If 0

Question

If  0 < x < 1 , which of the following expressions is greatest?

(A)  \frac{1}{\sqrt{x}}

(B)  \sqrt{x}

(C)  \frac{1}{π}  x

(D)  x^3

(E)  x^4

Source:  Nova GMAT

BrentGMATPrepNow · Accepted Answer

Since  \sqrt{0.25}=0.5 , a good number to test is  x = 0.25

We get: 
(A)  \frac{1}{\sqrt{0.25}}=\frac{1}{0.5}=2

(B)  \sqrt{0.25}=0.5

(C)  \frac{1}{π}(0.25)≈\frac{0.25}{3.14}=  some number less than 1

(D)  0.25^3=(\frac{1}{4})^3=\frac{1}{64}

(E)  0.25^4=(\frac{1}{4})^4=\frac{1}{256}

Answer: A

Cheers,
Brent

KSBGC · Answer

x is a fraction.

2 rules must be applied here .

1. If we square a fraction , it will be even smaller.

2. If we square root a fraction , it will be even larger from its original size.

Thus , D , E cancelled.

B represents square root only. Think about A. we will have a fraction. B represents only fraction but when we divide a integer by a fraction answer will be greater than original size. Therefore, between A and B , A is larger.

C is in multiplication form . we will get smaller answer.

Thus, A is the correct answer.

IanStewart · Answer

If 0 < x < 1, then 0 < √x < 1, so answer A, 1/√x, will be greater than 1. Every other answer is less than 1, so A is largest.

CrackverbalGMAT · Answer

Another method is to substitute values, which can take up some of your time, but will definitely get you there.

Since x is > 0 and x < 1, it has to be a decimal/fraction.

Take a value which is a perfect square, as some options have a root value. Let's take x = 0.25 = \(0.5^2\)

Option 1: \(\frac{1}{\sqrt{0.25}}= \frac{1}{0.5} = 2\)

Option 2: \(\sqrt{0.25} = 0.5\)

Option 3: \(\frac{x}{\sqrt{\pi}}= \frac{0.25}{\frac{22}{7}} = 0.08\)

Option 4: \(0.25^3 = 0.015\)

Option 5: \(0.25^4 = 0.015^2 = 0.000225\)

Option A

Arun Kumar

Key takeaways after doing this question.

(a) Squares, cubes or higher powers of a positive decimal/fraction, is lesser than the fraction. i.e \( x^n < x\)

(b) Square root of a fraction/decimal is larger than the fraction

(c) If n is a positive integer and n is divided by a fraction x, then the result is always greater than n i.e [m]\frac{n}{x}> n

Fdambro294 · Answer

Rule: Given X = a (+)Positive Proper Fraction that falls in the Range of: 0 < X < 1

(X)^2 < X < sqrt(X)

ex: X = 1/4

1/16 < 1/4 < 1/2
(X)^2 < X < sqrt(X)

-A- would give the Max Value

1 divided by the Square Root of a Proper Fraction will result in an Integer or Compound Fraction > 1

the only one that is close is -B- and we will still retain a Proper Fraction when we apply the Square Root to the Proper Fraction

-A- Correct Answer

chetan2u · Answer

I would also suggest taking a value for x, if you are not sure about how numbers behave in the range 0<x<1.

You can take x=0.01, so  \sqrt{0.01}=0.1=\frac{1}{10}

(A)  \frac{1}{\sqrt{x}}=\frac{1}{0.1}=10

(B)  \sqrt{x}=0.1

(C)  \frac{1}{π}*x=0.01*\frac{7}{22}

(D)  x^3=0.000001

(E)  x^4=0.00000001

Also, you can note the following
1) As the power increases in the range 0<x<1, the number becomes smaller... x^4<x^3<x^2<x<\sqrt{x} 
2) x in the denominator will give you a bigger value than x in the numerator... \frac{1}{x}>x 
If you know above two points, you can mark A as the answer.

A

flippedeclipse · Answer

The rule that if  0 < x < 1 ,  x^2 < x < \sqrt{x}  helps us here to eliminate D and E.

Looking at  \frac{1}{π}x , we can sub in  x=\frac{1}{4}  to quickly compare:   \frac{1}{π} * \frac{1}{4} = \frac{1}{4 π }  . We've made the fraction smaller, not larger. So this will be less than  \sqrt{x} .

That leaves  \sqrt{x}  and  \frac{1}{\sqrt{x}} . From the stem we can identify that  \sqrt{x}  will be a proper fraction, because x originally had a value between 0 and 1. Taking the reciprocal of a proper fraction will result in a value greater than itself. This means that  \frac{1}{\sqrt{x}}> \sqrt{x}  . Thus, our answer is A.

If 0 < x < 1, which of the following expressions is greatest?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions