Statement Independence Evaluation and Statement Combination Alıştırma Soruları — GMAT

Soru 1Soru →

When evaluating Statement (2) in a GMAT Data Sufficiency question, information introduced solely in Statement (1) may be used to analyze Statement (2) as long as Statement (1) is independently insufficient.

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Cevap: False

Cevap

The statement is false. Statement (2) must always be evaluated in complete isolation from Statement (1).

The statement is false because GMAT Data Sufficiency rules require that Statement (2) be evaluated completely independently of Statement (1). Information from Statement (1) cannot be used to help evaluate Statement (2) alone; the two statements are only combined during a joint evaluation stage if both statements have already been proven insufficient on their own.

Adım Adım Çözüm

1

Recall the fundamental rule of statement evaluation in GMAT Data Sufficiency.

Statement (1) and Statement (2) must each be evaluated independently of one another.

This determines whether either statement alone is sufficient to answer the question before any combination is attempted.

2

Analyze whether information from Statement (1) can be carried over when evaluating Statement (2).

No information, variables, or constraints given in Statement (1) may carry over into the evaluation of Statement (2).

Statement (2) must be tested on its own merits as if Statement (1) did not exist.

3

Identify when statement combination is allowed.

Combining Statement (1) and Statement (2) occurs only after both Statement (1) alone and Statement (2) alone have been determined to be insufficient.

Combining prematurely leads to false sufficiency conclusions.

Anahtar Kavram

Statement Independence in Data Sufficiency

Soru 2Soru →

In a GMAT Data Sufficiency problem, if Statement (1) alone does not provide sufficient information to answer the question stem, but Statement (2) alone provides sufficient information, combining both statements is required to establish overall sufficiency.

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Cevap: False

Cevap

The statement is False. Statement (2) alone is evaluated independently of Statement (1). If Statement (2) alone provides sufficient information to answer the question stem, it is sufficient by itself, and combining both statements is neither required nor correct.

The statement is false because Data Sufficiency evaluation follows a strict sequence: evaluate Statement 1 alone, evaluate Statement 2 alone (disregarding Statement 1), and only combine the statements if both individual statements are insufficient. When Statement 2 alone is sufficient, no statement combination is needed.

Adım Adım Çözüm

1

Evaluate Statement (1) independently

Statement (1) is determined to be insufficient.

Each statement must first be evaluated on its own merits without relying on information from the other statement.

2

Evaluate Statement (2) independently

Statement (2) is determined to be sufficient.

Statement (2) must be evaluated without carrying over any information or constraints from Statement (1).

3

Determine the Data Sufficiency decision logic outcome

Since Statement (2) alone is sufficient while Statement (1) alone is not, the evaluation stops and Statement (2) alone is the correct outcome.

Combining statements is reserved exclusively for scenarios where neither statement alone is sufficient to answer the question stem.

Anahtar Kavram

Statement Independence and Data Sufficiency Decision Logic

Soru 3Soru →

If $k$ and $m$ are non-zero real numbers, is $k > m$ ?

(1) $k^2 > m^2$
(2) $\frac{k - m}{m^2 + 1} > 0$

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.

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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 target question asks if

k > m

. Evaluating Statement (1) alone:

k^2 > m^2

indicates

|k| > |m|

, which permits cases where

k > m

(e.g.,

k=3, m=2

) and cases where

k < m

(e.g.,

k=-5, m=2

). Thus, Statement (1) is insufficient. Evaluating Statement (2) independently:

\frac{k - m}{m^2 + 1} > 0

. For any real number

m

,

m^2 + 1

is strictly positive. Multiplying both sides of the inequality by

(m^2 + 1)

preserves the inequality sign, yielding

k - m > 0

, which simplifies directly to

k > m

. This yields a definitive 'Yes' answer. Therefore, Statement (2) alone is sufficient.

Adım Adım Çözüm

1

Rephrase the target question

The target question asks whether the difference

(k - m)

is strictly positive.

Determining if

k > m

is algebraically equivalent to confirming

k - m > 0

.

2

Evaluate Statement (1) independently

Statement (1) states

k^2 > m^2

, which implies

|k| > |m|

. If

k = 3

and

m = 2

, then

9 > 4

and

3 > 2

(Yes). However, if

k = -5

and

m = 2

, then

25 > 4

but

-5 < 2

(No).

