Data Sufficiency Practice Questions — GMAT — Page 10

Question 181Question →

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

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

x = \sqrt{2}

(not an integer) or

x = 2

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

x = \frac{1}{3}

(not an integer) or

x = 1

(an integer), so Statement (2) is insufficient. Combining both statements establishes that

x = \frac{m}{3}

for an integer

m

, and

x^2 = \frac{m^2}{9} = k

for an integer

k

. This requires

m^2 = 9k

, which implies

m

must be a multiple of 3. Hence

x

must be an integer, giving a definitive 'Yes' answer. Thus, both statements together are sufficient.

Step-by-Step Solution

1

Analyze the question stem constraint

x

is constrained to be a real number (

x \in \mathbb{R}

). The target question is a Yes/No question: 'Is

x

an integer?'

Without an explicit integer constraint in the stem, non-integer real numbers must be tested as counterexamples.

2

Evaluate Statement (1) alone

If

x = \sqrt{2}

, then

x^2 = 2

(an integer), but

x

is not an integer (Answer: No). If

x = 2

, then

x^2 = 4

(an integer), and

x

is an integer (Answer: Yes). Statement (1) is NOT sufficient.

Statement (1) allows both integer and non-integer values for

x

.

3

Evaluate Statement (2) alone

If

x = \frac{1}{3}

, then

3x = 1

(an integer), but

x

is not an integer (Answer: No). If

x = 1

, then

3x = 3

(an integer), and

x

is an integer (Answer: Yes). Statement (2) is NOT sufficient.

Statement (2) allows fractional values with a denominator of 3 as well as integers.

4

Evaluate Statements (1) and (2) together

From Statement (2),

x = \frac{m}{3}

for some integer

m

. Substituting into Statement (1) yields

x^2 = \left(\frac{m}{3}\right)^2 = \frac{m^2}{9} = k

, where

k

is an integer. Thus,

m^2 = 9k

. Since

9k

is a multiple of 9,

m^2

is divisible by 9, which means

m

must be a multiple of 3. Let

m = 3p

for some integer

p

. Then

x = \frac{3p}{3} = p

, which guarantees that

x

is an integer. The answer is a definitive 'Yes'. Statements (1) and (2) together are SUFFICIENT.

Combining both conditions restricts

x

to rational numbers whose square is an integer, forcing

x

to be an integer.

Key Concept

Number Properties and Integer Constraints in Data Sufficiency

Estimated Time:1m 30s

Question 182Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

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

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

. Since

2n

is always even,

m + 2n

is even if and only if

m

is even. Rephrasing the question target yields: "Is

m

an even integer?" Statement (1) tells us

m + 3

is odd, which means

m

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

n

, which is irrelevant to whether

m

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

Step-by-Step Solution

1

Simplify the target expression in the question stem

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

Factoring the numerator simplifies the target expression for

m \neq n

.

2

Rephrase the target question using parity properties

Since

2n

is an even integer for any integer

n

,

m + 2n

is even if and only if

m

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

m

an even integer?"

Simplifying the target question eliminates the variable

n

from the requirement.

3

Evaluate Statement (1):

m + 3

is an odd integer

If

m + 3

is odd, then

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

. Thus,

m

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

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

4

Evaluate Statement (2):

n

is an odd integer

Statement (2) gives information about

n

, but gives no information about

m

. Since the rephrased question depends solely on

m

, Statement (2) alone is INSUFFICIENT.

Knowing

n

does not determine whether

m

is even.

Key Concept

Algebraic factoring and target rephrasing in Data Sufficiency questions

Question 183Question →

A regional logistics firm operates four distribution centers: Facility Alpha, Facility Beta, Facility Gamma, and Facility Delta. The table below displays the total tonnage of cargo processed and the average processing cost per ton (in dollars) at each facility during the last quarter:

Distribution Center	Total Tonnage Processed (tons)	Average Cost per Ton ($)
Facility Alpha	2,400	50
Facility Beta	1,600	40
Facility Gamma	$T$	80
Facility Delta	3,000	$C$

Is the overall average processing cost per ton across all four distribution centers combined less than $60?

(1) Facility Gamma processed 2,000 tons of cargo ( $T = 2,000$ ).
(2) Facility Delta's average cost per ton was $55 ($ C = 55$).

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 answer is the option stating that both statements together are sufficient, but neither alone is sufficient. By setting up the total cost and total tonnage expressions from the table, the question 'Is the overall average cost less than

60?' simplifies to the inequality 'Is

T + 150C < 11,800

?'. Neither statement alone supplies both variables (

T

and

C

), but together they specify

T = 2,000

and

C = 55

, giving

2,000 + 8,250 = 10,250 < 11,800$, which answers the question with a definitive 'Yes'.

Step-by-Step Solution

1

Set up the algebraic expression for the overall combined average processing cost.

Total Tonnage =

2,400 + 1,600 + T + 3,000 = 7,000 + T

.
Total Cost (

) =

(2,400 \times 50) + (1,600 \times 40) + (T \times 80) + (3,000 \times C) = 120,000 + 64,000 + 80T + 3,000C = 184,000 + 80T + 3,000C$.

Weighted average requires dividing total dollar expenditure across all facilities by total tonnage processed.

2

Rephrase the target question inequality.

Is

\frac{184,000 + 80T + 3,000C}{7,000 + T} < 60

?
Multiply by

(7,000 + T)

:

184,000 + 80T + 3,000C < 60(7,000 + T)

184,000 + 80T + 3,000C < 420,000 + 60T

20T + 3,000C < 236,000

Divide by 20:
Is

T + 150C < 11,800

?

