Data Sufficiency Practice Questions — GMAT

Question 1Question →

If $x$ is a nonzero real number, is $\frac{x^2 - 4x}{x} > 0$ ?

(1) $x > 5$
(2) $x^2 > 16$

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.

Simplifying the target expression

\frac{x^2 - 4x}{x}

for

x \neq 0

yields

x - 4

. Thus, the question target rephrases to 'Is

x > 4

?'. Statement (1) states

x > 5

, which guarantees

x > 4

, yielding a definitive 'Yes' answer. Statement (2) allows

x > 4

or

x < -4

, yielding both 'Yes' and 'No' outcomes. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Simplify the question stem expression

Since

x \neq 0

, factor out

x

from the numerator:

\frac{x(x - 4)}{x} = x - 4

.

Simplifying the expression reveals the underlying condition required by the question stem.

2

Rephrase the target question

The original question 'Is

\frac{x^2 - 4x}{x} > 0

?' simplifies to 'Is

x - 4 > 0

?', which is equivalent to 'Is

x > 4

?'.

Target rephrasing turns a fraction inequality into a simple comparison.

3

Evaluate Statement (1):

x > 5

If

x > 5

, then

x

is strictly greater than 4. The answer to 'Is

x > 4

?' is a definitive YES.

Since Statement (1) provides a conclusive 'Yes', Statement (1) alone is sufficient.

4

Evaluate Statement (2):

x^2 > 16

Taking the square root gives

|x| > 4

, meaning

x > 4

or

x < -4

.
- If

x = 5

, then

x > 4

(YES).
- If

x = -5

, then

x < 4

(NO).

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

Key Concept

Question Stem Simplification and Target Rephrasing

Question 2Question →

Is $x$ an integer?

(1) $3x$ is an integer.
(2) $5x$ 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.

Show answer & explanation

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

Answer

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

Evaluating Statement (1) alone shows

x

could be

\frac{1}{3}

(not an integer) or

1

(an integer), so Statement (1) is insufficient. Evaluating Statement (2) alone shows

x

could be

\frac{1}{5}

(not an integer) or

1

(an integer), so Statement (2) is insufficient. Taking both statements together,

3x = a

and

5x = b

for integers

a

and

b

. Subtracting

5x

from

2(3x)

yields

6x - 5x = x = 2a - b

. Because integers are closed under multiplication and subtraction,

2a - b

must be an integer, confirming that

x

is definitively an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently

If

3x = a

where

a

is an integer, then

x = \frac{a}{3}

. If

a = 1

,

x = \frac{1}{3}

(not an integer). If

a = 3

,

x = 1

(an integer).

Since

x

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

2

Evaluate Statement (2) independently

If

5x = b

where

b

is an integer, then

x = \frac{b}{5}

. If

b = 1

,

x = \frac{1}{5}

(not an integer). If

b = 5

,

x = 1

(an integer).

Since

x

can be either an integer or a non-integer, Statement (2) alone is not sufficient.

3

Evaluate Statements (1) and (2) together

Since

3x

and

5x

are both integers, their linear combination

2(3x) - 5x = 6x - 5x = x

must also be an integer.

The difference between two integers is always an integer, proving conclusively that

x

is an integer.

Key Concept

Linear combinations of real numbers and integer constraints in Data Sufficiency

Estimated Time:1m 0s

Question 3Question →

In a GMAT Data Sufficiency 'Value' question asking for the numerical value of $x + y$ , a statement that establishes that $x + y$ can equal either $5$ or $-5$ is considered sufficient.

True

False

Show answer & explanation

Answer: False

Answer

False

The assertion is false because GMAT 'Value' Data Sufficiency questions require finding a single, unique numerical value for the target expression. Obtaining two possible values (

5

or

-5

) means the value is not uniquely determined, rendering the statement insufficient.

Step-by-Step Solution

1

Identify the question type as a 'Value' Data Sufficiency problem.

The target requirement is to find one unique numerical value for

x + y

.

Value DS questions require absolute numerical uniqueness.

2

Evaluate the information provided by the statement.

The statement yields two possible values:

5

and

-5

.

Having multiple values means the exact value of

x + y

cannot be uniquely determined.

3

Determine sufficiency.

Since two distinct values exist, the statement is insufficient, making the assertion false.

Sufficiency in Value DS strictly demands a single unique outcome.

Key Concept

Value Data Sufficiency Decision Logic

Estimated Time:1m 0s

Question 4Question →

For all non-zero real numbers $a$ and $b$ with $|a| \neq |b|$ , the Data Sufficiency Yes/No target question "Is $\frac{a^2 - b^2}{ab} > 0$ ?" is algebraically equivalent to asking whether $a$ and $b$ have the same sign when $|a| > |b|$ , or different signs when $|a| < |b|$ .

True

False

Show answer & explanation

Answer: True

Answer

True. The rephrased question accurately state the necessary and sufficient conditions for the fraction

\frac{a^2 - b^2}{ab}

to be positive.

