Data Sufficiency Alıştırma Soruları — GMAT — Sayfa 2

Soru 21Soru →

A statement determining that a real variable $x$ satisfies the quadratic equation $x^2 - 5x + 6 = 0$ is sufficient to answer the Data Sufficiency question 'Is $x$ an integer?', but is insufficient to answer the Data Sufficiency question 'What is the value of $x$ ?'

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Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is True because both solutions

x = 2

and

x = 3

produce a definitive 'Yes' to the integer question, satisfying Yes/No sufficiency, whereas the existence of two distinct numerical values fails to satisfy Value sufficiency.

Solving the equation

x^2 - 5x + 6 = 0

gives

x = 2

and

x = 3

. Because both values are integers, the answer to 'Is

x

an integer?' is a definitive 'Yes' regardless of which root is chosen, making the statement sufficient for the Yes/No structure. However, because

x

can equal either

2

or

3

, a single unique value cannot be pinpointed, rendering the statement insufficient for the Value structure.

Adım Adım Çözüm

1

Solve the quadratic equation in the statement for all possible real values of

x

.

x^2 - 5x + 6 = 0 \Rightarrow (x - 2)(x - 3) = 0

, so

x = 2

or

x = 3

.

Identify the complete solution set permitted by the statement.

2

Evaluate the solution set against the Yes/No question 'Is

x

an integer?'.

If

x = 2

,

x

is an integer (Yes). If

x = 3

,

x

is an integer (Yes). The answer is conclusively 'Yes' in all cases.

In Yes/No Data Sufficiency, a statement is sufficient if it guarantees a single definitive outcome (always Yes or always No).

3

Evaluate the solution set against the Value question 'What is the value of

x

?'.

x

can be either

2

or

3

. Since two distinct values are possible, the value of

x

is not uniquely determined.

In Value Data Sufficiency, a statement is sufficient if and only if it leads to exactly one unique numerical value.

4

Compare the analytical results with the assertions made in the True/False statement.

The statement accurately classifies the equation as sufficient for the Yes/No question and insufficient for the Value question. Therefore, the statement is True.

Confirm alignment between analytical findings and the prompt statement.

Anahtar Kavram

Value vs. Yes/No Data Sufficiency Decision Logic

Soru 22Soru →

If $x$ is a positive integer, is $x$ an even number?

(1) $x$ is a prime number greater than 2.
(2) $x + 1$ is an odd number.

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.

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.

The correct option is the one stating that EACH statement ALONE is sufficient. Statement (1) establishes that

x

is an odd prime, yielding a definitive 'No' to the stem question, which makes it sufficient. Statement (2) establishes that

x

is even, yielding a definitive 'Yes' to the stem question, making it sufficient as well.

Adım Adım Çözüm

1

Analyze Statement (1):

x

is a prime number greater than 2.

All prime numbers greater than 2 are odd numbers (e.g., 3, 5, 7, 11). Therefore,

x

cannot be an even number. The question 'Is

x

an even number?' receives a definitive answer of 'No'. In Data Sufficiency, a statement that leads to a conclusive 'No' is SUFFICIENT.

In Yes/No Data Sufficiency, a consistent 'No' answer is just as sufficient as a consistent 'Yes' answer.

2

Analyze Statement (2):

x + 1

is an odd number.

If an integer plus 1 is odd, the integer itself must be even. Thus,

x

must be an even number. The question 'Is

x

an even number?' receives a definitive answer of 'Yes'. Therefore, Statement (2) is SUFFICIENT.

Direct algebraic or parity properties allow a unique 'Yes' determination.

3

Combine evaluations of both statements.

Since Statement (1) alone yields a definitive 'No' and Statement (2) alone yields a definitive 'Yes', each statement independently answers the question.

Both statements are sufficient on their own.

Anahtar Kavram

Definitive Yes/No Sufficiency Rule

Soru 23Soru →

If $x$ is a real number, is $x > 0$ ?

(1) $|x| = 5$
(2) $x + 3 > 5$

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.

Cevabı ve açıklamayı göster

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

Cevap

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

The choice stating that Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient, is correct because Statement (1) yields two possible values for

x

(

5

and

-5

), resulting in both 'Yes' and 'No' answers. In contrast, Statement (2) simplifies to

x > 2

, which guarantees that

x > 0

with a definitive 'Yes'.

Adım Adım Çözüm

1

Rephrase the target question

The target asks whether

x

lies to the right of

0

on the real number line (a Yes/No question).

Establishing clear sufficiency criteria requires identifying what yields a definitive 'Yes' or definitive 'No'.

2

Evaluate Statement (1):

|x| = 5

x

can be

5

or

-5

. If

x = 5

, the answer is Yes (

5 > 0

). If

x = -5

, the answer is No (

-5 \ngtr 0

).

Since Statement (1) allows both a 'Yes' and a 'No' response, it is NOT sufficient.

3

Evaluate Statement (2):

x + 3 > 5

Subtracting

3

from both sides gives

x > 2

. Any number strictly greater than

2

is automatically greater than

0

.

