Question Stem Simplification and Target Rephrasing Practice Questions — GMAT — Page 2

Question 21Question →

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.

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

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

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is 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

1

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

.

2

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.

3

Evaluate Statement (2):

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

.

Statement (2) is not sufficient.

If

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

Question 22Question →

If $a$ and $b$ are non-zero real numbers such that $a^2 \neq b^2$ , is $\frac{a^2 b + a b^2}{a^3 b - a b^3} > 0$ ?

(1) $a > b$
(2) $a + b > 0$

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

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

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

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

The target question asks whether

\frac{a^2 b + a b^2}{a^3 b - a b^3} > 0

. Factoring the numerator as

ab(a+b)

and the denominator as

ab(a+b)(a-b)

allows us to cancel common non-zero factors, simplifying the target question to 'Is

\frac{1}{a-b} > 0

?', which is equivalent to 'Is

a > b

?'. Statement (1) directly states that

a > b

, answering the rephrased question with a definitive 'Yes'. Statement (2) states that

a+b > 0

, which gives no information regarding whether

a > b

. Therefore, Statement (1) alone is sufficient, while Statement (2) alone is not.

Step-by-Step Solution

1

Rephrase the target question stem algebraically by factoring numerator and denominator.

Numerator:

a^2 b + a b^2 = a b(a + b)

. Denominator:

a^3 b - a b^3 = a b(a^2 - b^2) = a b(a + b)(a - b)

. The expression becomes

\frac{a b(a + b)}{a b(a + b)(a - b)}

.

Simplifying complex rational expressions before evaluating statements prevents misinterpretation of necessary conditions.

2

Cancel non-zero common terms in the simplified fraction.

Since

a b \neq 0

and

a + b \neq 0

(given

a^2 \neq b^2

), common terms cancel out to yield

\frac{1}{a - b} > 0

.

The sign of

\frac{1}{a - b}

is strictly positive if and only if

a - b > 0

, which is equivalent to

a > b

.

3

Evaluate Statement (1):

a > b

.

This directly yields

a - b > 0

, so

\frac{1}{a - b} > 0

. This provides a definitive 'Yes' answer.

Statement (1) alone provides sufficient information.

4

Evaluate Statement (2):

a + b > 0

.

Knowing

a + b > 0

gives no information about whether

a > b

or

a < b

(e.g., if

a=5, b=2

,

a > b

; if

a=2, b=5

,

a < b

).

Statement (2) alone is not sufficient to determine if

a > b

.

Key Concept

Question Stem Simplification and Target Rephrasing

Estimated Time:1m 40s

Question 23Question →

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 $a > b$ ?"

True

False

Show answer & explanation

Answer: True

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

1

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.

2

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

by

(b^2 + 3ab)

clears the fraction.

3

Subtract

3ab

from both sides.

a^2 > b^2

Eliminating the common linear term isolates the quadratic terms.

4

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

Question 24Question →

If $k$ is a positive integer, is $\frac{k^3 - 4k}{k + 2}$ an odd integer?

(1) $k^2 + 3k$ is an even integer.
(2) $3k + 1$ is an odd integer.

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

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

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

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

Rephrasing the target question

\frac{k^3 - 4k}{k + 2}

by factoring the numerator gives

\frac{k(k-2)(k+2)}{k+2} = k(k-2)

. Because

k

and

k-2

share the same parity, their product

k(k-2)

is odd if and only if

k

is odd. Thus, the question simplifies to: 'Is

k

an odd integer?' Statement (1) tells us

k^2 + 3k = k(k+3)

is even, which is true for all integers

k

, so it provides no information about whether

k

is odd or even (NOT sufficient). Statement (2) tells us

3k + 1

is odd, which implies

3k

is even, and therefore

k

must be even. Since

k

is definitively even, the answer to 'Is

k

an odd integer?' is a definitive 'No'. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

1

Rephrase and simplify the question stem algebraically.