The quotient

\frac{a^2 - b^2}{ab}

is positive when both

a^2 - b^2

and

ab

are positive, or when both are negative. If

a

and

b

have the same sign (

ab > 0

), then

a^2 - b^2 > 0

, which simplifies to

a^2 > b^2

or

|a| > |b|

. If

a

and

b

have different signs (

ab < 0

), then

a^2 - b^2 < 0

, which simplifies to

a^2 < b^2

or

|a| < |b|

. Hence, the statement correctly describes the target simplification.

Step-by-Step Solution

1

Set up the quotient inequality condition

The fraction

\frac{a^2 - b^2}{ab} > 0

holds when numerator

a^2 - b^2

and denominator

ab

have matching signs.

For any real fraction

\frac{N}{D} > 0

, either (

N > 0

and

D > 0

) or (

N < 0

and

D < 0

).

2

Analyze Case 1 where denominator

ab > 0

a

and

b

have the same sign. In this case,

\frac{a^2 - b^2}{ab} > 0 \implies a^2 - b^2 > 0 \implies |a| > |b|

.

When

ab > 0

, multiplying both sides of the inequality by

ab

preserves the inequality sign.

3

Analyze Case 2 where denominator

ab < 0

a

and

b

have different signs. In this case,

\frac{a^2 - b^2}{ab} > 0 \implies a^2 - b^2 < 0 \implies |a| < |b|

.

When

ab < 0

, multiplying both sides of the inequality by

ab

flips the inequality direction.

4

Combine the cases to evaluate the target rephrasing

The target question simplifies to: "Do

a

and

b

have the same sign with

|a| > |b|

, or opposite signs with

|a| < |b|

?"

Both cases together form the complete rephrased condition for the original statement to be true.

Key Concept

Algebraic Rephrasing of Quotients involving Absolute Values and Variable Signs

Question 5Question →

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

(1) $x^3$ is an integer.
(2) $x^5$ 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.

Show answer & explanation

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

Answer

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

Evaluating each statement independently leaves open the possibility that

x

is an irrational root like

\sqrt[3]{2}

or

\sqrt[5]{2}

, making neither statement alone sufficient. Combining both statements allows us to express

x

as

\frac{(x^3)^2}{x^5}

. Because both

x^3

and

x^5

are integers,

x

must be a rational number. For any rational number whose cube is an integer, its denominator must equal 1, which proves conclusively that

x

is an integer. Thus, both statements together are sufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is insufficient.

If

x = 2

, then

x^3 = 8

(an integer), so

x

is an integer (Yes). However, if

x = \sqrt[3]{2}

, then

x^3 = 2

(an integer), but

x

is not an integer (No). Since both Yes and No answers are possible, Statement (1) alone is not sufficient.

2

Evaluate Statement (2) independently.

Statement (2) is insufficient.

If

x = 2

, then

x^5 = 32

(an integer), so

x

is an integer (Yes). However, if

x = \sqrt[5]{2}

, then

x^5 = 2

(an integer), but

x

is not an integer (No). Since both Yes and No answers are possible, Statement (2) alone is not sufficient.

3

Combine Statement (1) and Statement (2).

The combined statements are sufficient.

Notice that

\frac{(x^3)^2}{x^5} = \frac{x^6}{x^5} = x

. Since

x^3

is an integer,

(x^3)^2

is also an integer. Since

x^5

is an integer,

x = \frac{(x^3)^2}{x^5}

is the ratio of two integers, meaning

x

must be a rational number. If

x = \frac{a}{b}

in simplest form where

a

and

b

are positive integers with

\gcd(a,b)=1

, then

x^3 = \frac{a^3}{b^3}

being an integer implies

b^3 = 1

, so

b = 1

. Thus,

x = a

, which means

x

must be an integer. This yields a definitive 'Yes' answer.

Key Concept

Integer constraints versus real numbers in Data Sufficiency and quotient relationships of exponent powers

Question 6Question →

If $x$ and $y$ are positive integers, is $x^3 - y^3$ divisible by $3$ ?

(1) $x + 2y$ is divisible by $3$ .
(2) $x^2 - y^2$ is divisible by $3$ .

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.

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 question asks whether

x^3 - y^3

is divisible by 3. By Fermat's Little Theorem or factoring

a^3 - a = a(a-1)(a+1)

(the product of three consecutive integers, always divisible by 3), any integer cubed has the same remainder when divided by 3 as the integer itself (

a^3 \equiv a \pmod 3

). Therefore,

x^3 - y^3

is a multiple of 3 if and only if

x - y

is a multiple of 3. Statement (1) specifies that

x + 2y

is divisible by 3. We can rewrite

x + 2y

as

(x - y) + 3y

. Because

3y

is automatically a multiple of 3,

(x - y)

must also be divisible by 3. This yields a definitive 'Yes', so Statement (1) alone is sufficient. Statement (2) states that

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

is divisible by 3. Because 3 is prime, this requires 3 to divide either

(x - y)

or

(x + y)

. If

x = 4, y = 1

, 3 divides

(x - y) = 3

, giving a 'Yes'. If

x = 2, y = 1

, 3 divides

(x + y) = 3

while

(x - y) = 1

, giving

x^3 - y^3 = 7

, which is a 'No'. Hence, Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the question stem using algebraic modular properties.