Statement (2) provides a definitive 'Yes' answer to the question

x > 0

, so Statement (2) ALONE is sufficient.

Anahtar Kavram

Absolute Value Definition and Single-Variable Inequality Simplification in Data Sufficiency

Soru 24Soru →

In a GMAT Data Sufficiency 'Value' question asking for the specific numerical value of a variable $x$ , a statement that narrows $x$ down to exactly two distinct numerical solutions provides sufficient information.

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Cevabı ve açıklamayı göster

Cevap: False

Cevap

False

The statement is False. GMAT Data Sufficiency questions are divided into two structural categories: Value questions and Yes/No questions. For a 'Value' question, a statement is sufficient only if it allows you to determine one single, unambiguous numerical value. If a statement results in two or more distinct possible values, the exact value of the target cannot be determined, so the statement is insufficient.

Adım Adım Çözüm

1

Identify the standard for sufficiency in a GMAT Data Sufficiency 'Value' question.

A statement must produce a single, unique numerical value for the target variable or expression.

Value questions require determinacy; if more than one value is possible from the statement alone, the specific value is unknown.

2

Evaluate the condition where a statement yields exactly two distinct numerical solutions.

Since two distinct values exist, the statement fails to provide a unique value, rendering it insufficient.

Having a finite set of two solutions (e.g.,

x = 2

or

x = -2

) does not satisfy the unique value requirement of Value Data Sufficiency questions.

Anahtar Kavram

Value Data Sufficiency Decision Logic

Soru 25Soru →

If $x$ is a real number, is $x > 0$ ?

(1) $|x| = -x$
(2) $x + |x| > 0$

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

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.

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.

The option stating that each statement alone is sufficient is correct. Statement (1) shows that

x \le 0

, which gives a definitive 'No' to whether

x > 0

. Statement (2) shows that

x > 0

, which gives a definitive 'Yes' to whether

x > 0

. Because both statements yield a clear, conclusive answer independently, each statement alone is sufficient.

Adım Adım Çözüm

1

Analyze Statement (1):

|x| = -x

By definition of absolute value,

|x| = -x

holds true if and only if

x \le 0

.

If

x \le 0

, then

x

is not strictly positive, so the answer to 'Is

x > 0

?' is a definitive 'No'.

2

Evaluate sufficiency of Statement (1)

Statement (1) provides a definitive 'No' answer to the question stem.

In Data Sufficiency, a statement that consistently yields a definitive 'No' is sufficient.

3

Analyze Statement (2):

x + |x| > 0

If

x \le 0

, then

|x| = -x

, so

x + |x| = x - x = 0

, which is not greater than

0

. If

x > 0

, then

|x| = x

, so

x + |x| = 2x > 0

.

The inequality

x + |x| > 0

is satisfied if and only if

x > 0

.

4

Evaluate sufficiency of Statement (2)

Statement (2) provides a definitive 'Yes' answer to the question stem.

Since Statement (2) guarantees

x > 0

, it is sufficient on its own.

5

Combine evaluations

Each statement alone provides a definitive answer to the question stem.

Therefore, EACH statement ALONE is sufficient.

Anahtar Kavram

Absolute Value Definition and Yes/No Data Sufficiency Decision Logic

Tahmini Süre:1m 0s

Soru 26Soru →

In a GMAT Data Sufficiency 'Value' question asking for the numerical value of the expression $xy$ , a statement that permits $x$ to be either $2$ or $-2$ while establishing that $y = \frac{8}{x}$ is insufficient to answer the question because $x$ itself is not uniquely determined.

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Cevabı ve açıklamayı göster

Cevap: False

Cevap

False. In Data Sufficiency 'Value' questions, sufficiency depends solely on whether the target expression can be uniquely determined. Because

xy = 8

under all permitted conditions, the statement is sufficient regardless of whether

x

has multiple values.

The claim presented in the stem is false. In GMAT Data Sufficiency 'Value' questions, a statement is sufficient if it yields a single, unambiguous numerical value for the requested target expression. It is not necessary for individual variables within the expression to be uniquely determined. Given

y = \frac{8}{x}

, multiplying both sides by

x

yields

xy = 8

. Whether

x = 2

or

x = -2

, the product

xy

remains constant at

8

. Thus, the statement is sufficient to answer the question, rendering the claim of insufficiency false.

Adım Adım Çözüm

1

Identify the precise target of the question stem.

The target quantity is the expression

xy

, not the individual variable

x

or

y

.

Evaluating Data Sufficiency requires focusing on the target expression as a single entity.

2

Test all cases allowed by the statement.

Case 1: If

x = 2

, then

y = \frac{8}{2} = 4

, so

xy = (2)(4) = 8

.
Case 2: If

x = -2

, then

y = \frac{8}{-2} = -4

, so

xy = (-2)(-4) = 8

.

Both permissible values of

x

must be tested alongside their corresponding

y

values.

3

Determine sufficiency based on target uniqueness.

Both cases yield the exact same value (

xy = 8

). The statement provides a unique numerical answer to the question.

Because the value of the target expression is uniquely determined, the statement is sufficient, making the claim in the stem false.

