Lesson: Introduction to Square Roots

Comment on Introduction to Square Roots

Can you please help me with this one ?

Is x > y ?
(1) ✓x > y
(2) x² > y
gmat-admin's picture

Tricky question! Just curious, what's the source?

TARGET QUESTION: Is x > y?

STATEMENT 1: ✓x > y
There are several values x and y that satisfy statement 1. Here are two:
CASE A: x = 9 and y = 2 (notice that ✓9 = 3, and 3 is greater than 2. In this case, x > y
CASE B: x = 1/4 and y = 0.3 (notice that ✓(1/4) = 1/2, and 1/2 is greater than 0.3. In this case, x < y
Since we cannot answer the TARGET QUESTION with certainty, statement 2 is NOT sufficient.

STATEMENT 2: x² > y
There are several values x and y that satisfy statement 2. Here are two:
CASE A: x = 3 and y = 2 (notice that 3² = 9, and 9 is greater than 2. In this case, x > y
CASE B: x = 3 and y = 4 (notice that 3² = 9, and 9 is greater than 4. In this case, x < y
Since we cannot answer the TARGET QUESTION with certainty, statement 2 is NOT sufficient.

STATEMENTS 1 and 2 COMBINED
It's easy to show values of x and y that satisfy BOTH statements where x > y. For example, the following values of x and y satisfy BOTH statements:
x = 100 and y = 1
x = 16 and y = 1
x = 25 and y = -3
x = 1 and y = -30
etc
In all of the above cases, x > y
So, the correct answer MAY be C.

That said, the answer will E if there are values of x and y that satisfy BOTH statements AND where x < y

Is it possible to find such values for x and y?
No.

The tricky part is finding finding a situation where ✓x > y YET x < y

Important concept: in most cases, the square root of a number is less than the original number. For example:
✓9 < 9
✓4 < 4
✓36 < 36
The only time ✓x is GREATER THAN x is when 0 < x < 1. For example:
✓(1/4) > 1/4
✓(1/25) > 1/25
etc

So, if 0 < x < 1 then ✓x > x
The problem is that, if 0 < x < 1 then x² < x
So, any value on this range will satisfy statement 1, but will NOT satisfy statement 2.
As such, we CANNOT find values for x and y that satisfy BOTH statements such that x < y

So, it MUST be the case that, when both statements are combined, x > y

Answer: C

Thanks Brent :)

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