Since

a^3 \equiv a \pmod 3

for any integer

a

,

x^3 - y^3 \equiv x - y \pmod 3

. Thus,

x^3 - y^3

is divisible by 3 if and only if

x - y

is divisible by 3.

Simplifying the target expression reduces the problem to determining whether 3 divides

(x - y)

.

2

Evaluate Statement (1):

x + 2y

is divisible by 3.

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

. Since

3y

is always a multiple of 3,

(x + 2y)

is divisible by 3 if and only if

(x - y)

is divisible by 3.

This guarantees a definitive 'Yes' to the target question. Statement (1) alone is SUFFICIENT.

3

Evaluate Statement (2):

x^2 - y^2

is divisible by 3.

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

. Since 3 is prime, 3 must divide

(x - y)

or

(x + y)

. If

x = 4, y = 1

, then

x^2 - y^2 = 15

(divisible by 3) and

x^3 - y^3 = 63

(divisible by 3) -> YES. If

x = 2, y = 1

, then

x^2 - y^2 = 3

(divisible by 3), but

x^3 - y^3 = 7

(NOT divisible by 3) -> NO.

Statement (2) yields both 'Yes' and 'No' cases. Statement (2) alone is NOT SUFFICIENT.

Key Concept

Divisibility and Modular Congruences in Integer Properties

Estimated Time:2m 0s

Question 7Question →

For all real numbers $x$ and $y$ such that $xy \neq 0$ and $|x| \neq |y|$ , the Data Sufficiency Yes/No target question "Is $\frac{x^2 y - x y^2}{x^3 y - x y^3} > 0$ ?" is algebraically equivalent to asking "Is $x + y > 0$ ?".

True

False

Show answer & explanation

Answer: True

Answer

The statement is True because algebraic factorization and cancellation simplify the target inequality directly to asking whether

x + y > 0

.

The statement is True. Factoring the given expression yields

\frac{xy(x - y)}{xy(x - y)(x + y)}

, which simplifies to

\frac{1}{x + y}

. The inequality

\frac{1}{x + y} > 0

is satisfied if and only if

x + y > 0

.

Step-by-Step Solution

1

Factor the numerator of the rational expression in the target question.

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

Extract the greatest common monomial factor

xy

.

2

Factor the denominator of the rational expression.

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

Extract

xy

and then apply the difference of squares formula

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

.

3

Cancel common factors between the numerator and denominator.

\frac{xy(x - y)}{xy(x - y)(x + y)} = \frac{1}{x + y}

Since

xy \neq 0

and

|x| \neq |y|

,

xy(x - y) \neq 0

, allowing safe cancellation.

4

Rephrase the target inequality

\frac{1}{x + y} > 0

.

\frac{1}{x + y} > 0 \iff x + y > 0

A unit fraction is positive if and only if its denominator is strictly positive.

Key Concept

Target stem simplification via algebraic factoring and difference of squares

Question 8Question →

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.

True

False

Show answer & explanation

Answer: False

Answer

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.

Step-by-Step Solution

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.

Key Concept

Statement Independence in Data Sufficiency

Question 9Question →

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.

True

False

Show answer & explanation

Answer: False

Answer

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.

Step-by-Step Solution

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.

Key Concept

Statement Independence and Data Sufficiency Decision Logic

Question 10Question →

Is the integer $n$ odd?

(1) $n$ is a multiple of 4.
(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.

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.

Statement (1) tells us that

n

is a multiple of 4, which guarantees that

n

is even. This allows us to answer the question 'Is

n

odd?' with a definitive and unambiguous 'No'. In Data Sufficiency Yes/No questions, a consistent 'No' answer means the statement is sufficient. Statement (2) allows

n

to be 6 (even) or 7 (odd), yielding both 'No' and 'Yes' responses, so it is insufficient. Therefore, the statement stating that Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient, is the correct choice.

Step-by-Step Solution

1

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

A statement is sufficient if it yields a consistent 'Yes' OR a consistent 'No'. It is insufficient only if it yields 'Yes' for some values and 'No' for others.

Data Sufficiency requires determinacy, not necessarily an affirmative response.

2

Evaluate Statement (1):

n

is a multiple of 4.

Any multiple of 4 can be written as

n = 4k

for some integer

k

, which is divisible by 2 and therefore even. Since

n

is even, the answer to 'Is

n

odd?' is a definitive 'No'.

Since the answer is always 'No', Statement (1) alone is SUFFICIENT.

3

Evaluate Statement (2):

n > 5

.

If

n = 6

,

n

is even (Answer: No). If

n = 7

,

n

is odd (Answer: Yes). Since both 'Yes' and 'No' are possible, Statement (2) alone is INSUFFICIENT.

Statement (2) does not fix the parity of

n

.

Key Concept

In Yes/No Data Sufficiency questions, a statement that conclusively answers 'No' to the question stem is SUFFICIENT.

Question 11Question →

A total of $30$ tourists visited a city, and each tourist visited Museum X, Museum Y, or both. How many of the tourists visited both Museum X and Museum Y?

(1) $20$ of the tourists visited Museum X.
(2) $15$ of the tourists visited Museum 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.