Simplifying the target question establishes a direct numerical threshold to test against both statements.

3

Evaluate Statement (1) alone.

Given

T = 2,000

, the target inequality becomes: Is

2,000 + 150C < 11,800 \implies 150C < 9,800 \implies C < 65.33

?
Since

C

is unknown, if

C = 50

, the answer is Yes (

9,500 < 11,800

). If

C = 70

, the answer is No (

12,500 \not< 11,800

).

Statement (1) leaves

C

undetermined, allowing both 'Yes' and 'No' outcomes.

4

Evaluate Statement (2) alone.

Given

C = 55

, the target inequality becomes: Is

T + 150(55) < 11,800 \implies T + 8,250 < 11,800 \implies T < 3,550

?
Since

T

is unknown, if

T = 2,000

, the answer is Yes (

10,250 < 11,800

). If

T = 4,000

, the answer is No (

12,250 \not< 11,800

).

Statement (2) leaves

T

undetermined, allowing both 'Yes' and 'No' outcomes.

5

Evaluate Statements (1) and (2) together.

Combining

T = 2,000

and

C = 55

into the rephrased inequality:

T + 150C = 2,000 + 150(55) = 2,000 + 8,250 = 10,250

.
Since

10,250 < 11,800

, we obtain a definitive 'Yes'.

Both variables are uniquely determined, resolving the target inequality definitively.

Key Concept

Weighted Average Calculation and Data Sufficiency Inequality Rephrasing

Estimated Time:2m 0s

Question 184Question →

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 $x$ .

True

False

Show answer & explanation

Answer: False

Answer

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.

Step-by-Step Solution

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.

Key Concept

Statement Combination and Solution Set Intersection

Question 185Question →

A medical research trial evaluated $100$ patients, each of whom experienced side effect A, side effect B, or both. Exactly $65$ patients experienced side effect A, and exactly $55$ patients experienced side effect B. What was the average (arithmetic mean) duration of side effect A, in days, among all patients who experienced side effect A?

(1) The average duration of side effect A for patients who experienced ONLY side effect A was $8$ days.
(2) The average duration of side effect A for patients who experienced BOTH side effects was $12$ days.

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 choice indicates that both statements combined provide sufficient data while neither is sufficient on its own. Using the standard inclusion-exclusion principle for two sets,

100 = 65 + 55 - N(\text{Both})

, giving exactly

20

patients with both side effects and

45

patients with only side effect A. Statement (1) provides the sum of durations for the

45

patients (

45 \times 8 = 360

), and Statement (2) provides the sum of durations for the

20

patients (

20 \times 12 = 240

). Neither statement alone gives both sums, but together they yield a total sum of

600

days for

65

patients, resulting in a single definitive mean of

\frac{600}{65}

days.

Step-by-Step Solution

1

Rephrase the question stem using overlapping sets principles.

Let

N(A) = 65

,

N(B) = 55

, and total patients

N(A \cup B) = 100

. Using the set formula

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

, we get

100 = 65 + 55 - N(A \cap B)

, so

N(A \cap B) = 20

.

Determining the exact count of patients in the overlap region (

A \cap B

) and the exclusive region (

A \text{ only}

) is necessary to set up the weighted average formula.

2

Calculate the subgroup sizes for patients experiencing side effect A.

Patients with ONLY side effect A =

N(A) - N(A \cap B) = 65 - 20 = 45

. Patients with BOTH side effects =

20

. Total patients with side effect A =

65

.

The total duration of side effect A across all

65

patients is the sum of durations from the

45

'only A' patients and the

20

'both' patients.

3

Evaluate Statement (1) independently.

Statement (1) states that the mean for the

45

'only A' patients is

8

days. The total duration for this group is

45 \times 8 = 360

days. However, the duration for the

20

'both' patients is unknown.

Without the total duration for the overlap group, the overall mean cannot be determined. Thus, Statement (1) alone is NOT sufficient.

4

Evaluate Statement (2) independently.

Statement (2) states that the mean for the

20

'both' patients is

12

days. The total duration for this group is

20 \times 12 = 240

days. However, the duration for the

45

'only A' patients is unknown.

Without the total duration for the exclusive A group, the overall mean cannot be determined. Thus, Statement (2) alone is NOT sufficient.

5

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

Combining both statements, the total duration for all

65

patients with side effect A is

360 + 240 = 600

days. The overall average duration is

\frac{600}{65} = \frac{120}{13}

days.

A unique numerical value is obtained for the target question, so both statements together are sufficient.

Key Concept

Combining overlapping set subgroup sizes with weighted arithmetic mean formulas to evaluate Data Sufficiency statements.

Estimated Time:2m 0s

Question 186Question →

If $a$ and $b$ are non-zero real numbers, what is the value of $\frac{a^2 + 9b^2}{ab}$ ?

(1) $a^2 - 3ab - 18b^2 = 0$
(2) $a > 0$ and $b > 0$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

Answer: 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 choice stating that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) produces two valid roots for the ratio of the variables, which lead to two different target values. Statement (2) specifies that both variables are positive, which eliminates the negative root and uniquely identifies the ratio and target value when combined with Statement (1).

Step-by-Step Solution

1

Rephrase the target expression in terms of the ratio

k = \frac{a}{b}

.

\frac{a^2 + 9b^2}{ab} = \frac{a^2}{ab} + \frac{9b^2}{ab} = \frac{a}{b} + 9\left(\frac{b}{a}\right) = k + \frac{9}{k}

. Finding the ratio

k = \frac{a}{b}

is sufficient to answer the question.

Dividing the numerator by the denominator simplifies the expression into a function of a single ratio variable.

