If a and b are real numbers such that a \neq b, what is the value of \frac{a + b}{a - b}? … — GMAT

If $a$ and $b$ are real numbers such that $a \neq b$ , what is the value of $\frac{a + b}{a - b}$ ?

(1) $a^2 + b^2 = 4ab$
(2) $a > b > 0$

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.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Cevap
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (1) allows us to determine that the square of the target expression

(a+b)/(a-b)

equals 3, which gives two possible values:

\sqrt{3}

and

-\sqrt{3}

. Statement (2) provides the condition

a > b > 0

, ensuring that both

a+b

and

a-b

are positive, so their quotient must be positive. Combining both statements eliminates

-\sqrt{3}

, uniquely determining that the expression equals

\sqrt{3}

. Thus, both statements together are sufficient.

Adım Adım Çözüm

Express the square of the target expression in terms of basic algebraic components.

Let

E = \frac{a + b}{a - b}

. Squaring both sides yields

E^2 = \frac{(a + b)^2}{(a - b)^2} = \frac{a^2 + 2ab + b^2}{a^2 - 2ab + b^2}

Rewriting the ratio in squared form allows substitution of expressions involving

a^2 + b^2

and

ab

Evaluate Statement (1) independently.

Substitute

a^2 + b^2 = 4ab

into the squared ratio:

E^2 = \frac{4ab + 2ab}{4ab - 2ab} = \frac{6ab}{2ab} = 3

. Taking the square root gives

E = \sqrt{3}

E = -\sqrt{3}

Because Statement (1) allows two distinct possible values for

E

, Statement (1) alone is NOT sufficient.

Evaluate Statement (2) independently.

Statement (2) states

a > b > 0

. This implies

a + b > 0

and

a - b > 0

, so

E > 0

, but no numerical value is specified.

Without quantitative equations, Statement (2) alone is NOT sufficient.

Evaluate Statements (1) and (2) combined.

From Statement (1),

E = \pm\sqrt{3}

. From Statement (2),

E > 0

. Combining both rules out

-\sqrt{3}

, leaving uniquely

E = \sqrt{3}

The two statements together establish a single, unique value for the target expression.

Anahtar Kavram

Evaluating algebraic ratios via squared identities and applying inequality sign constraints to eliminate redundant roots in Data Sufficiency.

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