Because both 'Yes' and 'No' outcomes are possible, Statement (1) alone is not sufficient.

3

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

Statement (2) states

\frac{k - m}{m^2 + 1} > 0

. Since

m

is a real number,

m^2 \ge 0

, which guarantees

m^2 + 1 \ge 1 > 0

. Multiplying the inequality by the strictly positive quantity

(m^2 + 1)

preserves the inequality direction, yielding

k - m > 0

, or

k > m

.

This yields a definitive 'Yes' answer, so Statement (2) alone is sufficient.

Anahtar Kavram

Statement Independence in Data Sufficiency

Soru 4Soru →

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

(1) $x = 5$
(2) $y = 3$

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.

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

Cevap

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

Neither statement alone provides values for both

x

and

y

. Evaluating each statement independently shows that one variable remains unknown in each case. When combined, Statement (1) gives

x = 5

and Statement (2) gives

y = 3

, allowing us to uniquely calculate

x + y = 8

. Thus, both statements together are sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently

x = 5

, but

y

is unknown, so

x + y

cannot be uniquely determined.

Statement (1) alone is not sufficient.

2

Evaluate Statement (2) independently

y = 3

, but

x

is unknown, so

x + y

cannot be uniquely determined.

Statement (2) alone is not sufficient. No information from Statement (1) can be carried over during independent evaluation.

3

Combine Statement (1) and Statement (2)

x = 5

and

y = 3

, yielding

x + y = 5 + 3 = 8

.

Combining both statements provides a single unique numerical answer.

Anahtar Kavram

Statement Independence Evaluation and Statement Combination

Soru 5Soru →

If $m$ and $n$ are positive integers, is $m$ an even integer?

(1) $m + n$ is an odd integer.
(2) $n = 5$

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.

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

Cevap

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

Neither statement alone determines the parity of

m

. However, combining Statement (1) (

m+n

is odd) and Statement (2) (

n=5

, an odd integer) proves that

m

must be an even integer. Therefore, both statements together are sufficient.

Adım Adım Çözüm

1

Analyze Statement (1) independently.

Statement (1) states that

m + n

is odd. For the sum of two integers to be odd, one integer must be even and the other must be odd. If

n

is odd,

m

is even (Yes). If

n

is even,

m

is odd (No). Since the parity of

n

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

Determining sufficiency requires a unique Yes or No answer.

2

Analyze Statement (2) independently, ignoring Statement (1).

Statement (2) states that

n = 5

. This gives no information about

m

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

Statement 2 must be evaluated strictly without carrying over information from Statement 1.

3

Combine Statement (1) and Statement (2).

From Statement (2),

n = 5

, which is an odd integer. Substituting

n = 5

into Statement (1),

m + 5

is odd. Since an odd integer added to an even integer yields an odd integer,

m

must be even. This provides a definitive 'Yes' answer.

Combining both statements provides enough information to uniquely answer the question.

Anahtar Kavram

Statement Independence and Combination in Data Sufficiency

Soru 6Soru →

If $a$ and $b$ are non-zero real numbers, is $a > b$ ?

(1) $a^2 b < a b^2$
(2) $\frac{a}{b} + \frac{b}{a} < 0$

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.

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

Cevap

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

The correct option states that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) simplifies to

ab(a - b) < 0

, which is insufficient by itself because the sign of

(a - b)

depends on whether

ab

is positive or negative. Statement (2) simplifies to

ab < 0

(since

a^2 + b^2 > 0

), which alone does not reveal whether

a

or

b

is the larger number. When evaluated together, knowing

ab < 0

allows us to divide

ab(a - b) < 0

by the negative value

ab

and flip the inequality sign, yielding

a - b > 0

(i.e.,

a > b

) definitively.

Adım Adım Çözüm

1

Analyze the target question and simplify Statement (1) algebraically.

Statement (1) can be rewritten as

a^2 b - a b^2 < 0

, which factors into

ab(a - b) < 0

.

Factoring isolates the relationship between

ab

and

(a - b)

.

2

Test Statement (1) independently.