2

Evaluate Statement (1) independently.

Divide

a^2 - 3ab - 18b^2 = 0

by

b^2

to get

\left(\frac{a}{b}\right)^2 - 3\left(\frac{a}{b}\right) - 18 = 0

, or

k^2 - 3k - 18 = 0

. Factoring yields

(k - 6)(k + 3) = 0

, so

k = 6

or

k = -3

. If

k = 6

, target value is

6 + \frac{9}{6} = 7.5

. If

k = -3

, target value is

-3 + \frac{9}{-3} = -6

.

Since two distinct numerical outcomes are possible for the target expression, Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

a > 0

and

b > 0

implies

k = \frac{a}{b} > 0

, but gives no numerical constraint on

k

.

Without an equation linking

a

and

b

, Statement (2) alone is NOT sufficient.

4

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

Statement (1) gives

k = 6

or

k = -3

. Statement (2) requires

k > 0

, eliminating

k = -3

and establishing

k = 6

uniquely. Thus, the target value is uniquely

7.5

.

Combining both statements uniquely determines the ratio

k

and therefore the target expression.

Key Concept

Rephrasing homogeneous algebraic expressions into single ratio variables and analyzing degree constraints in quadratic systems.

Question 187Question →

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

(1) $x^2 + y^2 = 25 - 2xy$
(2) $x^3 + y^3 = 125 - 3xy(x+y)$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Statement (2) ALONE is sufficient to determine that

x + y = 5

, but Statement (1) ALONE is not sufficient because it allows

x + y = 5

or

x + y = -5

.

Statement (2) alone allows us to factor the expression into

(x+y)^3 = 125

. Because real numbers have a unique real cube root,

x+y

must equal

5

. Thus, Statement (2) alone provides a single, definitive answer to the question stem. Statement (1) yields

(x+y)^2 = 25

, which leads to

x+y = 5

or

x+y = -5

, making it insufficient on its own.

Step-by-Step Solution

1

Rephrase the target question

The target is to find a unique numerical value for the expression

(x + y)

. We do not need individual values for

x

and

y

.

Data Sufficiency targets involving expressions often do not require solving for individual variables.

2

Evaluate Statement (1):

x^2 + y^2 = 25 - 2xy

Rearranging terms gives

x^2 + 2xy + y^2 = 25

, which factors as

(x + y)^2 = 25

. Taking the square root gives

x + y = 5

or

x + y = -5

.

Since there are two distinct real values for

x + y

, Statement (1) alone is insufficient.

3

Evaluate Statement (2):

x^3 + y^3 = 125 - 3xy(x+y)

Rearranging terms gives

x^3 + 3xy(x+y) + y^3 = 125

, which is the expanded form of

(x + y)^3 = 125

. Taking the cube root gives

x + y = 5

.

For real numbers, every real number has exactly one real cube root. Thus,

x + y = 5

uniquely. Statement (2) alone is sufficient.

Key Concept

Algebraic Rephrasing and Degree of Real Polynomial Identities

Estimated Time:2m 0s

Question 188Question →

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

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

(2) $x^2 + \frac{1}{x}$ 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.

The option stating that both statements together are sufficient is correct. Evaluating Statement (1) alone allows irrational solutions like

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

. Evaluating Statement (2) alone allows irrational roots of

x^3 - 3x + 1 = 0

. However, combining both statements reveals that

x^2 - x

is an integer. Subtracting the resulting linear system forces

x

to be a rational number. For any positive rational

x = \frac{p}{q}

in simplest form,

x + \frac{1}{x}

being an integer requires

p = 1

and

q = 1

, proving that

x = 1

, which is an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is NOT sufficient.

If

x = 1

, then

x + \frac{1}{x} = 2

(an integer), and

x

is an integer. However, if

x = \frac{3 + \sqrt{5}}{2} \approx 2.618

(a positive real non-integer), then

x + \frac{1}{x} = \frac{3 + \sqrt{5}}{2} + \frac{3 - \sqrt{5}}{2} = 3

(an integer). Since

x

can be an integer or a non-integer, Statement (1) alone does not uniquely answer the question.

2

Evaluate Statement (2) independently.

Statement (2) is NOT sufficient.

If

x = 1

, then

x^2 + \frac{1}{x} = 2

(an integer), and

x

is an integer. If

x^2 + \frac{1}{x} = 3

, multiplying by

x

yields

x^3 - 3x + 1 = 0

. Evaluating

f(x) = x^3 - 3x + 1

shows

f(1) = -1 < 0

and

f(2) = 3 > 0

, so by the Intermediate Value Theorem, there exists a real root

x \in (1, 2)

, which is positive but not an integer. Hence, Statement (2) alone is insufficient.

3

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

Both statements together are SUFFICIENT.

Let

x + \frac{1}{x} = a

and

x^2 + \frac{1}{x} = b

, where

a

and

b

are integers. Subtracting Statement (1) from Statement (2) gives

\left(x^2 + \frac{1}{x}\right) - \left(x + \frac{1}{x}\right) = b - a \implies x^2 - x = k

, where

k = b - a

is an integer. From Statement (1),

x^2 - ax + 1 = 0

, and from the difference,

x^2 - x - k = 0

. Subtracting these two quadratic equations gives

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

. For any positive real

x

,

a = x + \frac{1}{x} \ge 2

, so

1 - a \neq 0

. Thus

x = \frac{1 + k}{a - 1}

. Because

k

and

a

are integers,

x

must be a rational number. Let

x = \frac{p}{q}

in lowest terms, where

p

and

q

are positive integers with

\gcd(p, q) = 1

. Then

x + \frac{1}{x} = \frac{p}{q} + \frac{q}{p} = \frac{p^2 + q^2}{pq} = a \implies p^2 + q^2 = a p q

. Since

p^2 = q(ap - q)

,

q

must divide

p^2

. But

\gcd(p, q) = 1

, so

q = 1

. Similarly,

p

must divide

q^2

, so

p = 1

. Thus

x = 1

, which is an integer. Both statements together definitively answer YES.