Show answer & explanation

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

Answer

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. By applying the formula

\text{Total} = \text{Set A} + \text{Set B} - \text{Both}

, we see that knowing the total (

30

) along with both individual sets (

20

and

15

) allows us to solve for the overlap uniquely (

5

). Neither statement alone provides both set totals.

Step-by-Step Solution

1

Rephrase the question stem using the overlapping set formula.

The total number of tourists is given by

\text{Total} = N(X) + N(Y) - N(X \cap Y)

, where

\text{Total} = 30

. To find

N(X \cap Y)

, we need the sum

N(X) + N(Y)

.

Establishing the algebraic relation clarifies what specific information is required from the statements.

2

Evaluate Statement (1) independently.

Statement (1) gives

N(X) = 20

. Substituting into the equation gives

30 = 20 + N(Y) - N(X \cap Y) \Rightarrow N(X \cap Y) = N(Y) - 10

. Since

N(Y)

is unknown,

N(X \cap Y)

cannot be determined.

Determines whether Statement (1) alone yields a unique answer.

3

Evaluate Statement (2) independently.

Statement (2) gives

N(Y) = 15

. Substituting into the equation gives

30 = N(X) + 15 - N(X \cap Y) \Rightarrow N(X \cap Y) = N(X) - 15

. Since

N(X)

is unknown,

N(X \cap Y)

cannot be determined.

Determines whether Statement (2) alone yields a unique answer.

4

Evaluate both statements together.

Combining Statement (1) and Statement (2) provides

N(X) = 20

and

N(Y) = 15

. Plugging both into the formula:

30 = 20 + 15 - N(X \cap Y) \Rightarrow 30 = 35 - N(X \cap Y) \Rightarrow N(X \cap Y) = 5

. This gives a single, unique value.

Determines if combining both statements provides sufficient information.

Key Concept

Two-Group Overlapping Sets Principle

Estimated Time:1m 0s

Question 12Question →

A set consists of $5$ numbers. What is the arithmetic mean of the $5$ numbers?

(1) The sum of the $5$ numbers is $45$ .
(2) The median of the $5$ numbers is $9$ and the range of the $5$ numbers is $8$ .

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 mean of a dataset of

n

items is defined as the sum of all elements divided by

n

. Since

n = 5

, knowing that the sum of the

5

numbers is

45

immediately gives the mean as

45 / 5 = 9

. Thus, the first statement provides sufficient information on its own. Conversely, knowing only the median and range allows for multiple distinct sets of numbers with different total sums and arithmetic means, making the second statement insufficient.

Step-by-Step Solution

1

Rephrase the question stem

The arithmetic mean of

5

numbers is equal to

\frac{\text{Sum of numbers}}{5}

. Thus, knowing the sum of the

5

numbers is necessary and sufficient to find the mean.

By definition,

\text{Mean} = \frac{\text{Sum}}{n}

where

n = 5

.

2

Evaluate Statement (1)

Statement (1) gives

\text{Sum} = 45

. Therefore,

\text{Mean} = \frac{45}{5} = 9

. This gives a unique value.

Statement (1) alone provides all required information.

3

Evaluate Statement (2)

Consider two possible sets with median

9

and range

8

: Set X =

\{5, 9, 9, 9, 13\}

(Sum =

45

, Mean =

9

) and Set Y =

\{5, 5, 9, 13, 13\}

(Sum =

45

, Mean =

9

), but also Set Z =

\{5, 6, 9, 10, 13\}

(Sum =

43

, Mean =

8.6

). Since the mean is not uniquely determined, Statement (2) is not sufficient.

Median and range leave the remaining elements and the overall sum indeterminate.

Key Concept

Arithmetic Mean Definition in Data Sufficiency

Question 13Question →

In a cohort of $N$ financial analysts, every analyst tracks at least one of two asset classes: Equities or Bonds. The mean number of years of experience for analysts who track Equities is 12 years, and the mean number of years of experience for analysts who track Bonds is 18 years. Is the mean number of years of experience for all $N$ analysts in the cohort strictly greater than 15 years?

(1) The mean number of years of experience for analysts who track both Equities and Bonds is 15 years.
(2) The number of analysts who track Bonds only is strictly greater than the number of analysts who track Equities only.

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

Answer

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

Rephrasing the question stem shows that the overall mean is greater than 15 if and only if

3(y - x) + (15 - \bar{m})z > 0

, where

x

is analysts tracking Equities only,

y

is analysts tracking Bonds only,

z

is analysts tracking both, and

\bar{m}

is the mean experience of the overlap group. Statement (1) gives

\bar{m} = 15

, simplifying the target to

y > x

, which is insufficient alone. Statement (2) gives

y > x

, which is insufficient alone because

\bar{m}

is unknown. Combined, Statement (1) simplifies the target to

y > x

and Statement (2) confirms

y > x

, yielding a definitive 'Yes'.

Step-by-Step Solution

1

Define variables and set up the expression for the total sum of experience and overall mean.

Let

x

be the number of analysts tracking Equities only,

y

be the number of analysts tracking Bonds only, and

z

be the number of analysts tracking both. Total analysts

N = x + y + z

.
Sum of experience for Equities group:

S_E = 12(x+z)

.
Sum of experience for Bonds group:

S_B = 18(y+z)

.
Let

\bar{m}

be the mean experience of the

z

analysts tracking both, so their total experience is

z\bar{m}

.
Total experience

S_{total} = S_E + S_B - z\bar{m} = 12x + 18y + (30 - \bar{m})z

.