Anahtar Kavram

Target Expression Uniqueness vs. Variable Uniqueness in Value Questions

Tahmini Süre:2m 0s

Soru 27Soru →

If $r$ and $s$ are integers, is $r + s$ an even integer?

(1) $r^2 - s^2$ is an odd integer.
(2) $r$ 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.

Cevabı ve açıklamayı göster

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

Cevap

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

The correct response indicates that Statement (1) alone is sufficient while Statement (2) alone is not. Statement (1) establishes algebraically that

r + s

must be odd because

(r - s)(r + s)

is an odd integer, yielding a definitive 'No' answer to whether

r + s

is even. Statement (2) leaves the value and parity of

s

completely unrestricted, allowing for both 'Yes' and 'No' outcomes.

Adım Adım Çözüm

1

Rephrase the target question

The target is a Yes/No question: Is

r + s

even? A statement is sufficient if it yields a definitive 'Yes' or a definitive 'No'.

In Yes/No Data Sufficiency questions, proving that a condition is always false (a definitive 'No') is just as sufficient as proving it is always true.

2

Evaluate Statement (1):

r^2 - s^2

is an odd integer

Factor

r^2 - s^2 = (r - s)(r + s)

. Since the product of two integers is odd, both factors

(r - s)

and

(r + s)

must be odd integers. Thus,

r + s

is definitely odd, which means

r + s

is NOT even.

Since Statement (1) provides a definitive 'No' to the target question, Statement (1) alone is SUFFICIENT.

3

Evaluate Statement (2):

r

is an odd integer

If

r = 3

and

s = 1

, then

r + s = 4

(even

\rightarrow

Yes). If

r = 3

and

s = 2

, then

r + s = 5

(odd

\rightarrow

No).

Because Statement (2) allows both 'Yes' and 'No' answers depending on the value of

s

, Statement (2) alone is NOT sufficient.

Anahtar Kavram

Definitive Yes/No Decision Logic in Data Sufficiency

Soru 28Soru →

If $a$ and $b$ are real numbers, is $a^2 + b^2 < 4$ ?

(1) $(a - 2)^2 + (b - 2)^2 = 18$
(2) $a^2 - 6a + b^2 - 8b = -24$

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.

Cevabı ve açıklamayı göster

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

Cevap

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

Statement (1) permits values of

a^2 + b^2

ranging from 2 to 50, which can be either less than 4 (yielding a 'Yes') or greater than or equal to 4 (yielding a 'No'), making Statement (1) insufficient. Statement (2) defines a geometric locus of points where

a^2 + b^2 \ge 16

, which means

a^2 + b^2 < 4

is conclusively false for all possible values. Because Statement (2) yields a definitive 'No' answer, it is sufficient on its own. Consequently, Statement (2) alone is sufficient, but Statement (1) alone is not sufficient.

Adım Adım Çözüm

1

Understand the Data Sufficiency decision logic for Yes/No questions.

A statement is sufficient if it yields a definitive 'Yes' (always true) OR a definitive 'No' (always false). It is insufficient only if it allows both 'Yes' and 'No'.

Data Sufficiency requires determinism regarding the truth value of the target question.

2

Evaluate Statement (1):

(a - 2)^2 + (b - 2)^2 = 18

.

This equation represents a circle centered at

(2, 2)

with radius

r = \sqrt{18} = 3\sqrt{2}

. The distance from the origin to the center is

\sqrt{2^2 + 2^2} = 2\sqrt{2}

. The distance

d = \sqrt{a^2 + b^2}

of any point on the circle from the origin ranges from

|2\sqrt{2} - 3\sqrt{2}| = \sqrt{2}

to

2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}

. Thus,

a^2 + b^2 = d^2

ranges from

(\sqrt{2})^2 = 2

to

(5\sqrt{2})^2 = 50

.

Since

a^2 + b^2

can be

2

(which is

< 4

, giving 'Yes') or

50

(which is

\ge 4

, giving 'No'), Statement (1) cannot yield a single definitive Yes/No answer and is INSUFFICIENT.

3

Evaluate Statement (2):

a^2 - 6a + b^2 - 8b = -24

.

Completing the square gives

(a - 3)^2 - 9 + (b - 4)^2 - 16 = -24 \implies (a - 3)^2 + (b - 4)^2 = 1

. This is a circle centered at

(3, 4)

with radius

1

. The distance from the origin to the center

(3,4)

is

\sqrt{3^2 + 4^2} = 5

. The minimum distance from the origin to any point on this circle is

5 - 1 = 4

, so

d \ge 4

. Therefore,

a^2 + b^2 = d^2 \ge 4^2 = 16

.

Because

a^2 + b^2 \ge 16

for all points satisfying Statement (2), the inequality

a^2 + b^2 < 4

is NEVER true. Statement (2) yields a definitive 'No' answer, which makes it SUFFICIENT.

4

Conclude the final Data Sufficiency choice.

Statement (1) alone is insufficient, while Statement (2) alone is sufficient.

A definitive 'No' from Statement (2) satisfies sufficiency under standard Data Sufficiency decision logic.