Key Concept

Deduce integer constraints and rationality by combining non-linear algebraic expressions for real variables in Data Sufficiency.

Question 189Question →

If $p$ and $q$ are positive real numbers, is $p$ an integer?

(1) $p^2 + pq$ is an integer.
(2) $q^2 + pq$ 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: Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

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

p

is an integer.

The correct answer states that statements (1) and (2) together are not sufficient. Statement (1) alone allows

p = 0.5, q = 1.5

(giving

p^2+pq=1

) as well as

p = 1, q = 1

(giving

p^2+pq=2

). Statement (2) alone similarly allows

p = 0.5, q = 1.5

(giving

q^2+pq=3

) as well as

p = 1, q = 1

(giving

q^2+pq=2

). When both statements are evaluated together, adding the two equations reveals

p+q = \sqrt{(p^2+pq) + (q^2+pq)}

. Choosing

p^2+pq = 1

and

q^2+pq = 3

yields

p+q = 2

and

p = 0.5

, demonstrating that

p

does not have to be an integer even when both statements are satisfied.

Step-by-Step Solution

1

Analyze the Question Stem and Constraints

Variables

p

and

q

are positive real numbers (not constrained to integers). The goal is to determine if

p

must be an integer.

Recognizing that variables are real numbers prevents premature integer assumptions.

2

Evaluate Statement (1) independently:

p^2 + pq = p(p+q)

is an integer

Case 1: If

p = 1

and

q = 1

, then

p(p+q) = 1(2) = 2

(integer), so

p

IS an integer.
Case 2: If

p = 0.5

and

q = 1.5

, then

p(p+q) = 0.5(2) = 1

(integer), so

p

IS NOT an integer.
Statement (1) is INSUFFICIENT.

Testing non-integer real values tests whether the condition forces

p

to be an integer.

3

Evaluate Statement (2) independently:

q^2 + pq = q(p+q)

is an integer

Case 1: If

p = 1

and

q = 1

, then

q(p+q) = 1(2) = 2

(integer), so

p

IS an integer.
Case 2: If

p = 0.5

and

q = 1.5

, then

q(p+q) = 1.5(2) = 3

(integer), so

p

IS NOT an integer.
Statement (2) is INSUFFICIENT.

Statement (2) primarily constrains

q(p+q)

, leaving

p

unconstrained.

4

Evaluate Statements (1) and (2) Combined

Let

p(p+q) = k_1

and

q(p+q) = k_2

where

k_1, k_2

are positive integers.
Adding gives

(p+q)^2 = k_1 + k_2 \implies p+q = \sqrt{k_1 + k_2}

.
Thus,

p = \frac{k_1}{\sqrt{k_1 + k_2}}

.
Case 1: Let

k_1 = 1

and

k_2 = 3

. Then

p+q = \sqrt{4} = 2 \implies p = \frac{1}{2} = 0.5

(not an integer).
Case 2: Let

k_1 = 2

and

k_2 = 2

. Then

p+q = \sqrt{4} = 2 \implies p = \frac{2}{2} = 1

(an integer).
Since

p

can still be either an integer or a non-integer, both statements combined are INSUFFICIENT.

Algebraic combination yields

p = \frac{k_1}{\sqrt{k_1 + k_2}}

, which produces non-integers for appropriate choices of integer constants

k_1

and

k_2

.

Key Concept

Avoiding Implicit Integer Assumptions in Real-Valued Data Sufficiency

Estimated Time:2m 0s

Question 190Question →

A tech consulting firm billed a client for a project completed by senior developers and junior developers. Senior developers were billed at a constant hourly rate of $x$ , and junior developers were billed at a constant hourly rate of $y$ . What was the average (arithmetic mean) hourly rate billed across all developer hours on the project?

(1) The total amount billed for senior developer hours was $50\%$ greater than the total amount billed for junior developer hours.
(2) Senior developers worked $40\%$ fewer total hours on the project than junior developers did.

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 choice is the option stating that both statements together are not sufficient. Statement (1) provides the ratio of total revenues between the two developer types, which is insufficient to find the overall average rate. Statement (2) provides the ratio of hours worked between senior and junior developers, which is also insufficient on its own. Combining both statements allows us to find that senior developers earn

2.5

times the hourly rate of junior developers and to express the average hourly rate as

1.5625 y

. However, because neither statement provides an absolute dollar amount for

x

,

y

, or total revenue, the numerical average hourly rate cannot be calculated.

Step-by-Step Solution

1

Rephrase the target question mathematically.

Let

h_s

be the senior hours and

h_j

be the junior hours. Total billing is

x h_s + y h_j

, and total hours is

h_s + h_j

. The average hourly rate is

A = \frac{x h_s + y h_j}{h_s + h_j}

. A specific numerical value for

A

is required.

Clarifying the target prevents confusing relative ratios with absolute dollar values.

2

Evaluate Statement (1) independently.

Statement (1) states that

x h_s = 1.5 y h_j

. Total billing equals

2.5 y h_j

, so

A = \frac{2.5 y h_j}{h_s + h_j}

.

Without knowing the ratio of hours

\frac{h_s}{h_j}

or specific dollar values for

x

or

y

,

A

cannot be determined. Statement (1) alone is insufficient.

