If x and y are positive real numbers, is \frac{x + y}{2} - \sqrt{xy} > 1? (1) (\sqrt{x} + … — GMAT

If $x$ and $y$ are positive real numbers, is $\frac{x + y}{2} - \sqrt{xy} > 1$ ?

(1) $(\sqrt{x} + \sqrt{y})^2 = 9$ and $xy = 4$
(2) $x + y = 8$ and $xy = 9$

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.Answer
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

EACH statement ALONE is sufficient.

Rephrasing the question stem shows that

\frac{x + y}{2} - \sqrt{xy} > 1

is equivalent to

(\sqrt{x} - \sqrt{y})^2 > 2

. Statement (1) establishes that

(\sqrt{x} - \sqrt{y})^2 = 1

, giving a definitive 'No' to the target question. Statement (2) establishes that

(\sqrt{x} - \sqrt{y})^2 = 2

, which also gives a definitive 'No' since 2 is not strictly greater than 2. Thus, each statement alone is sufficient.

Step-by-Step Solution

Rephrase the target question stem algebraically.

The target question 'Is

\frac{x + y}{2} - \sqrt{xy} > 1

?' simplifies to 'Is

x + y - 2\sqrt{xy} > 2

?', which is equivalent to 'Is

(\sqrt{x} - \sqrt{y})^2 > 2

Since

x

and

y

are positive real numbers,

x + y - 2\sqrt{xy}

can be rewritten as the perfect square

(\sqrt{x} - \sqrt{y})^2

Evaluate Statement (1) using the rephrased target.

From

(\sqrt{x} + \sqrt{y})^2 = 9

and

xy = 4

, we have

\sqrt{xy} = 2

. Using the algebraic identity

(\sqrt{x} - \sqrt{y})^2 = (\sqrt{x} + \sqrt{y})^2 - 4\sqrt{xy}

, we find

(\sqrt{x} - \sqrt{y})^2 = 9 - 4(2) = 1

. Asking 'Is

1 > 2

?' yields a definitive 'No'.

A statement that yields a definitive 'No' to a Yes/No question is sufficient.

Evaluate Statement (2) using the rephrased target.

From

x + y = 8

and

xy = 9

, we have

\sqrt{xy} = 3

. Expanding

(\sqrt{x} - \sqrt{y})^2 = x + y - 2\sqrt{xy}

gives

8 - 2(3) = 2

. Asking 'Is

2 > 2

?' yields a definitive 'No'.

Since 2 is not strictly greater than 2, Statement (2) provides a definitive 'No' and is therefore sufficient.

Conclude the final sufficiency decision.

Because each statement independently provides a definitive 'No' answer, each statement alone is sufficient.

Both statements satisfy the Data Sufficiency requirements independently.

Key Concept

Question Stem Simplification and Target Rephrasing

Estimated Time:2m 0s

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Answer

Step-by-Step Solution

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