If the 5 numbers listed above are denoted

 
gmatophobia I think there will be a "-" sign in front of the 1st option A. thanks

­This makes sense to me except that I don't understand why each of the numbers aren't positive at the end (bc square of a negative should be positive). By that logic, you would have them all in reverse order; .2<.33<1<3<5, so I would have put x4=3 (answer A).
Can you please explain why these remain negative? 

chetan2u would you please be able to explain the steps of how you simplified these? For example, with A) I can't figure out on my own how you ended up with the negative square root of 3 as the final.

Hey there.

Squaring negatives is a little bit tricky.-3^2 is bigger than -2^2
You can factorize the negative out and square the positives then multiply the answer by the negative.

Looking forward to a fruitful discussion.

I don’t understand this approach but I will be glad if you shed some light on it. How you got negative square root 3

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Start with 1/3 inside the root in the numerator.

1/3 = 3^-1

The square root raises this to the 1/2, so the exponent of 3 above becomes -1*1/2 =

-1/2. So numerator is:

3^-(1/2)



Next, the denominator. 1/3 =

3^-1

So the division can be recast as:

3^-(1/2)/3^-1

Subtract the exponent of the denominator from the exponent of the numerator:

-(1/2)-(-1) = 1/2

So the resulting expression is:

-3^(1/2)

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On analysing the problem after my mock test I can think of 3 different ways to solve it -

1. Squaring each number... (I used this method in my mock)
Concept - The smaller the negative number, you are squaring, the larger its answer will be (when it becomes positive on squaring)... I confirmed this while I was giving the mock exam... (-1/2)^2 > (-1/4)^2... also... (-2)^2 > (-1)^2...

Using this concept below are the steps I carried out -
1. I squared each of the 5 numbers given - and in the order given received values of --> 3, 1/5, 1/3, 5, 1...
2. Ordering them from smallest to largest --> 1/5, 1/3, 1, 3, 5...
3. After rearranging them in step 2, We need the 4th largest number, this means on squaring, we need to find the 4th smallest number or the second largest number (applying the concept mentioned above) --> 1/3...
4. 1/3 is received by squaring the 3rd value, listed above... which is option C...

Happy that I got this right, but it took me around 4-5 mins to get the correct answer...

2. Estimation

This is pretty quick if you can keep your mind clear, and do basic calculations and estimations... Since I get such questions wrong, my mind kind of froze during the mock, and i ended up using the longer and surer way of getting the answer..

1. Using following values for estimation - √3 = 1.7, √5 = 2.2... we get approx values as below -
For the 1st value mentioned above --> -3/√3 = -3/1.7 = approx -1.8...
For the 2nd value mentioned above --> - √5/5 = approx -0.44
For the 3rd value mentioned above --> - √3/3 = approx -0.57
For the 4th value mentioned above --> - 5/√5 = approx -2.2
5th value is -1...

Now arranging them in the order of smallest to largest, we see x4 = -0.57, which is the 3rd value, which option C

3. The method used by Chetan2u above

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Harsha

Bunuel , I know this is an old question but shouldn't there be a negative sign in front of option A? Just to avoid any confusion for other people in the future. Thank you!

Yes. Edited. Thank you!

Responding to a pm:

We need to second greatest number which means that it will have the second smallest absolute value. If we make all these terms positive, we are looking for the second smallest number.

So we are comparing
\( I. \frac{\sqrt{\frac{1}{3}}}{\frac{1}{3}}, \quad II. \frac{\frac{1}{5}}{\sqrt{\frac{1}{5}}}, \quad III. \frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}}, \quad IV. \frac{\sqrt{\frac{1}{5}}}{\frac{1}{5}}, \quad V. 1\)

Between 0 and 1, sqrt(x) > x so numbers I and IV above are greater than 1 and II and III are less than 1.
So we have to compare II and III which are \( \frac{1}{\sqrt{5}} = \frac{1}{2.2}, \frac{1}{\sqrt{3}} = \frac{1}{1.7} \).

So II is the smallest and III is second smallest.

Answer (C)

Here is another discussion on a similar question and a discussion on comparing fractions in general.
https://www.youtube.com/watch?v=PQMxKCld65M
https://youtu.be/6AYRmxkAlrE