3

Evaluate Statement (2) independently.

Statement (2) states that

h_s = 0.6 h_j

. Thus

A = \frac{0.6 x h_j + y h_j}{1.6 h_j} = \frac{0.6 x + y}{1.6}

.

Without the specific values of

x

and

y

,

A

cannot be determined. Statement (2) alone is insufficient.

4

Evaluate Statements (1) and (2) together.

From (2), substitute

h_s = 0.6 h_j

into (1):

x (0.6 h_j) = 1.5 y h_j \implies 0.6 x = 1.5 y \implies x = 2.5 y

. Substituting

x = 2.5 y

and

h_s = 0.6 h_j

into

A

yields

A = \frac{(2.5 y)(0.6 h_j) + y h_j}{1.6 h_j} = \frac{1.5 y h_j + y h_j}{1.6 h_j} = \frac{2.5 y}{1.6} = 1.5625 y

.

Combining both statements allows us to express the average rate as a multiple of

y

(

1.5625 y

), but since no concrete dollar value for

y

or

x

is provided in either statement, the exact average dollar rate cannot be calculated. Thus, both statements together are insufficient.

Key Concept

Distinguishing relative ratios from absolute numerical values in Data Sufficiency

Estimated Time:2m 0s

Question 191Question →

If $n$ is a positive integer, is $n$ divisible by 18?

(1) $n^2$ is divisible by 54.
(2) $n^3$ is divisible by 108.

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) establishes that

n^2

is divisible by

54 = 2^1 \times 3^3

. Because the exponent of 3 in

n^2

is

2b

, the condition

2b \geq 3

implies

b \geq 1.5

, which forces

b \geq 2

since

b

must be an integer. Similarly,

2a \geq 1 \implies a \geq 1

. Therefore,

n

must contain at least

2^1 \times 3^2 = 18

as a factor, guaranteeing that

n

is divisible by 18 (Definitive Yes). Statement (2) establishes that

n^3

is divisible by

108 = 2^2 \times 3^3

, which only forces

a \geq 1

and

b \geq 1

, so

n

must only be a multiple of 6. Since

n = 6

yields a 'No' and

n = 18

yields a 'Yes', Statement (2) is insufficient.

Step-by-Step Solution

1

Rephrase the question target in terms of prime factorization

For a positive integer

n

to be divisible by

18 = 2^1 \times 3^2

, the prime factorization of

n

must contain at least one factor of 2 and at least two factors of 3.

Decomposing the target integer into prime factors establishes clear minimum exponent thresholds for evaluation.

2

Evaluate Statement (1):

n^2

is divisible by

54 = 2^1 \times 3^3

If

n = 2^a \times 3^b \times k

, then

n^2 = 2^{2a} \times 3^{2b} \times k^2

. Divisibility by

2^1 \times 3^3

requires

2a \geq 1 \implies a \geq 1

and

2b \geq 3 \implies b \geq 2

(since

a

and

b

are integers). Thus

n

must contain at least

2^1 \times 3^2 = 18

, giving a definitive YES.

Integer constraints on exponents force fractional lower bounds to round up to the next integer.

3

Evaluate Statement (2):

n^3

is divisible by

108 = 2^2 \times 3^3

If

n^3 = 2^{3a} \times 3^{3b} \times k^3

, then

3a \geq 2 \implies a \geq 1

and

3b \geq 3 \implies b \geq 1

. Thus

n

must be divisible by

2^1 \times 3^1 = 6

. Testing values: if

n = 6

,

6^3 = 216

(divisible by 108), but 6 is NOT divisible by 18 (NO). If

n = 18

,

18^3 = 5832

(divisible by 108), and 18 IS divisible by 18 (YES). Statement (2) is INSUFFICIENT.

Testing specific values demonstrates that Statement (2) allows both 'Yes' and 'No' outcomes.

Key Concept

Prime Factor Exponent Analysis under Integer Constraints in Data Sufficiency

Estimated Time:2m 0s

Question 192Question →

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.

Show answer & explanation

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

Answer

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.

Step-by-Step Solution

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).

Key Concept

Statement Independence Evaluation and Statement Combination with Real Number Sign Traps

Question 193Question →

If $m$ and $n$ are positive real numbers, is $m + n$ an integer?

(1) $m^2 - n^2$ is a prime number.
(2) $m - n$ is a positive 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: Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

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

The correct answer is that both statements combined are insufficient. A common high-level trap in GMAT Data Sufficiency is assuming variables must be integers when the stem specifies 'real numbers'. Combining

(m - n)(m + n) = p

(prime) with

m - n = k

(positive integer) gives

m + n = \frac{p}{k}

. If

m = 2.25

and

n = 0.25

, we get

m - n = 2

(integer) and

m^2 - n^2 = 5

(prime), but

m + n = 2.5

(not an integer). Conversely,

m = 2

and

n = 1

gives

m - n = 1

and

m^2 - n^2 = 3

(prime), where

m + n = 3

(an integer). Because both an affirmative and a negative answer are possible, the information remains insufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is NOT sufficient.

If

m^2 - n^2 = 2

(a prime number), we could have

m = 2

and

n = √2

, giving

m + n = 2 + √2

(not an integer). Alternatively,

m = 2

and

n = 1

gives

m^2 - n^2 = 3

(prime) and

m + n = 3

(an integer). Hence, Statement (1) alone is insufficient.

2

Evaluate Statement (2) independently.

Statement (2) is NOT sufficient.

If

m - n = 2

, we could have

m = 3.5

and

n = 1.5

, giving

m + n = 5

(an integer). Or we could have

m = 3.2

and

n = 1.2

, giving

m + n = 4.4

(not an integer). Hence, Statement (2) alone is insufficient.