If

a = 1

and

b = 2

, then

ab(a - b) = 2(-1) = -2 < 0

, giving

a < b

(No). If

a = 2

and

b = -1

, then

ab(a - b) = -2(3) = -6 < 0

, giving

a > b

(Yes).

Since Statement (1) permits both a 'Yes' and a 'No' response depending on the signs of

a

and

b

, it is NOT sufficient alone.

3

Evaluate Statement (2) strictly in isolation without using information from Statement (1).

Combining terms gives

\frac{a^2 + b^2}{ab} < 0

. Because

a^2 + b^2 > 0

for non-zero real numbers, the denominator must be negative, so

ab < 0

.

Knowing

ab < 0

implies

a

and

b

have opposite signs, but it does not specify which variable is positive and which is negative (e.g.,

a = 2, b = -1 \implies a > b

, whereas

a = -1, b = 2 \implies a < b

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

4

Combine Statement (1) and Statement (2).

From Statement (2),

ab < 0

. From Statement (1),

ab(a - b) < 0

. Dividing

ab(a - b) < 0

by the negative quantity

ab

requires flipping the inequality sign, yielding

a - b > 0

, or

a > b

.

This establishes a definitive 'Yes' answer to the question, making the statements together sufficient.

Anahtar Kavram

Statement Independence Evaluation and Statement Combination in Data Sufficiency

Soru 7Soru →

In a GMAT Data Sufficiency Value question asking for the unique numerical value of a variable $x$ , if Statement (1) alone restricts $x$ to a set of two distinct real numbers and Statement (2) alone also restricts $x$ to a set of two distinct real numbers, then combining Statement (1) and Statement (2) is guaranteed to determine a unique value for $x$ .

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Cevap: False

Cevap

The statement is false because the intersection of the solution sets of two statements can contain multiple values, failing to yield a unique numerical answer.

The statement is false. Combining two Data Sufficiency statements requires finding the values common to both solution sets. If both statements individually narrow the variable to the exact same pair of values, combining them does not eliminate either value. Consequently, the combined statements still leave multiple possibilities and remain insufficient.

Adım Adım Çözüm

1

Formulate the mathematical condition for combining Data Sufficiency statements

Evaluating Statement (1) and Statement (2) together requires finding the set of values that satisfy both statements simultaneously, which is the set intersection

S_1 \cap S_2

of their individual solution sets

S_1

and

S_2

.

Both statements are taken to be true at the same time when analyzing their combined sufficiency.

2

Analyze whether the intersection of two 2-element sets necessarily contains exactly one element

If

S_1 = \{2, 3\}

and

S_2 = \{2, 3\}

, then

S_1 \cap S_2 = \{2, 3\}

, which contains two elements rather than one.

Two independent DS statements can be algebraically redundant or share identical candidate roots without contradicting one another.

3

Determine the sufficiency outcome of the combined statements

Since

x

can still equal either

2

or

3

, the combined statements do not yield a single, unique value for

x

.

A Value-type Data Sufficiency question requires a single, unambiguous value for a condition to be deemed sufficient.

Anahtar Kavram

Statement Combination and Solution Set Intersection

Soru 8Soru →

If $x$ and $y$ are non-zero real numbers, is $\frac{x}{y} > 1$ ?

(1) $x^2 y > x y^2$
(2) $\frac{x - y}{x} > 0$

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.

Cevabı ve açıklamayı göster

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

Cevap

Statements (1) and (2) TOGETHER are NOT sufficient to determine whether

\frac{x}{y} > 1

.

The choice stating that statements (1) and (2) together are not sufficient is correct because combining both statements permits two valid cases: case 1 (

x=3, y=2

) yields a 'Yes' answer to the stem, while case 2 (

x=-1, y=2

) satisfies both statements but yields a 'No' answer to the stem.

Adım Adım Çözüm

1

Rephrase the target question

The condition

\frac{x}{y} > 1

is equivalent to

\frac{x - y}{y} > 0

, meaning

x - y

and

y

must have the same sign.

Simplifying the question target allows direct testing of signs.

2

Evaluate Statement (1) independently

Statement (1) states

x^2 y > x y^2 \iff x y (x - y) > 0

. If

x = 3, y = 2

, then

xy(x-y) = 6 > 0

and

\frac{x}{y} = 1.5 > 1

(Yes). If

x = -2, y = -3

, then

xy(x-y) = 6 > 0

and

\frac{x}{y} = \frac{2}{3} < 1

(No). Insufficient.

