## If x is a positive number and 1/2 the square root

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### If x is a positive number and 1/2 the square root

by Gmat_mission » Tue Mar 06, 2018 3:01 am
If x is a positive number and 1/2 the square root of x is equal to 2x, then x =

A. 1/16
B. 1/4
C. 1/2
D. 2
E. 8

[spoiler]OA=A[/spoiler].

How can I solve this PS question? Should I try number by number?

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by Vincen » Tue Mar 06, 2018 3:36 am
Hi Gmat_mission, welcome. Here is how I would solve it.

We know that x>0 and $$\frac{\sqrt{x}}{2}=2x.$$ This is equivalent to $$\sqrt{x}=4x\ \ \ \ \Rightarrow\ \ \ \ x=\left(4x\right)^2\ \Rightarrow\ \ x=16x^2$$ Since x>0, we can divide both sides by x and get $$1=16x\ \Rightarrow\ \ x=\frac{1}{16}.$$ If we see the options, we get that the correct asnwer is the option A.

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by [email protected] » Tue Mar 06, 2018 11:04 am
Gmat_mission wrote:If x is a positive number and 1/2 the square root of x is equal to 2x, then x =

A. 1/16
B. 1/4
C. 1/2
D. 2
E. 8

[spoiler]OA=A[/spoiler].

How can I solve this PS question? Should I try number by number?
We could also back-solve.

First, we know that if we're taking the square root of a number, and then cutting that number in half, the only way we'd end up with a larger than what we started with is if we start with a fraction between 0 and 1. (Otherwise taking the square root would reduce the value of the number, and then we'd reduce it again by multiplying it by 1/2. ) So only A, B, and C make sense. We can see pretty clearly that C won't work, as once we take the square root, we'll end up with a radical, and thus we can't end up with twice the original number, which does not contain a radical.

So we only have to test A or B. If the answer we test works, we're done. If it doesn't work, the answer must be the other.
Try B. In this case x = 1/4.
âˆš (1/4) = 1/2.
(1/2)* (1/2) = 1/4. This equal to x, not 2x, so Be is out.

At this point, the answer has to be A, but if you wanted to confirm
x = 1/16
âˆš (1/16) = 1/4
(1/2) * (1/4) = 1/8
if x = 1/16, then 2x = 1/8, and thus we have our answer.
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by [email protected] » Thu Mar 08, 2018 4:35 pm
Gmat_mission wrote:If x is a positive number and 1/2 the square root of x is equal to 2x, then x =

A. 1/16
B. 1/4
C. 1/2
D. 2
E. 8
We can create the equation:

(1/2)âˆšx = 2x

Squaring both sides of the equation, we have:

1/4(x) = 4x^2

x = 16x^2

1 = 16x

1/16 = x