Anahtar Kavram

Value vs. Yes/No Data Sufficiency Decision Logic

Soru 29Soru →

In a GMAT Data Sufficiency 'Yes/No' question asking whether $a^2 > b^2$ , a statement establishing that $a + b = 0$ (where $a \neq 0$ ) is sufficient to answer the question.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is True. In Yes/No Data Sufficiency, proving that a target condition is conclusively false constitutes a definitive 'No' answer, which makes the statement sufficient.

In Data Sufficiency Yes/No decision logic, sufficiency is attained whenever a statement leads to a single, unambiguous answer—either always 'Yes' or always 'No'. Here,

a + b = 0

forces

a^2 = b^2

, meaning

a^2 > b^2

is never true. Thus, the statement provides a definitive 'No' answer to the question, making it sufficient. Therefore, the assertion is correct (True).

Adım Adım Çözüm

1

Identify the target question type and sufficiency condition.

The target is a Yes/No Data Sufficiency question: 'Is

a^2 > b^2

?' Sufficiency requires either a definitive 'Yes' in all cases or a definitive 'No' in all cases.

Establishing decision logic boundaries is required before evaluating statement sufficiency.

2

Rephrase the given statement algebraically.

The statement gives

a + b = 0 \implies a = -b

(with

a \neq 0

).

Expressing one variable in terms of another allows direct substitution into the target inequality.

3

Substitute the expression into the target inequality

a^2 > b^2

.

(-b)^2 > b^2 \implies b^2 > b^2

, which simplifies to

0 > 0

. This statement is universally false for any non-zero real number

b

.

Evaluating the target inequality under the given constraint tests whether the outcome is deterministic.

4

Apply Data Sufficiency decision logic to the result.

Since

a^2 > b^2

is false for all allowable values, the answer to the question 'Is

a^2 > b^2

?' is a definitive 'No'. A definitive 'No' answer means the statement is sufficient.

In GMAT DS logic, both a definitive 'Yes' and a definitive 'No' satisfy the requirement for sufficiency.

Anahtar Kavram

Definitive No Sufficiency Rule in Yes/No Data Sufficiency

Soru 30Soru →

If $x$ and $y$ are real numbers, what is the value of $x + y$ ?

(1) $3x + 3y = 12$
(2) $x - y = 2$

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.

Cevabı ve açıklamayı göster

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

Cevap

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

The correct option correctly identifies that Statement (1) alone is sufficient because dividing

3x + 3y = 12

by 3 yields

x + y = 4

, directly answering the question without needing to solve for

x

and

y

individually. Statement (2) alone leaves

x + y

undetermined.

Adım Adım Çözüm

1

Analyze the target expression in the question stem.

The target to find is a single value for

x + y

.

Rephrasing the question stem clarifies what exact numerical value or algebraic combination is required.

2

Evaluate Statement (1) independently:

3x + 3y = 12

.

Factor out 3 to get

3(x + y) = 12

, then divide by 3 to obtain

x + y = 4

.

Statement (1) gives a unique value for the target expression

x + y

, so Statement (1) alone is sufficient.

3

Evaluate Statement (2) independently:

x - y = 2

.

If

x = 3

and

y = 1

, then

x - y = 2

and

x + y = 4

. If

x = 4

and

y = 2

, then

x - y = 2

and

x + y = 6

.

Multiple values for

x + y

are possible, so Statement (2) alone is not sufficient.

Anahtar Kavram

Question Stem Rephrasing and Linear Expression Combination

Tahmini Süre:1m 0s

Soru 31Soru →

If $x$ and $y$ are real numbers, is $|x - y| < |x + y|$ ?

(1) $x^3 y > 0$
(2) $|x| < y$

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.

Cevabı ve açıklamayı göster

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

Cevap

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

Rephrasing the stem by squaring both non-negative sides reveals that

|x - y| < |x + y|

is strictly equivalent to

xy > 0

. Statement (1) specifies

x^3 y > 0

. Because

x^3

and

x

always have the same sign,

x^3 y > 0

means

x

and

y

share the same sign, which guarantees

xy > 0

. Thus, Statement (1) alone provides a definitive 'Yes'. Statement (2) states

|x| < y

, which forces

y > 0

but allows

x

to be either positive or negative, so

xy

can be either positive or negative. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Adım Adım Çözüm

1

Rephrase the question stem by squaring both sides of the inequality.

Since both

|x - y|

and

|x + y|

are non-negative,

|x - y| < |x + y| \iff (x - y)^2 < (x + y)^2 \iff x^2 - 2xy + y^2 < x^2 + 2xy + y^2 \iff 0 < 4xy \iff xy > 0

. The question asks whether

x

and

y

have the same sign (and neither is zero).

Simplifying absolute value inequalities before analyzing statements prevents misinterpretation.

2

Evaluate Statement (1):

x^3 y > 0

.

Since

x^3

always has the exact same algebraic sign as

x

for all real numbers,

x^3 y > 0 \implies xy > 0

. This yields a definitive 'Yes' to the rephrased question.

Statement (1) alone provides enough information to answer the target question conclusively.

3

Evaluate Statement (2):

|x| < y

.