Testing positive and negative pairs shows Statement (1) alone yields both Yes and No.

3

Evaluate Statement (2) independently without carrying over Statement (1) information

Statement (2) states

\frac{x - y}{x} > 0

, meaning

x

and

x - y

have the same sign. If

x = 3, y = 2

,

\frac{1}{3} > 0

and

\frac{x}{y} = 1.5 > 1

(Yes). If

x = 2, y = -3

,

\frac{5}{2} > 0

and

\frac{x}{y} = -\frac{2}{3} < 1

(No). Insufficient.

Statement (2) evaluated strictly in isolation also yields both Yes and No.

4

Evaluate Statements (1) and (2) combined

From Statement (1),

xy(x-y) > 0

. From Statement (2),

\frac{x-y}{x} > 0 \implies x(x-y) > 0

. Combining these implies

y > 0

. Since

y > 0

, if

x > 0

, then

x - y > 0 \implies x > y > 0 \implies \frac{x}{y} > 1

(Yes). However, if

x < 0

, since

y > 0

,

x - y < 0

, satisfying

x(x-y) > 0

and

xy(x-y) > 0

. For example,

x = -1, y = 2

satisfies both statements but gives

\frac{x}{y} = -0.5 < 1

(No). Insufficient.

Even combined, two distinct cases exist (

x > y > 0

giving Yes, and

x < 0 < y

giving No).

Anahtar Kavram

Statement Independence Evaluation and Statement Combination with Real Number Sign Traps

Soru 9Soru →

In a GMAT Data Sufficiency Yes/No question asking whether $x > 0$ , if Statement (1) implies $x^3 - x = 0$ and Statement (2) implies $x^2 - 1 = 0$ , then combining both statements is sufficient to answer the question because combining them narrows the set of possible values of $x$ to a smaller set than either statement alone.

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Cevap: False

Cevap

False

The correct evaluation is False. In GMAT Data Sufficiency Yes/No questions, combining statements is sufficient only if every remaining value in the combined solution set yields the exact same answer (either all 'Yes' or all 'No'). Here, the combined set

\{-1, 1\}

contains

x = -1

(which yields 'No') and

x = 1

(which yields 'Yes'), so the combined statements remain insufficient.

Adım Adım Çözüm

1

Analyze Statement (1) independently.

Solving

x^3 - x = 0

yields

x(x - 1)(x + 1) = 0

, so

x \in \{-1, 0, 1\}

. Testing these values against the question 'Is

x > 0

?':

x = -1

gives No,

x = 0

gives No, and

x = 1

gives Yes. Because both Yes and No are possible, Statement (1) alone is insufficient.

Each statement must first be evaluated independently by checking if all allowed values yield a uniform answer.

2

Analyze Statement (2) independently.

Solving

x^2 - 1 = 0

yields

x \in \{-1, 1\}

. Testing these values:

x = -1

gives No and

x = 1

gives Yes. Because both Yes and No are possible, Statement (2) alone is insufficient.

Statement (2) must be evaluated strictly without carrying over any information or restrictions from Statement (1).

3

Evaluate the combined statement information.

Taking the intersection of the solution sets from both statements gives

\{-1, 0, 1\} \cap \{-1, 1\} = \{-1, 1\}

.

Combining statements requires finding all values that satisfy both Statement (1) and Statement (2) simultaneously.

4

Determine sufficiency of the combined set for a Yes/No question.

Within the combined set

\{-1, 1\}

,

x = -1

yields 'No' (

x \ngtr 0

) while

x = 1

yields 'Yes' (

x > 0

). Since a definitive Yes or No answer cannot be reached, the statements together remain insufficient.

For a Yes/No Data Sufficiency question, sufficiency requires that EVERY value in the combined solution set leads to the exact same answer.

Anahtar Kavram

Data Sufficiency Statement Combination for Yes/No Questions

Soru 10Soru →

If $x$ and $y$ are real numbers, is $x^2 + y^2 < 25$ ?