3

Evaluate Statements (1) and (2) combined.

Test for consistency with both integer and non-integer outcomes for

m + n

.

We know

m^2 - n^2 = (m - n)(m + n) = p

, where

p

is prime and

m - n = k

is a positive integer. Thus

k(m + n) = p

, so

m + n = \frac{p}{k}

.

4

Construct concrete counterexamples under both combined statements.

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

Case 1 (YES): Let

m = 2

and

n = 1

. Then

m^2 - n^2 = 3

(prime) and

m - n = 1

(positive integer). Here,

m + n = 3

, which IS an integer.
Case 2 (NO): Let

m = 2.25

and

n = 0.25

. Both are positive real numbers.

m - n = 2.25 - 0.25 = 2

(positive integer).

m^2 - n^2 = (2.25 - 0.25)(2.25 + 0.25) = 2 \times 2.5 = 5

(prime number). But

m + n = 2.25 + 0.25 = 2.5

, which is NOT an integer. Since

m + n

can be an integer or a non-integer, both statements together are not sufficient.

Key Concept

Implicit Integer Constraints vs. Real Number Definitions in Data Sufficiency

Question 194Question →

A dataset consists of $15$ distinct positive integers. Is the median of the dataset greater than its arithmetic mean?

(1) The sum of the $7$ smallest integers in the dataset is $42$ , and the sum of the $7$ largest integers in the dataset is $140$ .
(2) The dataset forms an arithmetic progression.

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.

The option stating that Statement (2) alone is sufficient while Statement (1) alone is not sufficient is correct because Statement (2) establishes that the dataset is an arithmetic progression, where median always equals mean. This provides a definitive 'No' to the question of whether the median is strictly greater than the mean, satisfying sufficiency. Statement (1) allows values of the median both above and below

13

, yielding both 'Yes' and 'No' outcomes.

Step-by-Step Solution

1

Rephrase the question stem mathematically.

Let the 15 distinct positive integers be ordered as

x_1 < x_2 < \dots < x_7 < x_8 < x_9 < \dots < x_{15}

. The median is

x_8

, and the arithmetic mean is

\mu = \frac{\sum_{i=1}^{15} x_i}{15}

. We need to determine whether

x_8 > \mu

.

Simplifying the question target clarifies what information is necessary for sufficiency.

2

Evaluate Statement (1) independently.

Statement (1) gives

\sum_{i=1}^7 x_i = 42

and

\sum_{i=9}^{15} x_i = 140

. The total sum is

42 + x_8 + 140 = 182 + x_8

, so the mean is

\mu = \frac{182 + x_8}{15}

. The inequality

x_8 > \frac{182 + x_8}{15}

simplifies to

14 x_8 > 182

, or

x_8 > 13

. If

x_8 = 10

(with

x_1 \dots x_7 = \{3,4,5,6,7,8,9\}

and

x_9 \dots x_{15} = \{11,12,13,14,15,16,59\}

), then

x_8 \le 13

yields 'No'. If

x_8 = 15

(with

x_1 \dots x_7 = \{1,2,3,4,5,13,14\}

and

x_9 \dots x_{15} = \{16,17,18,19,20,21,29\}

), then

x_8 > 13

yields 'Yes'.

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

3

Evaluate Statement (2) independently.

Statement (2) states that the elements form an arithmetic progression. In any set of numbers that forms an arithmetic progression, the distribution is perfectly symmetric, so the arithmetic mean is exactly equal to the median (

x_8 = \mu

). Therefore,

x_8 > \mu

is false, giving a definitive answer of 'No'.

In Data Sufficiency Yes/No questions, a definitive 'No' answer establishes that the statement IS sufficient.

Key Concept

Data Sufficiency evaluation of data distributions, comparing median and arithmetic mean in symmetric versus asymmetric datasets.

Question 195Question →

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

(1) $x^2 - y^2 = 12(x + y)$ and $x + y \neq 0$
(2) $x^2 + y^2 = 2xy + 144$

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 option stating that Statement (1) ALONE is sufficient while Statement (2) ALONE is not sufficient is correct. Factoring Statement (1) gives

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

. Because

x + y \neq 0

, dividing both sides by

(x + y)

establishes

x - y = 12

uniquely. In contrast, Statement (2) simplifies to

(x - y)^2 = 144

, yielding two possible values (

12

and

-12

), which makes Statement (2) insufficient on its own.

Step-by-Step Solution

1

Evaluate Statement (1) independently

The statement gives

x^2 - y^2 = 12(x + y)

. Factoring the difference of squares gives

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

. Since

x + y \neq 0

, divide both sides by

(x + y)

to get

x - y = 12

.

Because a unique numerical value for

x - y

is obtained, Statement (1) ALONE is sufficient.

2

Evaluate Statement (2) independently

Rearranging

x^2 + y^2 = 2xy + 144

yields

x^2 - 2xy + y^2 = 144

, which contracts to

(x - y)^2 = 144

. Taking the square root gives

x - y = 12

or

x - y = -12

.

Because there are two distinct potential values for

x - y

, Statement (2) ALONE is not sufficient.

3

Combine the evaluations

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

No combination of statements is required since Statement (1) alone uniquely determines the target value.

Key Concept

Algebraic Stem Simplification and Degree Ambiguity in Data Sufficiency

Estimated Time:2m 0s

Question 196Question →

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.

True

False

Show answer & explanation

Answer: False

Answer

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.

Step-by-Step Solution

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.

Key Concept

Data Sufficiency Statement Combination for Yes/No Questions

Question 197Question →

If $x$ and $y$ are real numbers such that $x + y \neq 0$ , what is the value of the product $xy$ ?