Overlapping sets counting principle requires subtracting the overlap sum so double-counted individuals are accounted for correctly.

2

Rephrase the target question stem inequality.

The overall mean is

\bar{M} = \frac{S_{total}}{N} = \frac{12x + 18y + (30 - \bar{m})z}{x + y + z}

.
The question asks if

\bar{M} > 15

:

\frac{12x + 18y + (30 - \bar{m})z}{x + y + z} > 15

12x + 18y + (30 - \bar{m})z > 15x + 15y + 15z

3y - 3x + (15 - \bar{m})z > 0 \iff 3(y - x) + (15 - \bar{m})z > 0

Simplifying the target question algebraically reveals the exact algebraic condition required to determine sufficiency.

3

Evaluate Statement (1) alone.

Statement (1) states

\bar{m} = 15

.
Substituting

\bar{m} = 15

into the rephrased inequality yields

3(y - x) + (15 - 15)z > 0 \iff 3(y - x) > 0 \iff y > x

.
Since we do not know whether

y > x

or

y \leq x

, Statement (1) alone is NOT sufficient.

Knowing the mean of the overlap eliminates the

z

term but leaves the relationship between

x

and

y

unknown.

4

Evaluate Statement (2) alone.

Statement (2) states

y > x

.
Without knowing the value of

\bar{m}

(the mean of the overlapping group), if

\bar{m}

is very large (e.g.,

\bar{m} = 50

), then

(15 - \bar{m})z

could be negative enough to make

3(y - x) + (15 - \bar{m})z < 0

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

Without information about the overlapping group's mean, the sign of

3(y - x) + (15 - \bar{m})z

cannot be determined.

5

Evaluate Statements (1) and (2) together.

From Statement (1), the condition simplifies to

y > x

.
From Statement (2), we are explicitly given that

y > x

.
Therefore, together the statements definitively prove that the overall mean is strictly greater than 15 years (Yes). Both statements together are SUFFICIENT.

Combining both statements satisfies the rephrased target condition completely.

Key Concept

Overlapping Sets and Weighted Averages in Data Sufficiency

Estimated Time:2m 0s

Question 14Question →

If $x$ is a real number, is $|x^2 - 4| < 5$ ?

(1) $|x - 1| < 2$
(2) $|x + 1| < 3$

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.

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 question target

|x^2 - 4| < 5

gives

-5 < x^2 - 4 < 5 \implies -1 < x^2 < 9

. Since

x^2 \ge 0

for all real numbers, the condition reduces to

x^2 < 9

, or

-3 < x < 3

.

Statement (1) states

|x - 1| < 2

, which translates to

-2 < x - 1 < 2 \implies -1 < x < 3

. Any number

x

in

(-1, 3)

automatically satisfies

-3 < x < 3

, giving a definitive 'Yes'. Hence, Statement (1) alone is sufficient.

Statement (2) states

|x + 1| < 3

, which translates to

-3 < x + 1 < 3 \implies -4 < x < 2

. If

x = 0

,

x

is in

(-3, 3)

('Yes'); if

x = -3.5

,

x

is not in

(-3, 3)

('No'). Because both 'Yes' and 'No' are possible, Statement (2) alone is not sufficient.

Therefore, the statement indicating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient, is the correct choice.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The target inequality

|x^2 - 4| < 5

is equivalent to

-5 < x^2 - 4 < 5

. Adding 4 yields

-1 < x^2 < 9

. Since

x^2 \ge 0

for all real numbers,

-1 < x^2

is always true. Thus, the target question simplifies to whether

x^2 < 9

, which means

-3 < x < 3

.

Simplifying the question stem simplifies evaluating the statement ranges.

2

Evaluate Statement (1):

|x - 1| < 2

.

Expanding the absolute value gives

-2 < x - 1 < 2

, which simplifies to

-1 < x < 3

. Every value of

x

in the interval

(-1, 3)

is strictly inside

(-3, 3)

. Thus, Statement (1) guarantees a definitive 'Yes'.

Since Statement (1) provides a single definitive answer ('Yes'), Statement (1) alone is sufficient.

3

Evaluate Statement (2):

|x + 1| < 3

.

Expanding the absolute value gives

-3 < x + 1 < 3

, which simplifies to

-4 < x < 2

. If

x = 0

, then

-3 < 0 < 3

is true ('Yes'). If

x = -3.5

, then

|(-3.5)^2 - 4| = |12.25 - 4| = 8.25 < 5

is false ('No').

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

Key Concept

Rephrasing absolute value inequality bounds in Data Sufficiency questions

Question 15Question →

A survey recorded the monthly electricity costs of 60 small business workshops. Each workshop operated during Shift X, Shift Y, or both shifts. Exactly 35 workshops operated during Shift X, and exactly 40 workshops operated during Shift Y. What was the median monthly electricity cost among all 60 workshops?