(1) $x + y = 7$
(2) $(x - 3)^2 + (y - 4)^2 = 0$

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.

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) allows multiple outcomes for

x^2 + y^2

(both values less than 25 and values greater than or equal to 25), so it is not sufficient. Statement (2) forces

x=3

and

y=4

because the sum of non-negative real squares can only be zero when each square term is zero. Substituting

x=3

and

y=4

yields

x^2 + y^2 = 25

, which conclusively answers 'No' to the question stem (

25 < 25

is false). Hence, Statement (2) alone is sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently

Statement (1) gives

x + y = 7

. If

x = 3.5

and

y = 3.5

, then

x^2 + y^2 = 12.25 + 12.25 = 24.5 < 25

(Yes). If

x = 7

and

y = 0

, then

x^2 + y^2 = 49 + 0 = 49 \not< 25

(No).

Since both 'Yes' and 'No' are possible, Statement (1) alone is insufficient.

2

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

Statement (2) states

(x - 3)^2 + (y - 4)^2 = 0

. Since the square of a real number is non-negative, the sum of two squared terms can equal zero if and only if each term is zero:

x - 3 = 0 \implies x = 3

and

y - 4 = 0 \implies y = 4

.

This uniquely fixes the values of

x

and

y

.

3

Calculate the target expression using values from Statement (2)

Substitute

x = 3

and

y = 4

into

x^2 + y^2

:

3^2 + 4^2 = 9 + 16 = 25

. The question asks if

x^2 + y^2 < 25

. Since

25 < 25

is false, the answer is a definitive 'No'.

A definitive 'No' answer means the statement provides enough information to answer the question, so Statement (2) alone is sufficient.

Anahtar Kavram

Statement Independence in Data Sufficiency and Definitive Yes/No Decision Rules

Tahmini Süre:2m 0s

Soru 11Soru →

In a GMAT Data Sufficiency problem asking for the unique value of an integer $x$ , Statement (1) establishes that $x \in \{3, 7\}$ and Statement (2) establishes that $|x - 5| = 2$ . Evaluating both statements together provides sufficient information to determine a unique value for $x$ .

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Cevap: False

Cevap

The statement is False. Combining both statements yields the intersection set {3, 7}, which still contains two possible values for x and therefore does not uniquely determine x.

The statement is false because solving Statement (2)'s absolute value equation

|x - 5| = 2

produces

x = 7

or

x = 3

, which is identical to the candidate set given by Statement (1). Combining two identical solution sets leaves the candidate set as

\{3, 7\}

, which does not yield a single unique value for

x

.

Adım Adım Çözüm

1

Evaluate Statement (1) solution set

Statement (1) restricts

x

to the set

\{3, 7\}

.

Statement (1) explicitly provides two candidate values for

x

.

2

Evaluate Statement (2) solution set independently

Statement (2) yields

|x - 5| = 2 \implies x = 7

or

x = 3

. The solution set is

\{3, 7\}

.

Solving the absolute value equation yields the two roots 3 and 7 without using any information from Statement (1).

3

Combine Statement (1) and Statement (2)

The intersection of the two solution sets is

\{3, 7\} \cap \{3, 7\} = \{3, 7\}

.

When combining statements in Data Sufficiency, valid values must satisfy both statements simultaneously.

4

Determine sufficiency of the combined statements

Since

x

can still be either 3 or 7, a single unique value is not determined. The combined statements are insufficient.

A Value Data Sufficiency question requires a single, unique numerical value to be considered sufficient.

Anahtar Kavram

Statement Combination and Redundancy in Data Sufficiency

Soru 12Soru →

A retailer sold standard boxes at $\$s$ each and premium boxes at $\$p$ each yesterday, generating a total revenue of $\$900$ . If the retailer sold $x$ standard boxes and $y$ premium boxes, where $x$ and $y$ are positive integers, did the retailer sell more standard boxes than premium boxes?

(1) Standard boxes cost $\$15$ each and premium boxes cost $\$25$ each.
(2) The average (arithmetic mean) price of all boxes sold yesterday was $\$18$ , and premium boxes cost $\$10$ more per box than standard boxes.

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.

