Signs, Roots, & Cubes

[ram.n.pai]

Hello, here is a "data sufficiency" problem that I believe is incorrectly evaluated by ManhattanGMAT


Is x > y?

(1) squareroot(x) > y (read this as "square root of x is greater than y")

(2) cube(x) > y (read this as "x cube is greater than y")

the correct answer as per ManhattanGMAT is "Both (1) and (2) are needed"

I think just (1) is sufficient, because if square root of x is greater than y, than both the positive and negative root of x is greater than y. Since the negative root of x is greater than y, certainly y is negative. Since y is negative and x is positive, x has to be greater than y.

The explanation provided by ManhattanGMAT does not take into consideration the negative root of x. Was that a mistake?

[sunny.jain]

Why dont u just do this:

x > Y(square)

y(square) = positive
X > some positive number ==> X has to be more positive

so Y = -3 ==> Y^2 = 9 ==> X has be greater than 9
similarly for Y = 3

But what if Y = 0.5
Y^2 = 0.25 ==> X can be 0.33
so even though X is greater than Y^2 , it is still less than Y.

So A is not sufficient at all.

Now they are saying:
X^3 > Y

so this is true for X,Y == 2,1 and 2,3
so this is also not sufficient.

But we can use this to discard the fraction condition arose in A.
so both together are sufficient.

[ram.n.pai]

I see a issue with your solution --

squareroot(x) > y does not imply x > y^2

for example if squareroot(x) is 2 and y is -3
2 is certainly greater than -3
however that does not imply, 4 is greater than 9.

[amoghgarg_dps]

This is the problem which arises when we consider the below mentioned two functions identical:
(#) y^2=x
(#) y = square root of x

These 2 functions are not identical but then also I have seen on various forums that they are being considered identical. For example: y^2=25 will have solutions as y=+5,-5 but y = square root of 25 will have one and only one solution and that is +5.

If we consider the above mentioned 2 functions as identical then the 1st solution given on this post sounds fair i.e. only option 'A' would do, but if we consider that the above mentioned 2 functions non identical [which should actually be the case going by real mathematics] we require both the options to solve this and hence in that case option 'C' should be the answer.

Thanks.

[ram.n.pai]

> If we consider the above mentioned 2 functions as identical then the 1st solution given on this
> post sounds fair i.e. only option 'A' would do, but if we consider that the above mentioned 2 >functions non identical [which should actually be the case going by real mathematics] we require > both the options to solve this and hence in that case option 'C' should be the answer.

Isn't it the other way round?

If the two functions are considered identical than solution C is the answer
and if the two functions are considered un-identical than solution A is the answer

In any case, the basic question is -- Should the two function be considered identical? I hope not.

[amoghgarg_dps]

I think that the way I have put it is right because in the very first solution suggested in this post, it is being considered that square root of x gives 2 solutions [y= square root of x] as does y^2 = x, hence being considered identical and we get the answer as option 'A'.
But yeah I agree that root remains the same that whether these 2 functions should be considered identical? I hope they shouldn't be.

Thanks.

[sunny.jain]

This is the problem which arises when we consider the below mentioned two functions identical:
(#) y^2=x
(#) y = square root of x

These 2 functions are not identical but then also I have seen on various forums that they are being considered identical. For example: y^2=25 will have solutions as y=+5,-5 but y = square root of 25 will have one and only one solution and that is +5.

This is wrong assumption, where did u read if
Y = sqrt(25) ==> y = +5

if Y = sqrt(25) ==> Y = +5 or -5

Trust me, its true. I have studied enough maths during my major and this was the basic assumption on every mathematical analysis.

[ram.n.pai]

ok. after a careful read through the explanation provided by ManhattanGMAT, i understand the reason why (1) is insufficient. It has to do with fractions.

squareroot(x) > y does not imply x > y, when it comes to fractions.
If not for fractions, squareroot(x) > y would have implied x > y.

[Ben Ku]

A quant problem should go in the appropriate Quant folder.