(1) For the workshops that operated ONLY during Shift X, the monthly electricity cost was $400 per workshop, and for the workshops that operated ONLY during Shift Y, the monthly electricity cost was$ 600 per workshop.

(2) For the workshops that operated during BOTH shifts, the median monthly electricity cost was $500, and the total monthly electricity cost for all 60 workshops combined was$ 30,000.

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

Answer

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

The correct option identifies that even when both statements are combined, the internal distribution of costs in the overlapping group allows for multiple overall medians (such as

500 and

600). Therefore, the statements together remain insufficient.

Step-by-Step Solution

1

Determine subgroup counts using the Principle of Inclusion-Exclusion.

Total workshops

N = 60

.

N(X) = 35

,

N(Y) = 40

. Since every workshop is in

X

or

Y

,

N(X \cup Y) = 60

. Thus,

N(X \cap Y) = 35 + 40 - 60 = 15

. Workshops in ONLY

X = 35 - 15 = 20

. Workshops in ONLY

Y = 40 - 15 = 25

.

Rephrasing the stem establishes the exact count of elements in each of the three disjoint set categories.

2

Evaluate Statement (1) alone.

The 20 ONLY

X

workshops cost

400 each, and the 25 ONLY

Y

workshops cost

600 each. However, the costs of the 15 BOTH workshops are completely unknown. The overall median (the average of the 30th and 31st values in sorted order) could be

400 (if all BOTH costs are low) or

600 (if all BOTH costs are high).

Without data on the overlapping subgroup, the median cannot be uniquely determined.

3

Evaluate Statement (2) alone.

Statement (2) provides the median (

500) and sum (

7,000) for the 15 BOTH workshops, as well as the total sum (

30,000), but gives no cost values for the ONLY

X

or ONLY

Y$ workshops.

Without specific values for the single-shift groups, Statement (2) is insufficient.

4

Evaluate Statements (1) and (2) together.

From Statement (1), the sum of ONLY

X

and ONLY

Y

workshops is

20(400) + 25(600) = 8,000 + 15,000 = 23,000

. From Statement (2), total sum is

30,000

, so the 15 BOTH workshops sum to

30,000 - 23,000 = 7,000

, with a median of

500

.
Case A: If 1 BOTH workshop has cost

0

and 14 have cost

500

(median =

500

, sum =

7,000

), the 60 sorted values consist of 1 zero, 20 values of

400

, 14 values of

500

, and 25 values of

600

. The 30th and 31st values are both

500

, so overall median =

500

.
Case B: If 7 BOTH workshops have cost

0

, 1 has cost

500

, and 7 have cost

642.85

(median =

500

, sum =

7,000

), the sorted values put the 20 values of

400

in positions 8–27, the single

500

at position 28, and the 25 values of

600

in positions 29–53. The 30th and 31st values are both

600

, so overall median =

600

.

Since the overall median can be

500

or

600

depending on how the costs within the overlapping group are distributed, both statements combined are insufficient.

Key Concept

Overlapping Sets and Data Sufficiency for Position-Based Measures (Median)

Question 16Question →

If $x$ and $y$ are real numbers such that $x \neq 3y$ , what is the value of $x + 3y$ ?

(1) $x^2 + 6xy + 9y^2 = 49$
(2) $x^2 - 9y^2 = 12(x - 3y)$

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.

Statement (1) simplifies to

(x + 3y)^2 = 49

, which means

x + 3y

could be

7

or

-7

, so it is not sufficient. Statement (2) factors as

(x - 3y)(x + 3y) = 12(x - 3y)

. Given

x \neq 3y

, dividing both sides by

(x - 3y)

uniquely determines

x + 3y = 12

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

Step-by-Step Solution

1

Analyze Statement (1) algebraically

The expression

x^2 + 6xy + 9y^2

is a perfect square trinomial equal to

(x + 3y)^2

. Therefore,

(x + 3y)^2 = 49

, which gives

x + 3y = 7

or

x + 3y = -7

.

Because there are two distinct valid values for

x + 3y

, Statement (1) alone does not provide a unique value.

2

Analyze Statement (2) algebraically

The left side

x^2 - 9y^2

factors as

(x - 3y)(x + 3y)

. The equation becomes

(x - 3y)(x + 3y) = 12(x - 3y)

.

Since the stem specifies

x \neq 3y

, we know

x - 3y \neq 0

, allowing us to divide both sides by

(x - 3y)

without dividing by zero.

3

Solve the simplified equation from Statement (2)

Dividing gives

x + 3y = 12

.

This yields a single, unique value for the target expression

x + 3y

, making Statement (2) alone sufficient.

Key Concept

Algebraic Stem Rephrasing and Quadratic Degree Evaluation in Data Sufficiency

Estimated Time:1m 30s

Question 17Question →

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

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

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.