The correct option is the one stating that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) allows multiple positive integer pairs for

(x, y)

such that

x > y

in some cases and

x < y

in others, making it insufficient on its own. Statement (2) alone leaves the base price

s

unknown, so the ratio of standard to premium boxes cannot be determined without Statement (1). Combining

s = 15

from Statement (1) with Statement (2) establishes that

\frac{x}{y} = \frac{7}{3}

, which definitively proves

x > y

.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

Statement (1) provides

s = 15

and

p = 25

, yielding the equation

15x + 25y = 900

, which simplifies to

3x + 5y = 180

. Testing positive integer solutions gives multiple valid pairs: for instance,

(x, y) = (55, 3)

yields

x > y

(Yes), while

(x, y) = (10, 30)

yields

x < y

(No). Because both Yes and No are possible, Statement (1) alone is NOT sufficient.

A statement must yield a single definitive Yes or No answer to be sufficient.

2

Evaluate Statement (2) independently without using information from Statement (1).

Statement (2) states that

p = s + 10

and the average price is

\frac{sx + py}{x + y} = 18

. Substituting

p = s + 10

gives

\frac{sx + (s + 10)y}{x + y} = 18

, which simplifies to

s + \frac{10y}{x + y} = 18

. Because the individual box price

s

is unknown,

\frac{y}{x + y}

can vary depending on

s

(e.g., if

s = 10

,

x < y

; if

s = 15

,

x > y

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

Information from Statement (1) must never be carried over when testing Statement (2) independently.

3

Combine Statements (1) and (2).

From Statement (1),

s = 15

. Substituting

s = 15

into the simplified average equation from Statement (2),

15 + \frac{10y}{x + y} = 18

, gives

\frac{10y}{x + y} = 3 \implies 10y = 3x + 3y \implies 7y = 3x \implies \frac{x}{y} = \frac{7}{3}

. Since

x

and

y

are positive integers,

\frac{x}{y} > 1

, which definitively proves that

x > y

(Yes).

Combining the known price from Statement (1) with the weighted average equation from Statement (2) uniquely determines the ratio of

x

to

y

.

Anahtar Kavram

Statement Independence and Statement Combination in Data Sufficiency

Soru 13Soru →

In GMAT Data Sufficiency, when evaluating whether Statement (2) alone is sufficient to answer the question stem, any information or numeric constraints introduced solely within Statement (1) must be completely disregarded.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True

The statement is true because standard GMAT Data Sufficiency procedure requires testing Statement (1) alone, then testing Statement (2) alone in complete isolation, and only combining their information if neither statement alone is sufficient.

Adım Adım Çözüm

1

Recall the structural rule for evaluating individual statements in GMAT Data Sufficiency.

Statement (1) and Statement (2) must each be evaluated in complete isolation before any combining phase.

The standard evaluation protocol requires testing whether each statement independently provides sufficient information to answer the question.

2

Analyze whether Statement (1) constraints can be active during Statement (2) standalone analysis.

Statement (1) constraints must be erased from memory when evaluating Statement (2).

Allowing Statement (1) facts to persist during the standalone assessment of Statement (2) causes statement carryover error, leading to incorrect sufficiency conclusions.

Anahtar Kavram

Statement Independence in Data Sufficiency

Soru 14Soru →

If $m$ and $n$ are positive integers, what is the value of $m + n$ ?

(1) $m^2 - n^2 = 21$
(2) $m - n = 3$

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.

The correct option identifies that neither statement alone provides enough information to determine a single value for

m + n

, but combining both statements allows us to substitute

m - n = 3

directly into the factored form

(m - n)(m + n) = 21

, yielding

m + n = 7

uniquely.

Adım Adım Çözüm

1

Analyze the question target and algebraic constraints

The target is to find a unique value for

m + n

, where

m

and

n

are positive integers.

Establishing positive integer constraints is critical when evaluating factor possibilities.

2

Evaluate Statement (1) independently

Statement (1) states

m^2 - n^2 = 21

, which factors to

(m - n)(m + n) = 21

. Positive integer factor pairs for

21

are

(1, 21)

and

(3, 7)

. If

m - n = 1

and

m + n = 21

, then

m + n = 21

. If

m - n = 3

and

m + n = 7

, then

m + n = 7

. Two values are possible, so Statement (1) alone is INSUFFICIENT.