(1) $x^3 + y^3 = 28(x + y)$
(2) $x^2 + y^2 = 20 - xy$

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.

Combining both statements provides two independent linear equations in terms of

(x+y)^2

and

xy

. Subtracting Statement (1) from Statement (2) eliminates the squared sum term

(x+y)^2

and yields a unique value of

-4

for the product

xy

, with real solutions existing for

x

and

y

. Therefore, both statements together are sufficient.

Step-by-Step Solution

1

Analyze Statement (1) algebraically.

(x + y)^2 - 3xy = 28

Factor

x^3 + y^3

as

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

. Since

x + y \neq 0

, divide both sides by

(x + y)

to obtain

x^2 - xy + y^2 = 28

. Express

x^2 + y^2

as

(x + y)^2 - 2xy

, yielding

(x + y)^2 - 3xy = 28

. Because

(x + y)^2

is unknown,

xy

can take multiple real values (e.g., if

(x+y)^2 = 16

,

xy = -4

; if

(x+y)^2 = 36

,

xy = 8/3

). Thus, Statement (1) alone is insufficient.

2

Analyze Statement (2) algebraically.

(x + y)^2 - xy = 20

Rearrange

x^2 + y^2 = 20 - xy

to

x^2 + xy + y^2 = 20

. Express

x^2 + y^2

as

(x + y)^2 - 2xy

, yielding

(x + y)^2 - xy = 20

. Again, since

(x + y)^2

is unknown,

xy

is not uniquely determined (e.g., if

(x+y)^2 = 16

,

xy = -4

; if

(x+y)^2 = 4

,

xy = -16

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

3

Combine both statements and solve the system of equations.

xy = -4

Let

u = (x + y)^2

and

v = xy

. Statement (1) gives

u - 3v = 28

and Statement (2) gives

u - v = 20

. Subtracting the first equation from the second yields

(u - v) - (u - 3v) = 20 - 28 \implies 2v = -8 \implies v = -4

. Hence,

xy = -4

uniquely.

4

Verify existence of real solutions for

x

and

y

.

Real solutions exist because the discriminant is positive.

With

u = 16

, we have

x + y = 4

or

x + y = -4

. For

x + y = 4

and

xy = -4

,

x

and

y

are roots of

t^2 - 4t - 4 = 0

, which has discriminant

16 - 4(1)(-4) = 32 > 0

. Thus real values of

x

and

y

exist.

Key Concept

System rephrasing in Data Sufficiency by substituting composite variables like

(x+y)^2

and

xy

.

Question 198Question →

If $r$ and $s$ are non-zero real numbers, what is the value of $\frac{r^2 + 2s^2}{rs}$ ?

(1) $r^2 - 3rs + 2s^2 = 0$
(2) $r + s = 6$

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 correct option states that Statement (1) alone is sufficient while Statement (2) is not. Rephrasing the target expression

\frac{r^2 + 2s^2}{rs}

as

\frac{r}{s} + 2\left(\frac{s}{r}\right)

shows that the target value depends exclusively on the ratio

t = \frac{r}{s}

. Statement (1) factors into

(r - s)(r - 2s) = 0

, giving

t = 1

or

t = 2

. Testing

t = 1

yields

1 + 2 = 3

, and testing

t = 2

yields

2 + 1 = 3

. Because both possible values of

t

evaluate to the same constant result (

3

), Statement (1) alone uniquely determines the target value and is sufficient. Statement (2) allows infinitely many ratios for

r/s

, giving different outcomes, and is therefore insufficient.

Step-by-Step Solution

1

Rephrase the target expression algebraically.

Dividing each term in the numerator by the denominator gives

\frac{r^2 + 2s^2}{rs} = \frac{r}{s} + 2\left(\frac{s}{r}\right)

. Letting

t = \frac{r}{s}

, the target expression is

t + \frac{2}{t}

.

Simplifying the question stem reveals that we only need to know the ratio

\frac{r}{s}

.

2

Evaluate Statement (1):

r^2 - 3rs + 2s^2 = 0

.

Divide the entire equation by

s^2

(since

s \neq 0

) to get

\left(\frac{r}{s}\right)^2 - 3\left(\frac{r}{s}\right) + 2 = 0

, or

t^2 - 3t + 2 = 0

. Factoring yields

(t - 1)(t - 2) = 0

, so

t = 1

or

t = 2

.
If

t = 1

,

t + \frac{2}{t} = 1 + 2 = 3

.
If

t = 2

,

t + \frac{2}{t} = 2 + \frac{2}{2} = 3

.
In both cases, the target expression equals

3

.

Although there are two solutions for the ratio

t

, both lead to the identical value of

3

for the target expression. Thus, Statement (1) alone is sufficient.

3

Evaluate Statement (2):

r + s = 6

.

If

r = 2

and

s = 4

, then

\frac{r}{s} = \frac{1}{2}

, giving a target value of

\frac{1}{2} + 4 = 4.5

.
If

r = 3

and

s = 3

, then

\frac{r}{s} = 1

, giving a target value of

1 + 2 = 3

.
Since different pairs of

r

and

s

yield different values for the expression, Statement (2) alone is not sufficient.

Knowing only the linear sum of two variables does not uniquely fix their ratio.

Key Concept

Question Stem Rephrasing and Symmetry in Homogeneous Quadratic Expressions

Question 199Question →

A logistics center processes packages using two sorting machines, Machine X and Machine Y, each operating continuously at its own constant hourly rate. On a given operational shift, Machine X processed packages for $t_x$ hours and Machine Y processed packages for $t_y$ hours. What percentage of the total number of packages processed by both machines combined during the shift was processed by Machine X?

