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

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

(1) $u^2 - v^2 = 4(u - v)^2$
(2) $u^2 + v^2 = 5uv$

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.

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

Statement (1) allows factoring the difference of squares into

(u - v)(u + v) = 4(u - v)^2

. Because

u \neq v

, we can divide by

(u - v)

to determine directly that

\frac{u + v}{u - v} = 4

, which provides a unique value. Statement (2) gives

\left(\frac{u + v}{u - v}\right)^2 = \frac{7}{3}

, which yields two possible values (

\pm\sqrt{\frac{7}{3}}

) and is therefore not sufficient. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Rephrase the target question and examine constraints.

The target expression is

\frac{u + v}{u - v}

. The stem specifies

u \neq v

, so

u - v \neq 0

, making the fraction well-defined.

Establishing non-zero denominator conditions allows division by

(u - v)

when evaluating statements.

Evaluate Statement (1) independently.

Rewrite the left side of

u^2 - v^2 = 4(u - v)^2

(u - v)(u + v) = 4(u - v)^2

. Since

u - v \neq 0

, divide both sides by

(u - v)

to get

u + v = 4(u - v)

. Divide both sides by

(u - v)

again to obtain

\frac{u + v}{u - v} = 4

This yields a single, definitive numerical value for the target expression. Statement (1) is sufficient.

Evaluate Statement (2) independently.

Expand

(u + v)^2 = u^2 + v^2 + 2uv

and

(u - v)^2 = u^2 + v^2 - 2uv

. Substituting

u^2 + v^2 = 5uv

gives

(u + v)^2 = 7uv

and

(u - v)^2 = 3uv

. Taking the ratio gives

\left(\frac{u + v}{u - v}\right)^2 = \frac{7uv}{3uv} = \frac{7}{3}

(since

uv \neq 0

). Taking the square root yields

\frac{u + v}{u - v} = \sqrt{\frac{7}{3}}

-\sqrt{\frac{7}{3}}

Because two distinct numerical values are possible, Statement (2) alone does not yield a unique answer. Statement (2) is not sufficient.

Factoring difference of squares and avoiding degree miscounts in quadratic ratios

Question