide by n I know that it should be like this -1/n>10 and then 1/n < -10
But what I want to ask is when we divide by n, which we know is a negative number, why can't we proceed like this, -1<10n (n<0) 1/n>10 since -1/(a negative number) = positive number

KarishmaB · Answer

If we know -1 < 10n (n is a negative number so the right hand side is negative too. n could take values such as -1/20, -1/50 etc) To process it easily, we can just divide both sides by 10 (n gets separated out) -1/10 < n Since n < 0, we get -1/10 < n < 0 If instead, we divide both sides by n, we get -1/n > 10 How did you get 1/n > 10?

gmexamtaker1 · Answer

Well, to be frank with you I was really confused between handling inequalities when there is a variable and when we have a specific number what I did was to consider n<0 but then imagined in my head that n is eg.-2 so i both changed the sign of inequality and the sign of the fraction to positive, which is wrong so instead of just dividing -1<10n by n(n<0) and getting -1/n>10 I would divide by n and simultaneously think that n is a number like eg.-2 so I would also change the sign of -1/n to 1/n which is wrong . thank you  VeritasKarishma  for your reply.

KarishmaB · Answer

Yes, you don't have to change the sign of the fraction. Note that -1/n is a positive number because n is negative. When you make it 1/n, you are making it negative again.

SeriousSLOBO · Answer

Why exactly do you multiply by -10/n? What is the reasoning behind this decision. Thanks.

Bunuel · Answer

The questions asks:  ... the reciprocal of n  must be . So, we multiply by -10/n to get the the reciprocal of n (1/n) in the inequality and get the answer.

Hope it's clear.

avigutman · Answer

Video solution from Quant Reasoning starts at 0:28 Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=-PJmTbC1vLQ

kadamhari825 · Answer

hey  Bunuel ,

can you please elaborate why we did not flip the sign after multiplying by-10/n?

Thanks in advance.

Bunuel · Answer

I think this is explained in the solution, no?

"As  n<0 , then  -\frac{10}{n}>0 , and we don't have to switch signs".

BrentGMATPrepNow · Answer

STRATEGY : As with all GMAT Problem Solving questions, we should immediately ask ourselves,   Can I use the answer choices to my advantage?   
In this case, the answer is a resounding yes, because we can just find a value of n that satisfies the given information, and then take the reciprocal of n. 
From here, we should give ourselves about 20 seconds to identify a faster approach. 
In this case, we can probably perform some algebra, but that will likely take longer than testing a value

We're told that   the square of n is less than 1/100  
In other words: n² < 1/100
So, for example, it could be the case that n = 1/20, since (1/20)² = 1/400, and 1/400 < 1/100

Important: The question also tells us that   n denotes a number  to the left of 0  on the number line    
In other words, n is NEGATIVE, which means n = 1/20 doesn't satisfy this condition.
However, n = -1/20 meets all of the criteria since -1/20 is negative, and (-1/20)² = 1/400, and 1/400 < 1/100

If n = -1/20, then the reciprocal of n = -20/1 = -20
Only answer choice A works.

Kinshook · Answer

Asked: 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

n < 0
n^2 < 1/100
n > -1/10
1/n < -10

IMO A

erwr43534 · Answer

Hi Bunuel, I don't understand what you did after figuring out that -1/10 < n < 0. I would think that I only need to get the reciprocal so I would do -10 < 1/n < 0.

Do signs change?

Bunuel · Answer

This is not correct.

We have  -\frac{1}{10}<n<0 .

Say n = -1/100, so 1/n = -100:  -\frac{1}{10}<-\frac{1}{100}<0 . Would it be correct to write -10 < -100 < 0? No.

We multiply  -\frac{1}{10}<n<0  by -10/n to get the the reciprocal of n (1/n) in the inequality and get the answer.

Hope it's clear.

deepak.brijendra · Answer

Kindly explain how come square of a number is less than ZERO.....

Bunuel · Answer

The square of a number is always greater than or equal to 0. Can you specify which post you are referring to where it suggests that the square of a number is less than 0?

erwr43534 · Answer

Hi Bunuel!

Thank you for your response. If you multiple n by -10/n, wouldn't it just become -10?

Bunuel · Answer

Yes:

(-\frac{1}{10})*(-\frac{10}{n})<n*(-\frac{10}{n})<0*(-\frac{10}{n})

\frac{1}{n}<-10<0 .

thegmatmaster · Answer

Hello Bunuel, thank you for your continued support. I’m curious about how you realized that multiplying by -10/n would give you the reciprocal of n. Is this a math trick or rule, or did you notice that this algebraic manipulation would leave 1/n in the inequality? Thank you!

Bunuel · Answer

We have  -\frac{1}{10}<n<0  and want to evaluate 1/n. How can we do this? If we multiply by -10/n, we'll get 1/n on the left side, and n is eliminated in the middle. So, it's a simple algebraic trick.

If n denotes a number to the left of 0 on the number line such that

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