A single unique value cannot be determined from Statement (1) alone.

3

Evaluate Statement (2) independently without carrying over Statement (1) data

Statement (2) states

m - n = 3

. Infinitely many positive integer pairs

(m, n)

satisfy this equation (e.g.,

(4, 1) \implies m+n=5

;

(5, 2) \implies m+n=7

). Thus, Statement (2) alone is INSUFFICIENT.

Statement (2) must be evaluated strictly isolated from Statement (1).

4

Combine Statement (1) and Statement (2)

Substitute

m - n = 3

into

(m - n)(m + n) = 21

:

3(m + n) = 21 \implies m + n = 7

. A single unique value is determined.

Combining both statements resolves the ambiguity present in each statement individually.

Anahtar Kavram

Statement Independence Evaluation and Algebraic Combination in Data Sufficiency

Tahmini Süre:1m 30s

Soru 15Soru →

In a GMAT Data Sufficiency question asking for the unique value of a variable $x$ , if Statement (1) alone limits the possible values of $x$ to the set $\{2, 5\}$ and Statement (2) alone limits the possible values of $x$ to the set $\{5, 8\}$ , then the two statements combined are sufficient to answer the question.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True. Combining two consistent Data Sufficiency statements yields a solution set equal to the intersection of their individual solution sets. The intersection of

\{2, 5\}

and

\{5, 8\}

is

\{5\}

, which uniquely identifies

x

and is therefore sufficient.

The statement is true because taking the intersection of the two solution sets

\{2, 5\}

and

\{5, 8\}

leaves only the single value

x = 5

, satisfying the GMAT Data Sufficiency requirement for a unique value.

Adım Adım Çözüm

1

Evaluate Statement (1) alone

Statement (1) yields

x \in \{2, 5\}

, which contains two possible values and is insufficient on its own.

A Value Data Sufficiency question requires a single unique numerical value for sufficiency.

2

Evaluate Statement (2) alone independently

Statement (2) yields

x \in \{5, 8\}

, which contains two possible values and is insufficient on its own.

Statement (2) must be evaluated without carrying over any constraints from Statement (1).

3

Combine Statement (1) and Statement (2)

The combined solution set is

\{2, 5\} \cap \{5, 8\} = \{5\}

.

Combining statements requires identifying values that satisfy both statements simultaneously.

4

Determine sufficiency of the combined solution set

Since

x = 5

is the only value in the combined set, the statements together provide a single unique answer.

A single unique result satisfies the criterion for Data Sufficiency.

Anahtar Kavram

Statement Combination and Solution Set Intersection

Soru 16Soru →

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?

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.

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.

Adım Adım Çözüm

1

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.

2

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

or

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.

3

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.

Anahtar Kavram

Statement Independence & Stem Constraint Application

Soru 17Soru →

In GMAT Data Sufficiency, when evaluating whether Statement (2) alone is sufficient to answer the question stem, any algebraic conditions or variable constraints introduced exclusively within Statement (1) must be completely disregarded.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True. Statement (2) must be evaluated in strict isolation from Statement (1).

Each statement in a Data Sufficiency question must be evaluated independently first. Statement (1) constraints cannot be referenced or assumed during the standalone evaluation of Statement (2).

Adım Adım Çözüm

1

Recall the fundamental procedural rules of GMAT Data Sufficiency.

Statement (1) and Statement (2) must be evaluated independently before any combining step is attempted.

Independent evaluation determines whether either statement alone answers the question stem.

2

Apply the rule of independence specifically to the evaluation phase of Statement (2).

When evaluating Statement (2), Statement (1) is treated as non-existent.

Carrying over constraints from Statement (1) creates a false sense of sufficiency for Statement (2) alone.

Anahtar Kavram

Statement Independence in Data Sufficiency

Soru 18Soru →

A manufacturing facility produced two types of products, Type X and Type Y. In July, the total cost to produce $N$ units of Type X and $M$ units of Type Y was $4,200. What was the unit manufacturing cost of a Type X product?

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 (2) alone provides sufficient data because the total cost for Type X products is given as

2,800 for

N = 70

units. Dividing

2,800 by 70 gives a unique unit manufacturing cost of $40 for Type X products, completely independent of Statement (1) and without needing information regarding Type Y.

