If x and y are positive real numbers, is (x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4… — GMAT

If $x$ and $y$ are positive real numbers, is $(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4$ ?

(1) $x^2 + y^2 = 2xy + 9$
(2) $\sqrt{x} + \sqrt{y} = 5$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
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.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Rephrasing the target question stem shows that

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4

is equivalent to

4 + \frac{(x - y)^2}{xy} > 4

, which simplifies to

x \neq y

for positive real numbers

x

and

y

. Statement (1) rearranges to

(x - y)^2 = 9

, which proves

x \neq y

and provides a definitive 'Yes' answer. Statement (2) allows cases where

x = y = 6.25

as well as

x = 1, y = 16

, leaving the answer uncertain. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Step-by-Step Solution

Rephrase the target question stem algebraically.

The target question asking whether

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4

simplifies to asking whether

x \neq y

Expanding

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) = 1 + \frac{x}{y} + \frac{y}{x} + 1 = 2 + \frac{x^2 + y^2}{xy} = 4 + \frac{(x - y)^2}{xy}

. Since

x, y > 0

, the quantity

\frac{(x - y)^2}{xy} > 0

holds if and only if

(x - y)^2 > 0

, which means

x \neq y

Evaluate Statement (1):

x^2 + y^2 = 2xy + 9

Statement (1) is sufficient.

Rearranging yields

x^2 - 2xy + y^2 = 9 \implies (x - y)^2 = 9

. Since

(x - y)^2 = 9 \neq 0

, it must be that

x \neq y

. This gives a definitive 'Yes' answer to the rephrased question.

Evaluate Statement (2):

\sqrt{x} + \sqrt{y} = 5

Statement (2) is not sufficient.

x = 6.25

and

y = 6.25

, then

\sqrt{6.25} + \sqrt{6.25} = 2.5 + 2.5 = 5

. Here

x = y

, so the answer to 'Is

x \neq y

?' is 'No'. If

x = 1

and

y = 16

, then

\sqrt{1} + \sqrt{16} = 1 + 4 = 5

. Here

x \neq y

, so the answer is 'Yes'. Because both 'Yes' and 'No' are possible, Statement (2) is insufficient.

Key Concept

Data Sufficiency Question Stem Simplification

Estimated Time:1m 30s

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