(1) If Machine X had operated for $20\%$ more time and Machine Y had operated for $20\%$ less time, Machine X would have processed $50\%$ more packages than Machine Y.

(2) The ratio of the hourly processing rate of Machine X to the hourly processing rate of Machine Y was $3:2$ , and the total operating time of both machines combined was $10$ hours.

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) alone is sufficient because translating the percentage modifications into an equation gives

1.2 P_x = 1.5 \times (0.8 P_y)

, which simplifies to

1.2 P_x = 1.2 P_y

, or

P_x = P_y

. This proves Machine X processed exactly

50\%

of the total packages. Statement (2) alone is not sufficient because knowing

t_x + t_y = 10

and

r_x / r_y = 3/2

allows infinitely many different package output ratios depending on how the

10

hours are split between the two machines.

Step-by-Step Solution

1

Define target variable algebraically from question stem

Let

r_x

and

r_y

be the constant hourly processing rates of Machine X and Machine Y, respectively. The total packages processed by Machine X is

P_x = r_x t_x

, and by Machine Y is

P_y = r_y t_y

. The target percentage processed by Machine X is

\frac{P_x}{P_x + P_y} \times 100\% = \frac{1}{1 + \frac{P_y}{P_x}} \times 100\%

. Finding the ratio

\frac{P_x}{P_y}

is necessary and sufficient.

Rephrasing the question stem shows that finding the single ratio of total packages

P_x / P_y

is sufficient to answer the question.

2

Evaluate Statement (1)

Under the hypothetical scenario in Statement (1), the new operating time for Machine X is

1.2 t_x

and for Machine Y is

0.8 t_y

. The new package totals are

r_x (1.2 t_x) = 1.2 P_x

and

r_y (0.8 t_y) = 0.8 P_y

. The statement specifies that

1.2 P_x = 1.5 \times (0.8 P_y)

. Simplifying the right-hand side:

1.5 \times 0.8 = 1.2

, so

1.2 P_x = 1.2 P_y \implies P_x = P_y

. Thus,

\frac{P_x}{P_y} = 1

, which means Machine X processed

\frac{P_x}{P_x + P_x} = 50\%

of the total packages.

Algebraic simplification yields a unique value for the target percentage, making Statement (1) sufficient.

3

Evaluate Statement (2)

Statement (2) gives

\frac{r_x}{r_y} = \frac{3}{2}

and

t_x + t_y = 10

. The ratio of total packages processed is

\frac{P_x}{P_y} = \left(\frac{r_x}{r_y}\right) \left(\frac{t_x}{t_y}\right) = \frac{3}{2} \left(\frac{t_x}{t_y}\right)

. Since

t_x

and

t_y

can take any positive real values summing to

10

(e.g.,

t_x = 2, t_y = 8 \implies \frac{P_x}{P_y} = \frac{3}{8}

vs.

t_x = 6, t_y = 4 \implies \frac{P_x}{P_y} = \frac{9}{4}

), the percentage varies.

Without individual values for

t_x

and

t_y

, multiple target percentages are possible, so Statement (2) is not sufficient.

Key Concept

Question Stem Rephrasing and Algebraic Simplification of Combined Rates

Estimated Time:2m 30s

Question 200Question →

If $r$ and $s$ are positive integers, is $r^2 - s^2$ divisible by 8?

(1) $r$ and $s$ are both odd integers.
(2) $r + s$ is divisible by 4.

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: EACH statement ALONE is sufficient.

Answer

Each statement alone is sufficient to answer the question with a definitive 'Yes'.

The correct response identifies that each statement alone gives a definitive 'Yes' answer. Statement (1) shows that the square of any odd integer is

1 \pmod 8

, so subtracting two odd squares gives

0 \pmod 8

. Statement (2) shows

r+s

is a multiple of 4, which forces

r

and

s

to share parity, making

r-s

even and their product

(r+s)(r-s)

a multiple of

4 \times 2 = 8

.

Step-by-Step Solution

1

Rephrase the question target using algebraic factorization.

Note that

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

. The question asks whether

(r + s)(r - s)

is a multiple of 8.

Factoring difference of squares isolates the sum and difference components to easily test parity and divisibility.

2

Evaluate Statement (1):

r

and

s

are both odd integers.

Any odd integer can be written as

2k + 1

. Squaring gives

(2k + 1)^2 = 4k^2 + 4k + 1 = 4k(k + 1) + 1

. Since

k(k + 1)

is the product of two consecutive integers, it is always even, making

4k(k + 1)

a multiple of 8. Thus, any odd integer squared leaves a remainder of 1 when divided by 8. Therefore,

r^2 \equiv 1 \pmod 8

and

s^2 \equiv 1 \pmod 8

, so

r^2 - s^2 \equiv 1 - 1 = 0 \pmod 8

.

Statement (1) alone yields a definitive 'Yes', so Statement (1) is sufficient.

3

Evaluate Statement (2):

r + s

is divisible by 4.

If

r + s

is divisible by 4, then

r + s = 4m

for some integer

m

. Since

r + s

is even,

r

and

s

must have the same parity (either both even or both odd). Consequently, their difference

r - s = (r + s) - 2s = 4m - 2s = 2(2m - s)

is an even integer (a multiple of 2). Multiplying

(r + s)

by

(r - s)

yields

(\text{multiple of } 4) \times (\text{multiple of } 2) = \text{multiple of } 8

.

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

Key Concept

Divisibility Rules of Difference of Squares and Parity of Integers

Estimated Time:2m 0s

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