Since

k

is a positive integer,

k + 2 > 0

. Factoring the numerator gives

k^3 - 4k = k(k^2 - 4) = k(k - 2)(k + 2)

. Dividing by

k + 2

yields

\frac{k(k - 2)(k + 2)}{k + 2} = k(k - 2)

.

Simplifying the target expression reduces cognitive complexity before evaluating statements.

2

Determine the parity condition for

k(k - 2)

to be an odd integer.

The factors

k

and

k - 2

differ by 2, so they have the same parity. If

k

is even,

k(k - 2)

is (even)(even) = even. If

k

is odd,

k(k - 2)

is (odd)(odd) = odd. Thus, the target question 'Is

k(k - 2)

an odd integer?' simplifies to 'Is

k

an odd integer?'

Rephrasing the target question to 'Is

k

odd?' establishes a clear, minimal sufficiency criterion.

3

Evaluate Statement (1):

k^2 + 3k

is an even integer.

Factor

k^2 + 3k

as

k(k + 3)

. The terms

k

and

k + 3

differ by 3, meaning one is always even and the other is odd. Their product is ALWAYS even regardless of whether

k

is odd or even. Therefore, Statement (1) provides no information about whether

k

is odd. NOT sufficient.

A statement that holds true for all integers provides no distinguishing information about

k

.

4

Evaluate Statement (2):

3k + 1

is an odd integer.

If

3k + 1

is odd, then

3k

must be even. Since 3 is odd,

3k

is even if and only if

k

is even. Hence,

k

is definitely even. This gives a definitive 'No' to the rephrased question 'Is

k

an odd integer?'. A definitive 'No' means the statement IS sufficient.

In Data Sufficiency Yes/No questions, a statement that yields a conclusive 'No' is sufficient.

Key Concept

Data Sufficiency Question Stem Simplification and Parity Analysis

Estimated Time:1m 30s

Question 25Question →

If $m$ and $n$ are positive integers such that $m \neq n$ , is $\frac{m^2 + mn - 2n^2}{m - n}$ an even integer?

(1) $m + 3$ is an odd integer.
(2) $n$ is an odd integer.

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

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

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

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

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

Answer

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

By factoring the numerator of the expression in the question stem,

\frac{m^2 + mn - 2n^2}{m - n} = \frac{(m + 2n)(m - n)}{m - n} = m + 2n

. Since

2n

is always even,

m + 2n

is even if and only if

m

is even. Rephrasing the question target yields: "Is

m

an even integer?" Statement (1) tells us

m + 3

is odd, which means

m

must be even (giving a definitive "Yes" answer). Statement (2) provides information about

n

, which is irrelevant to whether

m

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

Step-by-Step Solution

1

Simplify the target expression in the question stem

\frac{m^2 + mn - 2n^2}{m - n} = \frac{(m + 2n)(m - n)}{m - n} = m + 2n

Factoring the numerator simplifies the target expression for

m \neq n

.

2

Rephrase the target question using parity properties

Since

2n

is an even integer for any integer

n

,

m + 2n

is even if and only if

m

is an even integer. The rephrased question target is: "Is

m

an even integer?"

Simplifying the target question eliminates the variable

n

from the requirement.

3

Evaluate Statement (1):

m + 3

is an odd integer

If

m + 3

is odd, then

m = \text{odd} - 3 = \text{even}

. Thus,

m

is definitively even. Statement (1) alone is SUFFICIENT.

Subtracting an odd integer from an odd integer yields an even integer.

4

Evaluate Statement (2):

n

is an odd integer

Statement (2) gives information about

n

, but gives no information about

m

. Since the rephrased question depends solely on

m

, Statement (2) alone is INSUFFICIENT.

Knowing

n

does not determine whether

m

is even.

Key Concept

Algebraic factoring and target rephrasing in Data Sufficiency questions

Question Stem Simplification and Target Rephrasing

Answer

Step-by-Step Solution

Key Concept

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes