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0.888, sq. root (0.888) and (0.888)^2 - In-equality

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by gmattesttaker2 » Wed Feb 19, 2014 6:51 pm
Hello,

Can you please tell me how to solve this:

If x = 0.888, y = sq. root (0.888) and z = (0.888)^2 then which of the following is true?

(A) x <y <z
(B) x <z <y
(C) y <x <z
(D) y <z <x
(E) z <x <y

OA: E

After looking at the right answer choice I was thinking that for any number x between 0 and 1, the above in-equality should hold true since sq. root x would be greater than x which would in-turn be greater than the x^2. Is this correct?

Also, is there a different way to solve this kind of problem?

Thanks a lot,
Sri
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by Brent@GMATPrepNow » Wed Feb 19, 2014 7:23 pm
gmattesttaker2 wrote:Hello,

Can you please tell me how to solve this:

If x = 0.888, y = √(0.888) and z = (0.888)² then which of the following is true?

(A) x < y <z
(B) x <z <y
(C) y <x <z
(D) y <z <x
(E) z < x <y

OA: E

After looking at the right answer choice I was thinking that for any number x between 0 and 1, the above in-equality should hold true since sq. root x would be greater than x which would in-turn be greater than the x^2. Is this correct?

Also, is there a different way to solve this kind of problem?

Thanks a lot,
Sri
You're absolutely right, Sri.

If 0 < k < 1, then: k² < k < √k

Since 0 < 0.888 < 1, then: (0.888)² < 0.888 < √(0.888)
This means that [spoiler]z < x < y[/spoiler]

Answer: E

Cheers,
Brent
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by [email protected] » Thu Feb 20, 2014 12:58 am
Hi Sri,

Your logic is correct. This problem solving question is built on a couple of Number Properties, ones that specifically deal with positive fractions.

Any positive fraction between 0 and 1, when squared, GETS SMALLER.
Any positive fraction between 0 and 1, when square-rooted, GETS BIGGER.

With those two factoids, you can easily answer the given question.

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by Brent@GMATPrepNow » Thu Feb 20, 2014 8:16 am
We can also take what Rich said and expand it to include other roots and powers to get the following rule:

If 0 < k < 1, then ... k� < k³ < k² < k < √k < cuberoot(k) <fourthroot(k) <...

Cheers,
Brent
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by Abhishek009 » Thu Feb 20, 2014 9:34 am
gmattesttaker2 wrote:If x = 0.888, y = sq. root (0.888) and z = (0.888)^2 then which of the following is true?
X = 888/1000

y = √ 888 / 1000 => √ x

z = (888/1000)² =>x²


Now the problem gets easier for us , we have defined both y and z in terms of x

Thus we have -

X , √ x and x²


The follow the form -

If 0 < x < 1, then: x² < x < √x

The answer will obviously be (E)
Abhishek
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by Matt@VeritasPrep » Thu Feb 20, 2014 2:34 pm
gmattesttaker2 wrote:Hello,

Can you please tell me how to solve this:

If x = 0.888, y = sq. root (0.888) and z = (0.888)^2 then which of the following is true?

(A) x <y <z
(B) x <z <y
(C) y <x <z
(D) y <z <x
(E) z <x <y
One last note here: if you're stuck, try setting up an inequality and approximating if you have to to solve it.

For instance, say I'm comparing .888 and .888².

If I think .888² > .888, I can write that, then test it.

.888² > .888

divide both sides by .888 ...

.888 > 1

What!? Whoops, that didn't work. So now I know that .888 > .888², because assuming the opposite gave me a false result.

This sort of approach is REALLY helpful when you have to deal with messier speculative inequalities (such as "Which is bigger, √5^(√7) or √7^(√5)?", to give one I remember from a number theory test).
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