For all positive real numbers a and b, the Data Sufficiency Yes/No target question "Is \fr… — GMAT

For all positive real numbers $a$ and $b$ , the Data Sufficiency Yes/No target question "Is $\frac{a^2 + 3ab}{b^2 + 3ab} > 1$ ?" is algebraically equivalent to asking "Is $a > b$ ?"

Answer: Answer

Answer

True. The target question simplifies directly to "Is

a > b

The statement is true because cross-multiplying the strictly positive denominator

b^2 + 3ab

and subtracting the shared term

3ab

reduces

\frac{a^2 + 3ab}{b^2 + 3ab} > 1

directly to

a^2 > b^2

. Since

a

and

b

are both positive,

a^2 > b^2

is logically equivalent to

a > b

Step-by-Step Solution

Determine the sign of the denominator.

Since

a > 0

and

b > 0

b^2 + 3ab > 0

Knowing the denominator is strictly positive ensures we can cross-multiply without reversing the inequality sign.

Cross-multiply the inequality by the denominator.

a^2 + 3ab > b^2 + 3ab

Multiplying both sides of

\frac{a^2 + 3ab}{b^2 + 3ab} > 1

(b^2 + 3ab)

clears the fraction.

Subtract

3ab

from both sides.

a^2 > b^2

Eliminating the common linear term isolates the quadratic terms.

Take the square root of both sides.

a > b

Because

a

and

b

are positive,

a^2 > b^2

holds if and only if

a > b

Key Concept

Data Sufficiency Question Stem Simplification

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