Since

|x| \ge 0

, this implies

y > 0

. However,

x

can be positive (e.g.,

x=2, y=3 \implies xy=6 > 0

, Yes) or negative (e.g.,

x=-2, y=3 \implies xy=-6 < 0

, No).

Because Statement (2) allows both 'Yes' and 'No' answers, it is not sufficient.

Anahtar Kavram

Rephrasing absolute value inequalities and analyzing product sign constraints in Data Sufficiency

Soru 32Soru →

In a GMAT Data Sufficiency 'Yes/No' question regarding whether a real variable $x$ is positive, a statement establishing that $x^2 - 4x + 3 = 0$ is sufficient, whereas in a 'Value' question asking for the exact numerical value of $x$ , the exact same statement is insufficient.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is True.

The statement accurately reflects GMAT Data Sufficiency decision rules: a 'Yes/No' question requires only that all possible cases produce a uniform 'Yes' or uniform 'No' answer (which occurs here since both

x=1

and

x=3

are positive), while a 'Value' question strictly requires a single unique value (which fails here due to having two solutions).

Adım Adım Çözüm

1

Analyze the mathematical implications of the given statement.

Solving

x^2 - 4x + 3 = 0

yields

(x-1)(x-3) = 0

, so

x = 1

or

x = 3

.

Determine the exact solution set allowed by the statement.

2

Evaluate sufficiency for a 'Yes/No' Data Sufficiency question stem.

Since both

x = 1

and

x = 3

are strictly greater than

0

, the answer to 'Is

x > 0

?' is unconditionally 'Yes'. Thus, the statement is sufficient.

In 'Yes/No' DS logic, any statement that yields a definitive, consistent 'Yes' (or a definitive, consistent 'No') is sufficient.

3

Evaluate sufficiency for a 'Value' Data Sufficiency question stem.

Since

x

can be either

1

or

3

, a single unique value for

x

cannot be determined. Thus, the statement is insufficient.

In 'Value' DS logic, a statement is sufficient if and only if it leads to exactly one numerical value.

4

Compare the conclusions with the given assertion.

The assertion correctly identifies that the statement is sufficient for 'Yes/No' logic and insufficient for 'Value' logic.

Confirm total alignment between the assertion and GMAT DS decision logic.

Anahtar Kavram

Value vs. Yes/No Data Sufficiency Decision Logic

Soru 33Soru →

In Data Sufficiency, simplifying the question target "Is $x^2 < x$ ?" yields the algebraically equivalent question target "Is $0 < x < 1$ ?" for all real numbers $x$ .

Doğru

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Cevap: True

Cevap

The statement is true because solving the inequality

x^2 < x

yields the exact range

0 < x < 1

.

Rephrasing the question stem target

x^2 < x

algebraically leads directly to

x(x - 1) < 0

, which holds true if and only if

x

is strictly between

0

and

1

. Therefore, the simplified target 'Is

0 < x < 1

?' is completely equivalent.

Adım Adım Çözüm

1

Rearrange the inequality to set one side to zero.

x^2 - x < 0

Standard algebraic procedure for solving non-linear inequalities requires comparing a factored expression to zero.

2

Factor the quadratic expression.

x(x - 1) < 0

Factoring isolates the roots (

x = 0

and

x = 1

) that define the boundary intervals on the real number line.

3

Determine the interval where the product of the factors is negative.

The product is negative between the roots, which corresponds to

0 < x < 1

.

When

x < 0

, both factors are negative (product is positive). When

x > 1

, both factors are positive (product is positive). Only when

0 < x < 1

is

x

positive and

(x - 1)

negative.

Anahtar Kavram

Simplifying Data Sufficiency question stems by finding equivalent inequality ranges reduces complex targets to direct boundary checks.

Tahmini Süre:45s

Soru 34Soru →

If $x$ is a non-zero real number, is $x$ an integer?

(1) $x + \frac{6}{x}$ is an integer.

(2) $x^2 + \frac{36}{x^2}$ is an 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.

Cevabı ve açıklamayı göster

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

Cevap

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

The correct response establishes that neither statement individually nor both statements combined are sufficient to determine whether the real number is an integer. Testing specific real numbers demonstrates that integer values such as 2 and irrational values such as 3 plus the square root of 3 satisfy both algebraic conditions, producing affirmative and negative answers to the question.

Adım Adım Çözüm

1

Analyze Statement (1):

x + \frac{6}{x} = k

for some integer

k

.

If

x = 2

, then

2 + \frac{6}{2} = 5

(an integer), and

x

IS an integer. If

x = 3 + \sqrt{3}

, then

x + \frac{6}{x} = 3 + \sqrt{3} + \frac{6(3 - \sqrt{3})}{6} = 6

(an integer), but

x

is NOT an integer.

Since Statement (1) yields both a 'Yes' and a 'No' answer, it is not sufficient.

2

Analyze Statement (2):

x^2 + \frac{36}{x^2} = m

for some integer

m

.

