If x and y are positive real numbers, what is the value of the ratio \frac{x}{y}? (1) \fra… — GMAT

If $x$ and $y$ are positive real numbers, what is the value of the ratio $\frac{x}{y}$ ?

(1) $\frac{x + 2y}{y} = 5$

(2) $x^2 - 9y^2 = 0$

Which of the following correctly describes the sufficiency of the statements?

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.

EACH statement ALONE is sufficient.

Evaluating each statement independently demonstrates sufficiency for both. Statement (1) simplifies directly to

\frac{x}{y} = 3

. Statement (2) factors to

(x-3y)(x+3y)=0

, and since

x

and

y

are positive real numbers as stated in the question stem,

x+3y

cannot equal zero, leaving

x-3y=0

, or

\frac{x}{y} = 3

. Because each statement independently yields a unique value for the ratio, each statement alone is sufficient.

Evaluate Statement (1) independently

Simplifying

\frac{x + 2y}{y} = 5

gives

\frac{x}{y} + 2 = 5

, so

\frac{x}{y} = 3

This yields a single unique value for the target ratio, so Statement (1) alone is sufficient.

Evaluate Statement (2) independently without carrying over info from Statement (1)

Factoring

x^2 - 9y^2 = 0

yields

(x - 3y)(x + 3y) = 0

, meaning

x = 3y

x = -3y

. Since the stem specifies

x

and

y

are positive real numbers,

x + 3y > 0

, so

x = -3y

is impossible. Thus,

x = 3y

, which gives

\frac{x}{y} = 3

Because of the domain constraint in the stem, Statement (2) independently provides a single unique ratio of 3, making Statement (2) alone sufficient.

Conclude overall Data Sufficiency decision

Since Statement (1) alone is sufficient and Statement (2) alone is sufficient, each statement alone is sufficient.

Matching standard GMAT Data Sufficiency evaluation rules.

Statement Independence & Stem Constraint Application

Question