Adım Adım Çözüm

1

Rephrase the target question

Let

C_x

be the unit cost of Type X and

C_y

be the unit cost of Type Y. Total cost is

N \cdot C_x + M \cdot C_y = 4200

. The target question asks for the unique value of

C_x

.

Clarify what specific variable needs to be solved for.

2

Evaluate Statement (1) independently

Statement (1) tells us

C_y = 30

and

N = 2M

. Substituting into the total cost equation gives

N \cdot C_x + (N/2) \cdot 30 = 4200

, which simplifies to

N(C_x + 15) = 4200

. Since

N

is unknown,

C_x

cannot be uniquely determined. Statement (1) alone is NOT sufficient.

Determine if Statement (1) yields a single unique numerical value for

C_x

.

3

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

Statement (2) states that the total cost for Type X products is

N \cdot C_x = 2800

, and

N = 70

. Solving for

C_x

gives

C_x = 2800 / 70 = 40

. This uniquely answers the question stem directly. Statement (2) alone IS sufficient.

Test Statement (2) strictly in isolation.

Anahtar Kavram

Evaluating Statement Independence in Data Sufficiency

Soru 19Soru →

In a GMAT Data Sufficiency question asking for the specific numerical values of two variables $x$ and $y$ , if Statement (1) provides the equation $2x + 3y = 12$ and Statement (2) provides the equation $4x + 6y = 24$ , combining both statements is sufficient to determine unique values for $x$ and $y$ .

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The statement is False.

The correct answer is False because Statement (2) is a scalar multiple of Statement (1). Linearly dependent equations do not reduce the number of unknown variables or narrow down the solution set to a single unique pair

(x, y)

. Therefore, combining both statements does not yield sufficient information to answer the question stem.

Adım Adım Çözüm

1

Analyze Statement (1) equation

Statement (1) provides

2x + 3y = 12

, which defines a line with infinitely many valid coordinate pairs

(x, y)

.

A single linear equation in two variables has infinitely many solutions.

2

Analyze Statement (2) equation and test for linear independence

Dividing Statement (2),

4x + 6y = 24

, by

2

yields

2x + 3y = 12

, which is identical to Statement (1).

Linearly dependent equations provide redundant data and do not provide a distinct second constraint.

3

Evaluate the combination of Statement (1) and Statement (2)

Combining both statements produces no new information beyond

2x + 3y = 12

, leaving infinitely many solution pairs.

Determining unique values for two variables requires two linearly independent equations. Because these statements are dependent, combining them remains insufficient.

Anahtar Kavram

Linear Independence in Statement Combination

Soru 20Soru →

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

(1) $x + y = 10$
(2) $3x - 5y = 0$

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.

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.

Evaluating Statement (2) independently of Statement (1) shows that the equation

3x - 5y = 0

can be rearranged to

3x = 5y

. Dividing both sides by

3y

(since

y \neq 0

) yields

\frac{x}{y} = \frac{5}{3}

. This gives a single, unique value for the target ratio without needing any information from Statement (1). Meanwhile, Statement (1) alone (

x + y = 10

) allows infinitely many pairs of non-zero real numbers

(x, y)

, resulting in different values for the ratio

\frac{x}{y}

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

Adım Adım Çözüm

1

Evaluate Statement (1) independently

Statement (1) gives

x + y = 10

. If

x = 5

and

y = 5

, then

\frac{x}{y} = 1

. If

x = 8

and

y = 2

, then

\frac{x}{y} = 4

.

Since multiple values are possible for the ratio

\frac{x}{y}

, Statement (1) alone is not sufficient.

2

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

Statement (2) gives

3x - 5y = 0 \implies 3x = 5y \implies \frac{x}{y} = \frac{5}{3}

.

Since

y \neq 0

, dividing both sides by

3y

yields a single, unique value of

\frac{5}{3}

for the ratio

\frac{x}{y}

.

3

Conclude Data Sufficiency determination

Statement (2) alone is sufficient, whereas Statement (1) alone is not sufficient.

Statement (2) independently answers the question stem completely.

Anahtar Kavram

Evaluating Statement 2 independently of Statement 1 in Data Sufficiency

Statement Independence Evaluation and Statement Combination

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