If

x = 2

, then

2^2 + \frac{36}{2^2} = 4 + 9 = 13

(an integer), so

x

IS an integer. If

x = \sqrt{2}

, then

(\sqrt{2})^2 + \frac{36}{(\sqrt{2})^2} = 2 + 18 = 20

(an integer), but

x

is NOT an integer.

Since Statement (2) yields both a 'Yes' and a 'No' answer, it is not sufficient.

3

Analyze Statements (1) and (2) combined.

Note that

\left(x + \frac{6}{x}\right)^2 = x^2 + 12 + \frac{36}{x^2}

. Thus

x^2 + \frac{36}{x^2} = \left(x + \frac{6}{x}\right)^2 - 12

. Whenever

x + \frac{6}{x}

is an integer

k

,

x^2 + \frac{36}{x^2} = k^2 - 12

is automatically an integer. Therefore, Statement (2) provides no extra constraints.

Both

x = 2

(integer) and

x = 3 + \sqrt{3}

(non-integer) satisfy both statements simultaneously, so the combined statements remain insufficient.

Anahtar Kavram

Evaluating algebraic expressions for non-integer real number solutions in Data Sufficiency

Soru 35Soru →

For all real numbers $x$ such that $x \neq 0$ and $x \neq 1$ , the Data Sufficiency Yes/No target question "Is $\frac{x^2 - x}{|x - 1|} > 0$ ?" is algebraically equivalent to asking "Is $x > 0$ ?".

Doğru

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Cevap: False

Cevap

The statement is False.

The statement is False because simplifying the target inequality

\frac{x(x - 1)}{|x - 1|} > 0

yields

x < 0

or

x > 1

, which is not equivalent to

x > 0

.

Adım Adım Çözüm

1

Factor the numerator of the expression in the target inequality.

\frac{x^2 - x}{|x - 1|} = \frac{x(x - 1)}{|x - 1|}

.

Factoring allows for analyzing the individual signs of the linear factors.

2

Analyze the denominator to simplify the inequality.

Since

|x - 1| > 0

for all

x \neq 1

, multiplying both sides of

\frac{x(x - 1)}{|x - 1|} > 0

by

|x - 1|

gives the equivalent inequality

x(x - 1) > 0

.

Multiplying an inequality by a strictly positive quantity preserves the direction of the inequality sign.

3

Solve the quadratic inequality

x(x - 1) > 0

.

The product

x(x - 1)

is positive when both factors have the same sign, yielding the solution set

x < 0

or

x > 1

.

A product of two real terms is positive when both terms are positive or both terms are negative.

4

Compare the rephrased target (

x < 0

or

x > 1

) with the proposed target (

x > 0

).

The range

0 < x < 1

makes

x > 0

true but makes

x(x - 1) > 0

false. Thus, the targets are not equivalent.

Two target questions are algebraically equivalent if and only if they yield identical truth values for all values in the domain.

Anahtar Kavram

Question Stem Simplification and Target Rephrasing

Soru 36Soru →

If $k$ is a non-zero real number, is $k$ an integer?

(1) $\frac{12}{k}$ is an integer.

(2) $k^2 - 3k$ is an 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.

Cevabı ve açıklamayı göster

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

Cevap

Both statements together are sufficient to answer the question, but neither statement alone is sufficient.

The correct response demonstrates that neither statement individually restricts

k

exclusively to integer values, but combining them enforces that

k

is rational (from the reciprocal condition) and has a denominator of 1 (from the quadratic condition), proving conclusively that

k

must be an integer.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

Statement (1) states that

\frac{12}{k} = m

for some non-zero integer

m

, so

k = \frac{12}{m}

. If

m = 1

,

k = 12

(an integer). If

m = 24

,

k = 0.5

(not an integer). Since

k

can be either an integer or a non-integer, Statement (1) is NOT sufficient.

Testing specific values reveals that

k

can be fractional while fulfilling the condition.

2

Evaluate Statement (2) independently.

Statement (2) states that

k^2 - 3k = n

for some integer

n

. If

k = 4

,

k^2 - 3k = 4

(an integer). However, if

k = \frac{3 + \sqrt{13}}{2}

,

k^2 - 3k = 1

(an integer), but

k

is irrational. Since

k

can be an integer or irrational, Statement (2) is NOT sufficient.

Quadratic expressions with integer outputs can have irrational inputs.

3

Evaluate Statements (1) and (2) together.

From Statement (1),

k

must be rational. Let

k = \frac{p}{q}

in lowest terms, where

p

and

q

are coprimes (

\gcd(p, q) = 1

) and

q \ge 1

. Substitute

k = \frac{p}{q}

into Statement (2):

k^2 - 3k = \frac{p^2 - 3pq}{q^2}

. For this expression to be an integer,

q^2

must divide

p(p - 3q)

. Since

\gcd(p, q) = 1

,

q

shares no common prime factors with

p

or

p - 3q

. Thus

\gcd(p^2 - 3pq, q^2) = 1

, which means

q^2

can divide

p^2 - 3pq

if and only if

q^2 = 1

, so

q = 1

. Since

q = 1

,

k = p

, which guarantees

k

is an integer. Thus, the combined statements yield a definitive 'Yes'.