Show answer & explanation

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

Answer

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

Combining Statement (1) and Statement (2) allows substituting

x - y = 3

into

xy(x - y) = 12

, yielding

xy = 4

. Solving the system

x - y = 3

and

xy = 4

gives two valid real solution pairs:

(4, 1)

and

(-1, -4)

. The quotient

\frac{x}{y}

evaluates to

4

in the first case and

\frac{1}{4}

in the second case. Because a unique numerical value for

\frac{x}{y}

cannot be determined, the statements together are not sufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Factor Statement (1):

xy(x - y) = 12

. Many pairs of real numbers

(x, y)

satisfy this, such as

(4, 1)

which gives

\frac{x}{y} = 4

, or

(3, 1)

which gives

3(1)(2) = 6 \neq 12

. For

(4, 1)

,

\frac{x}{y} = 4

. For

(x, y) = (2, -1)

,

2(-1)(3) = -6 \neq 12

. However, choosing

x = \frac{3 + \sqrt{17}}{2}

and

y = \frac{-3 + \sqrt{17}}{2}

gives a different ratio. Since

\frac{x}{y}

can take infinitely many values, Statement (1) is not sufficient.

A single equation with two variables generally allows infinitely many solutions.

2

Evaluate Statement (2) independently.

Statement (2) gives

x - y = 3

. If

x = 4

and

y = 1

, then

\frac{x}{y} = 4

. If

x = 5

and

y = 2

, then

\frac{x}{y} = 2.5

. Since multiple values for

\frac{x}{y}

exist, Statement (2) is not sufficient.

A linear difference condition alone does not fix the ratio of two unknown quantities.

3

Evaluate Statements (1) and (2) combined.

Substitute

x - y = 3

into

xy(x - y) = 12

to get

3xy = 12 \implies xy = 4

. Express

y

as

x - 3

and substitute into

xy = 4

:

x(x - 3) = 4 \implies x^2 - 3x - 4 = 0 \implies (x - 4)(x + 1) = 0

. This yields two real solutions for

x

:

x = 4

or

x = -1

. If

x = 4

, then

y = 1

and

\frac{x}{y} = \frac{4}{1} = 4

. If

x = -1

, then

y = -4

and

\frac{x}{y} = \frac{-1}{-4} = \frac{1}{4}

.

Because two distinct values for

\frac{x}{y}

(

4

and

\frac{1}{4}

) are possible, combined sufficiency fails.

Key Concept

Degree of non-linear algebraic systems and non-unique solutions in Data Sufficiency

Estimated Time:2m 0s

Question 18Question →

A commercial law firm charges clients for services rendered by its Senior Partners and Associate Partners. Last month, the firm billed a total of $450,000 across all client accounts. What was the ratio of the total amount billed for Senior Partner hours to the total amount billed for Associate Partner hours?

(1) Senior Partners billed at an hourly rate that was 50 percent higher than the hourly rate billed by Associate Partners.
(2) Senior Partners accounted for 40 percent of the total number of hours billed by the firm last month.

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

Answer

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

The correct option is the statement indicating both statements together are sufficient, but neither alone is sufficient. Statement (1) provides the ratio of hourly billing rates (

3:2

), and Statement (2) provides the ratio of total hours billed (

2:3

). Multiplying rate ratio by hours ratio gives the ratio of total amounts billed (

1:1

), yielding a single unique value.

Step-by-Step Solution

1

Rephrase the target question mathematically

Let

R_S

and

R_A

be the hourly rates for Senior and Associate Partners, respectively. Let

H_S

and

H_A

be the hours billed by Senior and Associate Partners, respectively. Total billing ratio is

\frac{\text{Senior Billing}}{\text{Associate Billing}} = \frac{R_S \cdot H_S}{R_A \cdot H_A} = \left(\frac{R_S}{R_A}\right) \cdot \left(\frac{H_S}{H_A}\right)

.

Total revenue for each group is the product of hourly rate and hours billed.

2

Evaluate Statement (1) independently

Statement (1) gives

R_S = 1.50 R_A \implies \frac{R_S}{R_A} = \frac{3}{2}

. Without

\frac{H_S}{H_A}

, the target ratio cannot be evaluated.

Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently

Statement (2) gives

H_S = 0.40 (H_S + H_A) \implies 0.60 H_S = 0.40 H_A \implies \frac{H_S}{H_A} = \frac{2}{3}

. Without

\frac{R_S}{R_A}

, the target ratio cannot be evaluated.

Statement (2) alone is NOT sufficient.

4

Evaluate Statements (1) and (2) combined

Multiplying the two ratios yields

\frac{\text{Senior Billing}}{\text{Associate Billing}} = \left(\frac{3}{2}\right) \cdot \left(\frac{2}{3}\right) = 1

. The ratio is uniquely determined as

1:1

.

Both statements together are SUFFICIENT.

Key Concept

Data Sufficiency stems asking for a ratio can be solved by finding the relative ratios of constituent factors (Rate Ratio × Quantity Ratio) without needing absolute numerical quantities.

Estimated Time:2m 0s

Question 19Question →

An industrial bakery uses two automated mixing machines, Machine X and Machine Y, operating simultaneously at their respective constant rates, to prepare batches of dough. How many hours would it take Machine X, working alone at its constant rate, to complete a production order of $1,200$ batches of dough?

(1) Working together at their respective constant rates, Machine X and Machine Y can complete the production order of $1,200$ batches of dough in $8$ hours.
(2) Machine Y's constant production rate is $50\%$ greater than Machine X's constant production rate.