Combining rationality from the first condition with the algebraic divisibility constraint of the second condition eliminates all non-integer possibilities.

Anahtar Kavram

Data Sufficiency logic with integer constraints and rational number representations

Tahmini Süre:2m 0s

Soru 37Soru →

At a technology firm, $60$ software engineers work on Project Alpha, Project Beta, or both. Exactly $40$ engineers work on Project Alpha, and exactly $35$ engineers work on Project Beta. Is the average (arithmetic mean) years of experience of all $60$ engineers combined greater than $7$ years?

(1) The average years of experience of the engineers who work ONLY on Project Alpha is $8$ years, and the average years of experience of the engineers who work ONLY on Project Beta is $5$ years.
(2) The average years of experience of all $40$ engineers on Project Alpha is $8.75$ years.

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.

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient.

Cevap

Both statements together are sufficient to answer the question definitively with a 'Yes', but neither statement alone is sufficient.

Combining both statements provides the exact sum of values across all three mutually exclusive sub-groups (Only Alpha, Only Beta, and Both), yielding a unique overall average of 7.5 years, which yields a definitive 'Yes' answer.

Adım Adım Çözüm

1

Determine the sizes of the non-overlapping and overlapping groups.

Group sizes: Only Alpha = 25, Only Beta = 20, Both = 15.

Using the overlapping sets formula

N(A \cup B) = N(A) + N(B) - N(A \cap B)

, we get

60 = 40 + 35 - N(A \cap B)

, which yields

N(A \cap B) = 15

. Thus,

N(\text{Only Alpha}) = 40 - 15 = 25

and

N(\text{Only Beta}) = 35 - 15 = 20

.

2

Evaluate Statement (1) alone.

Statement (1) is NOT sufficient.

Statement (1) gives

S_{\text{Only A}} = 25 \times 8 = 200

and

S_{\text{Only B}} = 20 \times 5 = 100

. The sum of experience for the

15

overlap engineers (

S_{\text{Both}}

) remains unknown. Total mean

= \frac{300 + S_{\text{Both}}}{60}

. If

S_{\text{Both}} = 150

(mean

10

), total mean

= 7.5 > 7

(Yes). If

S_{\text{Both}} = 30

(mean

2

), total mean

= 5.5 \le 7

(No). Hence, Statement (1) alone is insufficient.

3

Evaluate Statement (2) alone.

Statement (2) is NOT sufficient.

Statement (2) gives the total experience of all

40

engineers on Project Alpha:

S_{\text{Alpha}} = 40 \times 8.75 = 350

. This means

S_{\text{Only A}} + S_{\text{Both}} = 350

. Total mean

= \frac{350 + S_{\text{Only B}}}{60}

. Since

S_{\text{Only B}}

is unknown, the total mean could be

7.5

(if

S_{\text{Only B}} = 100

) or

6.17

(if

S_{\text{Only B}} = 20

). Hence, Statement (2) alone is insufficient.

4

Evaluate Statements (1) and (2) together.

Statements (1) and (2) together are SUFFICIENT.

From Statement (1),

S_{\text{Only A}} = 200

and

S_{\text{Only B}} = 100

. From Statement (2),

S_{\text{Only A}} + S_{\text{Both}} = 350

, which implies

200 + S_{\text{Both}} = 350 \implies S_{\text{Both}} = 150

. The combined total experience is

S_{\text{Total}} = 200 + 100 + 150 = 450

. The overall mean is

\frac{450}{60} = 7.5

years. Since

7.5 > 7

, we get a definitive 'Yes'.

Anahtar Kavram

Data Sufficiency evaluation for combined weighted averages and overlapping set partitions

Soru 38Soru →

If $x$ and $y$ are real numbers, what is the value of $x^3 - y^3$ ?

(1) $x - y = 2$
(2) $x^2 + y^2 = 10$

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.

Cevabı ve açıklamayı göster

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

Cevap

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

Combining Statement (1) and Statement (2) allows us to determine the exact value of

xy = 3

. Substituting

x - y = 2

,

x^2 + y^2 = 10

, and

xy = 3

into the factored form

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

yields

2 \times (10 + 3) = 26

. Because this value is uniquely determined, both statements together are sufficient while neither statement alone is sufficient.

Adım Adım Çözüm

1

Rephrase the target expression using algebraic identities.

The target expression

x^3 - y^3

factors into

(x - y)(x^2 + xy + y^2)

.

To find

x^3 - y^3

, we need the values of

(x - y)

and

(x^2 + xy + y^2)

, or equivalently,

(x - y)

,

(x^2 + y^2)

, and

xy

.

2

Evaluate Statement (1) alone.

Statement (1) gives

x - y = 2

.

Without knowing

x^2 + y^2

or

xy

,

x^3 - y^3

can take infinitely many values (for example, if

x=2, y=0

,

x^3-y^3=8

; if

x=3, y=1

,

x^3-y^3=26

). Thus, Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) alone.

Statement (2) gives

x^2 + y^2 = 10

.

Without knowing

x - y

or

xy

,

x^3 - y^3

can take multiple values (for example, if

x=\sqrt{10}, y=0

,

x^3-y^3=10\sqrt{10}

; if

x=3, y=1

,

x^3-y^3=26

). Thus, Statement (2) alone is NOT sufficient.

4

Combine Statement (1) and Statement (2).

Square Statement (1):

(x - y)^2 = 2^2 \implies x^2 - 2xy + y^2 = 4

. Substitute

x^2 + y^2 = 10

:

10 - 2xy = 4 \implies 2xy = 6 \implies xy = 3

.

By squaring

x - y

and using

x^2 + y^2

, we isolate a unique value for the product term

xy

.

5

Substitute known values into the target expression.

x^3 - y^3 = (x - y)(x^2 + y^2 + xy) = 2 \times (10 + 3) = 26

.

Since the expression evaluates to a single, unique numerical value (

26

), the two statements together are sufficient.

Anahtar Kavram

Algebraic Expression Rephrasing and Solution Set Invariance in Data Sufficiency

Tahmini Süre:2m 0s

Soru 39Soru →

If $a$ and $b$ are real numbers, what is the value of $a^2 - b^2$ ?

(1) $a + b = 7$
(2) $a - b = 3$

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.

Cevabı ve açıklamayı göster

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

Cevap

Both statements together are sufficient, but neither statement alone is sufficient.

Factoring the target expression yields

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

. Neither statement alone gives both factors, so neither statement alone is sufficient. Combining Statement (1), which gives

a + b = 7

, and Statement (2), which gives

a - b = 3

, allows us to calculate

a^2 - b^2 = 7 \times 3 = 21

. Therefore, both statements together are sufficient.

Adım Adım Çözüm

1

Rephrase the target question stem algebraically.

Recognize that

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

. The objective is to find the value of the product

(a + b)(a - b)

.

Simplifying the target expression reduces the problem to finding the sum and difference of the variables.

2

Evaluate Statement (1) independently.

Statement (1) states

a + b = 7

. Knowing only

a + b

leaves

a - b

unknown, so

a^2 - b^2 = 7(a - b)

can take infinitely many values. Statement (1) is insufficient.

A single linear equation with two variables cannot determine the value of a two-factor product unless the second factor is fixed.

3

Evaluate Statement (2) independently.

Statement (2) states

a - b = 3

. Knowing only

a - b

leaves

a + b

unknown, so

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

can take infinitely many values. Statement (2) is insufficient.

A single difference equation does not uniquely determine the sum of two variables.

4

Evaluate Statements (1) and (2) together.

Combining both statements gives

a + b = 7

and

a - b = 3

. Substituting these values into the rephrased target expression yields

a^2 - b^2 = (7)(3) = 21

. A unique numerical value is obtained. Statements combined are sufficient.

Having both component factors allows direct computation of their product without needing to solve for individual values of

a

and

b

first.

Anahtar Kavram

Algebraic Rephrasing and Difference of Squares in Data Sufficiency

Tahmini Süre:1m 0s

Soru 40Soru →

For all non-zero real numbers $m$ and $n$ with $m \neq n$ , the Data Sufficiency target question "Is $\frac{m^2 + n^2}{2mn} > 1$ ?" is algebraically equivalent to the simplified target question "Are $m$ and $n$ of opposite signs?".

Doğru

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Cevabı ve açıklamayı göster

Cevap: False

Cevap

False. The target question "Is

\frac{m^2 + n^2}{2mn} > 1

?" simplifies to "Do

m

and

n

have the same sign?", not opposite signs.

The statement is false. Rephrasing

\frac{m^2 + n^2}{2mn} > 1

yields

\frac{(m-n)^2}{2mn} > 0

. Because

(m-n)^2 > 0

when

m \neq n

, the ratio is positive if and only if

mn > 0

, which requires

m

and

n

to have the same sign, not opposite signs.

Adım Adım Çözüm

1

Subtract 1 from both sides of the inequality to consolidate into a single rational expression.

\frac{m^2 + n^2}{2mn} - 1 > 0 \implies \frac{m^2 - 2mn + n^2}{2mn} > 0

Consolidating terms onto one side avoids cross-multiplying by

2mn

, which has an unknown sign.

2

Factor the numerator quadratic expression.

\frac{(m - n)^2}{2mn} > 0

Recognize that

m^2 - 2mn + n^2

is a perfect square binomial

(m - n)^2

.

3

Determine the sign conditions of the numerator and denominator.

Since

m \neq n

,

(m - n)^2 > 0

is strictly positive. Therefore,

\frac{(m - n)^2}{2mn} > 0

holds if and only if

2mn > 0 \implies mn > 0

.

A fraction with a positive numerator is strictly greater than zero if and only if its denominator is also strictly positive.

4

Translate the condition

mn > 0

into verbal sign properties.

mn > 0

implies that

m

and

n

share the SAME non-zero sign (both positive or both negative).

Comparing this condition to the statement in the prompt reveals that the claim of 'opposite signs' is incorrect.

Anahtar Kavram

Question Stem Simplification and Target Rephrasing

Data Sufficiency

Cevap

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Anahtar Kavram

Sık Yapılan Hatalar

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