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

Answer

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

The question asks for the time Machine X needs to produce

1,200

batches alone, which requires finding Machine X's individual rate,

r_X

. Statement (1) gives the combined rate

r_X + r_Y = 150

batches/hour, which is insufficient on its own because

r_Y

is unknown. Statement (2) gives the relative relationship

r_Y = 1.5 r_X

, which is insufficient on its own because no absolute rate is given. Combining both statements yields

r_X + 1.5 r_X = 150

, giving a unique value

r_X = 60

batches/hour and a single solution

T_X = 20

hours. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Step-by-Step Solution

1

Define the target variable and rephrase the question stem.

Let

r_X

be Machine X's rate (batches per hour) and

r_Y

be Machine Y's rate. The required time for Machine X working alone is

T_X = \frac{1,200}{r_X}

. The target is to determine the exact value of

r_X

.

Rephrasing simplifies the Data Sufficiency goal from finding time to finding a unique numerical value for

r_X

.

2

Evaluate Statement (1) independently.

Statement (1) states

(r_X + r_Y) \times 8 = 1,200

, which simplifies to

r_X + r_Y = 150

batches per hour.

Since there are two unknown rates (

r_X

and

r_Y

) and only one equation,

r_X

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

3

Evaluate Statement (2) independently.

Statement (2) states

r_Y = r_X + 0.50 r_X = 1.5 r_X = \frac{3}{2} r_X

.

This establishes a proportional relationship between the two rates but gives no absolute values.

r_X

can take infinitely many positive values, so Statement (2) alone is NOT sufficient.

4

Evaluate Statement (1) and Statement (2) together.

Substitute

r_Y = 1.5 r_X

into

r_X + r_Y = 150

:

r_X + 1.5 r_X = 150 \implies 2.5 r_X = 150 \implies r_X = 60

batches per hour.
Then

T_X = \frac{1,200}{60} = 20

hours.

Combining the statements yields a single linear equation with one variable, giving a unique value for

r_X

and thus for

T_X

. Both statements together are sufficient.

Key Concept

Data Sufficiency evaluation for combined work rate problems with linear algebraic systems

Question 20Question →

A high-tech manufacturing firm produced a batch of $150$ electronic assemblies. Each assembly in the batch underwent standard quality testing by Inspector Alpha, Inspector Beta, or both. Exactly $90$ assemblies were tested by Inspector Alpha and $105$ assemblies were tested by Inspector Beta. What was the average (arithmetic mean) testing duration per assembly for the entire batch of $150$ assemblies?

(1) The average testing duration for assemblies tested only by Inspector Alpha was $12$ minutes, and the average testing duration for assemblies tested only by Inspector Beta was $18$ minutes.
(2) The average testing duration for assemblies tested by both Inspector Alpha and Inspector Beta was $25$ minutes.

Which of the following statements provides sufficient information to answer the question?

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

Answer

Both statements together are sufficient to determine a unique average duration, but neither statement alone is sufficient.

The correct response identifies that both statements combined provide all required subgroup averages. Using the overlapping set counts established in the rephrased question stem (

45

Alpha-only,

60

Beta-only, and

45

Both), the three subgroup averages supplied across both statements allow for a unique calculation of the overall weighted average.

Step-by-Step Solution

1

Apply the Principle of Inclusion-Exclusion to find the number of assemblies in each mutually exclusive subgroup.

\text{Total} = N(\text{Alpha}) + N(\text{Beta}) - N(\text{Both}) \implies 150 = 90 + 105 - N(\text{Both}) \implies N(\text{Both}) = 45

. Thus,

N(\text{Alpha only}) = 90 - 45 = 45

,

N(\text{Beta only}) = 105 - 45 = 60

, and

N(\text{Both}) = 45

.

Before evaluating the statements, rephrasing the stem by partitioning the set into three non-overlapping groups simplifies the weighted average equation.

2

Evaluate Statement (1) alone.

Statement (1) provides

\text{Mean}(\text{Alpha only}) = 12

minutes and

\text{Mean}(\text{Beta only}) = 18

minutes. However,

\text{Mean}(\text{Both})

remains unknown. The overall sum of testing durations cannot be determined. Statement (1) is NOT sufficient.

We cannot compute a weighted mean of three groups if one group's mean is completely missing.

3

Evaluate Statement (2) alone.

Statement (2) provides

\text{Mean}(\text{Both}) = 25

minutes. However,

\text{Mean}(\text{Alpha only})

and

\text{Mean}(\text{Beta only})

are unknown. Statement (2) is NOT sufficient.

Without the average durations for the single-inspector groups, the total testing duration cannot be computed.

4

Evaluate Statements (1) and (2) combined.

Combining both statements gives all three subgroup means:

12

minutes for

45

assemblies,

18

minutes for

60

assemblies, and

25

minutes for

45

assemblies. Total testing duration

= 45(12) + 60(18) + 45(25) = 540 + 1080 + 1125 = 2745

minutes. The overall average duration is

\frac{2745}{150} = 18.3

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

Knowing all subgroup sizes and all subgroup means enables exact calculation of the overall mean.

Key Concept

Overlapping Sets and Weighted Averages in Data Sufficiency

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Data Sufficiency

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept