Data Sufficiency Practice Questions — GMAT — Page 8

Question 141Question →

If $r$ is a real number, is $r^3 < r$ ?

(1) $|r - 0.5| < 0.5$
(2) $|r| < 1$

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

r^3 < r

shows that the inequality holds true if and only if

r < -1

or

0 < r < 1

. Statement (1) bounds

r

precisely within the interval

(0, 1)

, ensuring a definitive 'Yes' answer. Statement (2) defines the range

(-1, 1)

, which includes values where the inequality holds (such as

0.5

) and values where it fails (such as

-0.5

), making it insufficient.

Step-by-Step Solution

1

Rephrase the target inequality algebraically.

The inequality

r^3 < r

rearranges to

r^3 - r < 0

, which factors as

r(r - 1)(r + 1) < 0

.

Simplifying the stem isolates the specific intervals on the number line where the statement holds true.

2

Determine the valid number ranges for the rephrased question.

The expression

r(r - 1)(r + 1)

is strictly negative when

r < -1

or when

0 < r < 1

.

Testing sign changes across key boundary points (

r = -1

,

r = 0

,

r = 1

) identifies the target ranges.

3

Evaluate Statement (1):

|r - 0.5| < 0.5

.

This absolute value inequality expands to

-0.5 < r - 0.5 < 0.5

, which simplifies to

0 < r < 1

.

Since every value in the range

0 < r < 1

satisfies

r^3 < r

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

4

Evaluate Statement (2):

|r| < 1

.

This absolute value inequality simplifies to

-1 < r < 1

.

If

r = 0.5

, then

r^3 = 0.125 < 0.5

(Yes). However, if

r = -0.5

, then

r^3 = -0.125 > -0.5

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

Key Concept

Data Sufficiency evaluation of cubic inequalities and absolute value range constraints

Question 142Question →

Each of the $100$ employees at Company K works in Division X, Division Y, or both divisions. The arithmetic mean age of the employees in Division X is $35$ years, and the arithmetic mean age of the employees in Division Y is $45$ years. Is the arithmetic mean age of all $100$ employees at Company K greater than $40$ years?

(1) Exactly $40$ employees work in Division X and exactly $70$ employees work in Division Y.
(2) The arithmetic mean age of the employees who work in both Division X and Division Y is $40$ years.

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

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

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

EACH statement ALONE is sufficient.

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

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 definitively, but neither statement alone is sufficient.

Combining both statements establishes the exact sizes of all three disjoint subsets (Division X only = 30, Division Y only = 60, both divisions = 10) and the average value of the overlapping subset (40 years). Calculating the total age sum gives 4,150 years, resulting in an exact overall mean of 41.5 years. This provides a definitive 'Yes' answer to whether the mean is greater than 40 years.

Step-by-Step Solution

1

Formulate the algebraic expressions for the total sum of ages.

Let

x

be the number of employees in Division X only,

y

be the number in Division Y only, and

z

be the number in both divisions.

x + y + z = 100

. Let

S_x, S_y, S_z

be the sum of ages of employees in Division X only, Division Y only, and both divisions, respectively. Then

S_x + S_z = 35(x + z)

and

S_y + S_z = 45(y + z)

. The total sum of ages of all

100

employees is

S_{total} = S_x + S_y + S_z = 35x + 45y + 80z - S_z

.

Because employees in both divisions contribute to the averages of both Division X and Division Y, simply adding

35(x+z)

and

45(y+z)

counts

S_z

twice.

2

Evaluate Statement (1) independently.

Statement (1) states

x + z = 40

and

y + z = 70

. Since

x + y + z = 100

, we find

z = (40 + 70) - 100 = 10

,

x = 30

, and

y = 60

. Thus,

S_{total} = 35(40) + 45(70) - S_z = 4550 - S_z

. The overall mean age is

\frac{4550 - S_z}{100} = 45.5 - \frac{S_z}{100}

. Depending on the value of

S_z

(the sum of ages of the

10

overlap employees), the overall mean can be greater than

40

or less than or equal to

40

.

Without knowing

S_z

or the average age of the overlap group, the overall mean cannot be uniquely bounded. Thus, Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

Statement (2) states

\frac{S_z}{z} = 40 \implies S_z = 40z

. Substituting into

S_{total}

gives

S_{total} = 35(x+z) + 45(y+z) - 40z = 35x + 45y + 40z

. The overall mean age is

\frac{35x + 45y + 40z}{x + y + z}

. The condition

\frac{35x + 45y + 40z}{x + y + z} > 40

simplifies to

35x + 45y > 40x + 40y \iff 5y > 5x \iff y > x

.

Statement (2) provides no information about whether

y > x

(whether more employees work exclusively in Division Y than in Division X). Thus, Statement (2) alone is NOT sufficient.

4

Evaluate Statements (1) and (2) together.

From Statement (1),

x = 30

,

y = 60

, and

z = 10

. From Statement (2),

S_z = 40(10) = 400

. Since

y = 60 > x = 30

, the condition

y > x

holds. Substituting these values into

S_{total}

yields

S_{total} = 35(30) + 45(60) + 40(10) = 1050 + 2700 + 400 = 4150

. The arithmetic mean age of all

100

employees is

\frac{4150}{100} = 41.5

years, which is strictly greater than

40

.

Combining both statements yields a unique and definitive 'Yes' answer to the question stem.

Key Concept

Weighted averages in overlapping sets with double-counted sums

Question 143Question →

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

(1) $a^2 + b^2 + 4a - 6b = -13$
(2) $ab = -6$

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.

Statement (1) can be rewritten by completing the square as

(a + 2)^2 + (b - 3)^2 = 0

. Since

a

and

b

are constrained to be real numbers,

(a + 2)^2 \geq 0

and

(b - 3)^2 \geq 0

. The only way their sum can equal

0

is if

a = -2

and

b = 3

. This uniquely determines

(a + b)^2 = (-2 + 3)^2 = 1

, making Statement (1) alone sufficient. Statement (2) gives

ab = -6

, which allows multiple values for

(a + b)^2

(e.g.,

1

or

25

), making it insufficient. Hence, the option stating that Statement (1) alone is sufficient while Statement (2) alone is not is the correct choice.

Step-by-Step Solution

1

Analyze Statement (1):

a^2 + b^2 + 4a - 6b = -13

Rearrange and group terms:

(a^2 + 4a) + (b^2 - 6b) = -13

. Complete the square for both variables by adding

4

and

9

to both sides:

(a^2 + 4a + 4) + (b^2 - 6b + 9) = -13 + 4 + 9

, which simplifies to

(a + 2)^2 + (b - 3)^2 = 0

.

Grouping and completing the square reveals the sum-of-squares structure.

2

Evaluate the real number constraint on Statement (1)

Because

a

and

b

are real numbers,

(a + 2)^2 \geq 0

and

(b - 3)^2 \geq 0

. The sum of two non-negative terms can equal zero if and only if both terms are independently zero:

a + 2 = 0 \implies a = -2

and

b - 3 = 0 \implies b = 3

.

The sum of non-negative real squares equaling zero forces each squared term to be zero.

3

Calculate the target expression using values from Statement (1)

(a + b)^2 = (-2 + 3)^2 = (1)^2 = 1

. Since this yields a single unique value, Statement (1) ALONE is sufficient.

A unique value for the target expression establishes sufficiency.

4

Analyze Statement (2):

ab = -6

If

a = 2

and

b = -3

, then

ab = -6

and

(a + b)^2 = (2 - 3)^2 = 1

. If

a = 1

and

b = -6

, then

ab = -6

and

(a + b)^2 = (1 - 6)^2 = 25

. Multiple values are possible.

Testing specific valid cases proves that Statement (2) does not yield a unique result.

5

Conclude Data Sufficiency determination

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

Statement (1) produces a unique answer, whereas Statement (2) does not.

Key Concept

Completing the Square and Trivial Inequality for Real Squares in Data Sufficiency

Question 144Question →

Among a group of $50$ healthcare professionals, each professional has certified training in either Telemedicine, Robotic Surgery, or both. Exactly $30$ professionals are certified in Telemedicine and exactly $35$ are certified in Robotic Surgery. What is the average (arithmetic mean) years of experience of the professionals who are certified in BOTH Telemedicine and Robotic Surgery?

(1) The average years of experience of all $30$ professionals certified in Telemedicine is $9$ years, and the average years of experience of all $35$ professionals certified in Robotic Surgery is $8$ years.
(2) The average years of experience of the $15$ professionals certified ONLY in Telemedicine is $10$ years, and the average years of experience of the $20$ professionals certified ONLY in Robotic Surgery is $8$ years.

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

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

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

EACH statement ALONE is sufficient.

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

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 one stating that BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. Rephrasing the stem shows there are

15

professionals in the 'Telemedicine Only' group,

20

in the 'Robotic Surgery Only' group, and

15

in 'Both'. Statement (1) gives total Telemedicine experience sum

S_T = 270

, but cannot separate

S_{T_{only}}

from

S_B

. Statement (2) gives

S_{T_{only}} = 150

, but gives no total experience bound. Together,

S_B = 270 - 150 = 120

, giving a unique mean of

\frac{120}{15} = 8

years.

Step-by-Step Solution

1

Rephrase the question stem using overlapping set formulas to find subgroup counts.

Let

N = 50

,

T = 30

, and

R = 35

. Using

N = T + R - B

, we get

50 = 30 + 35 - B \implies B = 15

(professionals in Both). Thus, 'Telemedicine Only' count is

30 - 15 = 15

, and 'Robotic Surgery Only' count is

35 - 15 = 20

. Target: Find the mean experience of the

15

professionals in Both,

\mu_B = \frac{S_B}{15}

.

Simplifying the stem establishes the exact numerical count of professionals in each of the three distinct subgroups: Telemedicine Only (

15

), Robotic Surgery Only (

20

), and Both (

15

).

2

Evaluate Statement (1) independently.

Statement (1) gives total experience

S_T = 30 \times 9 = 270

and

S_R = 35 \times 8 = 280

. Since

S_T = S_{T_{only}} + S_B = 270

,

S_B

depends on

S_{T_{only}}

, which is unknown. Multiple values of

S_B

are possible. NOT sufficient.

Knowing total group averages does not isolate how experience is divided between single-category members and dual-category members.

3

Evaluate Statement (2) independently.

Statement (2) gives

S_{T_{only}} = 15 \times 10 = 150

and

S_{R_{only}} = 20 \times 8 = 160

. Without knowledge of

S_T

,

S_R

, or total group experience,

S_B

can take any real value. NOT sufficient.

Knowing only the single-category subgroup totals provides no boundary or equation for the overlapping group.

4

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

From Statement (1),

S_T = S_{T_{only}} + S_B = 270

. From Statement (2),

S_{T_{only}} = 150

. Substituting gives

150 + S_B = 270 \implies S_B = 120

. Thus,

\mu_B = \frac{120}{15} = 8

years. SUFFICIENT.

Combining both statements yields a unique value for the total experience sum of the overlapping group.

Key Concept

Data Sufficiency with Overlapping Sets and Weighted Averages

Estimated Time:2m 0s

Question 145Question →

A seminar was attended by $100$ professionals, each of whom speaks at least one of two languages: Spanish or French. Exactly $60$ of the professionals speak Spanish, and exactly $50$ speak French. If all $100$ professionals took a language proficiency examination scored on a scale from $0$ to $100$ , is the average (arithmetic mean) score of all $100$ professionals greater than $75$ ?

(1) The average score of the professionals who speak only Spanish is $80$ , and the average score of the professionals who speak only French is $70$ .
(2) The average score of the professionals who speak both Spanish and French is $85$ .

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 response identifies that both statements together are sufficient while neither statement alone is sufficient. By using the principle of inclusion-exclusion on the overlapping set sizes, we find that there are 50 professionals who speak only Spanish, 40 who speak only French, and 10 who speak both. Statement (1) alone is insufficient because the average score of the 10 dual-language professionals remains unknown, allowing the overall average to fall either above or below 75. Statement (2) alone is insufficient because it provides no score data for 90 of the 100 professionals. When both statements are combined, all three disjoint subgroup averages are known, giving an exact overall average of 76.5, which is definitively greater than 75.

Step-by-Step Solution

1

Rephrase the question stem using overlapping set principles

Let

S

be Spanish speakers (

60

) and

F

be French speakers (

50

). Since every professional speaks at least one language,

|S \cup F| = 100

. By the inclusion-exclusion principle,

|S \cup F| = |S| + |F| - |S \cap F| \implies 100 = 60 + 50 - |S \cap F|

, so

|S \cap F| = 10

. The group partitions into: Spanish only =

60 - 10 = 50

, French only =

50 - 10 = 40

, and Both =

10

.

Deconstructing the total population into three mutually exclusive subgroups establishes the exact weights for computing the overall weighted average score.

2

Evaluate Statement (1) alone

Statement (1) provides average score for Spanish-only (

80

) and French-only (

70

). The total score sum is

50(80) + 40(70) + 10(A_{both}) = 6800 + 10(A_{both})

. The overall average is

68 + 0.1(A_{both})

. Depending on

A_{both}

(

0 \le A_{both} \le 100

), the overall average can range from

68

to

78

. For instance, if

A_{both} = 70

, overall average is

75

(not

>75

); if

A_{both} = 100

, overall average is

78

(

>75

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

Since the score of the overlap group is unknown, the overall average cannot be uniquely tested against the threshold of 75.

3

Evaluate Statement (2) alone

Statement (2) provides

A_{both} = 85

, but gives no information about the average scores of the Spanish-only (

50

people) or French-only (

40

people) groups. Thus, the overall average score could be very low or very high. Statement (2) alone is NOT sufficient.

Without data on 90% of the population, statement (2) alone leaves the overall mean undetermined.

4

Evaluate Statements (1) and (2) together

Combining both statements gives:

50

people with average

80

,

40

people with average

70

, and

10

people with average

85

. Overall total score sum

= 50(80) + 40(70) + 10(85) = 4000 + 2800 + 850 = 7650

. Overall average

= 7650 / 100 = 76.5

. Since

76.5 > 75

, we get a definitive YES answer.

Having full weighted average data for all three disjoint components of the set yields a single, precise overall mean.

Key Concept

Weighted Average across Mutually Exclusive Partitions of Overlapping Sets

Estimated Time:2m 0s

Question 146Question →

If $x$ is a real number, is $|x^3 - x| > x^2 - 1$ ?

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

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 correct option is the one stating that Statement (2) alone is sufficient, but Statement (1) alone is not sufficient. Simplifying the stem target

|x^3 - x| > x^2 - 1

into

|x||x^2 - 1| > x^2 - 1

reveals that the inequality holds for every real number except

x = 1

and

x = -1

(where both sides equal zero). Statement (1) states

|x - 1| > 0

, meaning

x \neq 1

, but it permits

x = -1

, which yields a 'No' answer, making Statement (1) insufficient. Statement (2) states

|x + 2| < 1

, which expands to

-3 < x < -1

. Because this interval strictly excludes both

x = -1

and

x = 1

, every value of

x

in this range satisfies the stem inequality, yielding a definitive 'Yes'.

Step-by-Step Solution

1

Rephrase the question stem target algebraically.

The target inequality

|x^3 - x| > x^2 - 1

simplifies to

|x(x^2 - 1)| > x^2 - 1

, which is equivalent to

|x| \cdot |x^2 - 1| > x^2 - 1

.

Factoring allows analysis of the critical values where

x^2 - 1 = 0

.

2

Analyze the conditions under which

|x| \cdot |x^2 - 1| > x^2 - 1

holds true.

Case 1: If

x^2 - 1 = 0

(meaning

x = 1

or

x = -1

), LHS =

0

and RHS =

0

, giving

0 > 0

, which is FALSE. Case 2: If

x^2 - 1 < 0

(meaning

-1 < x < 1

), LHS is positive and RHS is negative, so non-negative > negative is TRUE. Case 3: If

x^2 - 1 > 0

(meaning

x > 1

or

x < -1

), dividing both sides by

x^2 - 1 > 0

yields

|x| > 1

, which is TRUE. Thus, the inequality holds for ALL real numbers EXCEPT

x = 1

and

x = -1

. The question simplifies to: Is

x \neq 1

and

x \neq -1

?

Simplifying the target reveals that the inequality is true everywhere except at the two boundary roots

x = 1

and

x = -1

.

3

Evaluate Statement (1):

|x - 1| > 0

.

This implies

x \neq 1

. However,

x

could equal

-1

. If

x = -1

,

|-1 - 1| = 2 > 0

is satisfied, but

|(-1)^3 - (-1)| = 0

and

(-1)^2 - 1 = 0

, giving

0 > 0

(NO). If

x = 2

,

|2 - 1| = 1 > 0

is satisfied, and

|8 - 2| = 6 > 3

(YES). Since both 'Yes' and 'No' are possible, Statement (1) is NOT sufficient.

Statement (1) rules out

x = 1

but permits

x = -1

.

4

Evaluate Statement (2):

|x + 2| < 1

.

Solving the absolute value inequality gives

-1 < x + 2 < 1

, which simplifies to

-3 < x < -1

. In this range,

x

cannot be

1

or

-1

because the upper boundary is strictly less than

-1

. Thus, for all

x \in (-3, -1)

,

x \neq 1

and

x \neq -1

is guaranteed, yielding a definitive 'YES'. Statement (2) is SUFFICIENT.

The open interval

(-3, -1)

excludes both

x = -1

and

x = 1

.

Key Concept

Question Stem Simplification with Absolute Value Inequalities

Estimated Time:2m 0s

Question 147Question →

The table below displays the number of employees and the average salary per employee across three departments at Company Z:

Department	Number of Employees	Average Salary ($)
Marketing	20	60,000
Engineering	$E$	80,000
Sales	30	50,000

What was the overall average salary per employee across all three departments combined?

(1) The total number of employees across all three departments combined is 100.
(2) The total annual payroll for the Engineering department is $4,000,000.

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: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.

The question target requires finding the weighted average salary across all departments, which reduces to finding the value of the unknown employee count

E

. Statement (1) yields

20 + E + 30 = 100

, so

E = 50

. Statement (2) yields

E \times 80,000 = 4,000,000

, so

E = 50

. Since each statement independently determines

E

, each statement alone is sufficient.

Step-by-Step Solution

1

Rephrase the question stem target

Overall average salary =

\frac{20(60,000) + E(80,000) + 30(50,000)}{20 + E + 30} = \frac{2,700,000 + 80,000E}{50 + E}

. Finding the value of

E

is sufficient to answer the question.

The overall average salary is a weighted average that depends solely on the single unknown parameter

E

.

2

Evaluate Statement (1) independently

Total employees =

20 + E + 30 = 100 \implies E = 50

. Since

E

is uniquely determined, the overall average salary can be calculated.

Statement (1) provides a direct linear equation in

E

.

3

Evaluate Statement (2) independently

Engineering total payroll =

E \times 80,000 = 4,000,000 \implies E = 50

. Since

E

is uniquely determined, the overall average salary can be calculated.

Statement (2) gives the total department cost, which divided by the known average salary gives

E

directly.

4

Conclude sufficiency classification

Each statement alone gives

E = 50

, making each statement individually sufficient.

Both Statement (1) and Statement (2) yield a unique value for

E

independently.

Key Concept

Weighted average determination from tabular data using Data Sufficiency logic

Estimated Time:1m 0s

Question 148Question →

For a real number $p$ , is $p$ an integer?

(1) $p^3 - p$ is a positive prime number.
(2) $p^2$ is an integer.

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

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

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

EACH statement ALONE is sufficient.

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

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) is sufficient because factoring

p^3 - p

gives

(p-1)p(p+1)

, which represents the product of three consecutive integers whenever

p

is an integer. The product of any three consecutive integers must be divisible by 6. Since no prime number is a multiple of 6,

p

cannot be an integer if

p^3 - p

is prime. This provides a definitive 'No' answer to the question 'Is

p

an integer?', establishing sufficiency. Statement (2) is insufficient because

p

could be an integer like 3 (yielding 'Yes') or an irrational number like

\sqrt{3}

(yielding 'No').

Step-by-Step Solution

1

Analyze Statement (1):

p^3 - p

is a positive prime number.

p

cannot be an integer, yielding a definitive 'No' to the question stem.

Factor

p^3 - p

as

(p-1)p(p+1)

. If

p

were an integer, this product would represent three consecutive integers. Any three consecutive integers contain at least one even factor and exactly one multiple of 3, making their product divisible by 6. A prime number is a positive integer greater than 1 with no positive divisors other than 1 and itself, so no prime number can be a multiple of 6. Thus, no integer

p

can make

p^3 - p

a prime number. Since

p^3 - p

is given to be a positive prime,

p

must be a non-integer real number. A definitive 'No' answer establishes that Statement (1) alone is sufficient.

2

Analyze Statement (2):

p^2

is an integer.

p

could be an integer or a non-integer, so the question cannot be answered uniquely.

If

p = 3

,

p^2 = 9

(an integer), and

p

is an integer (Yes). If

p = \sqrt{3}

,

p^2 = 3

(an integer), but

p

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

3

Determine the overall Data Sufficiency choice.

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

Statement (1) provides a conclusive answer on its own, whereas Statement (2) remains ambiguous.

Key Concept

Number Properties and Integer Constraints in Data Sufficiency

Question 149Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

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.

Neither statement alone provides values for both

x

and

y

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

x = 5

and Statement (2) gives

y = 3

, allowing us to uniquely calculate

x + y = 8

. Thus, both statements together are sufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently

x = 5

, but

y

is unknown, so

x + y

cannot be uniquely determined.

Statement (1) alone is not sufficient.

2

Evaluate Statement (2) independently

y = 3

, but

x

is unknown, so

x + y

cannot be uniquely determined.

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

3

Combine Statement (1) and Statement (2)

x = 5

and

y = 3

, yielding

x + y = 5 + 3 = 8

.

Combining both statements provides a single unique numerical answer.

Key Concept

Statement Independence Evaluation and Statement Combination

Question 150Question →

If $x$ and $y$ are positive real numbers, is $(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4$ ?

(1) $x^2 + y^2 = 2xy + 9$
(2) $\sqrt{x} + \sqrt{y} = 5$

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Rephrasing the target question stem shows that

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4

is equivalent to

4 + \frac{(x - y)^2}{xy} > 4

, which simplifies to

x \neq y

for positive real numbers

x

and

y

. Statement (1) rearranges to

(x - y)^2 = 9

, which proves

x \neq y

and provides a definitive 'Yes' answer. Statement (2) allows cases where

x = y = 6.25

as well as

x = 1, y = 16

, leaving the answer uncertain. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Step-by-Step Solution

1

Rephrase the target question stem algebraically.

The target question asking whether

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4

simplifies to asking whether

x \neq y

.

Expanding

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) = 1 + \frac{x}{y} + \frac{y}{x} + 1 = 2 + \frac{x^2 + y^2}{xy} = 4 + \frac{(x - y)^2}{xy}

. Since

x, y > 0

, the quantity

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

holds if and only if

(x - y)^2 > 0

, which means

x \neq y

.

2

Evaluate Statement (1):

x^2 + y^2 = 2xy + 9

.

Statement (1) is sufficient.

Rearranging yields

x^2 - 2xy + y^2 = 9 \implies (x - y)^2 = 9

. Since

(x - y)^2 = 9 \neq 0

, it must be that

x \neq y

. This gives a definitive 'Yes' answer to the rephrased question.

3

Evaluate Statement (2):

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

.

Statement (2) is not sufficient.

If

x = 6.25

and

y = 6.25

, then

\sqrt{6.25} + \sqrt{6.25} = 2.5 + 2.5 = 5

. Here

x = y

, so the answer to 'Is

x \neq y

?' is 'No'. If

x = 1

and

y = 16

, then

\sqrt{1} + \sqrt{16} = 1 + 4 = 5

. Here

x \neq y

, so the answer is 'Yes'. Because both 'Yes' and 'No' are possible, Statement (2) is insufficient.

Key Concept

Data Sufficiency Question Stem Simplification

Estimated Time:1m 30s

Question 151Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

Both statements together are sufficient to determine that

x

is an integer, but neither statement alone is sufficient.

The correct response is that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) permits irrational values like

x = -1 + \sqrt{2}

, making it insufficient on its own. Statement (2) permits irrational values like

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

, making it insufficient on its own. Combining both statements allows us to express

x

as the quotient of two integers

b/a

when

x^2 + 2x \neq 0

, which mathematically forces

x

to be an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently:

x^2 + 2x

is an integer.

Statement (1) is NOT sufficient.

Let

x^2 + 2x = 1

. Solving

x^2 + 2x - 1 = 0

gives

x = -1 + \sqrt{2}

, which is a real non-integer. However, if

x = 1

,

x^2 + 2x = 3

is also an integer. Since

x

could be an integer or a non-integer, Statement (1) alone is insufficient.

2

Evaluate Statement (2) independently:

x^3 + 2x^2

is an integer.

Statement (2) is NOT sufficient.

Let

x^3 + 2x^2 = 1

. The equation

x^3 + 2x^2 - 1 = 0

factors as

(x + 1)(x^2 + x - 1) = 0

. Setting

x^2 + x - 1 = 0

yields non-integer root

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

. For this non-integer value,

x^3 + 2x^2 = 1

, which is an integer. Since

x

can also be the integer

-1

, Statement (2) alone is insufficient.

3

Evaluate Statements (1) and (2) together.

Both statements together are SUFFICIENT.

From Statement (1), let

x^2 + 2x = a

, where

a

is an integer. From Statement (2), let

x^3 + 2x^2 = b

, where

b

is an integer. Notice that

x^3 + 2x^2 = x(x^2 + 2x) = x \cdot a = b

. Case 1: If

a = 0

, then

x^2 + 2x = 0 \implies x(x + 2) = 0 \implies x = 0

or

x = -2

, both of which are integers. Case 2: If

a \neq 0

, then

x = \frac{b}{a}

, meaning

x

is a rational number. Let

x = \frac{m}{n}

in lowest terms where

\gcd(m, n) = 1

and

n > 0

. Substituting

x = \frac{m}{n}

into

x^2 + 2x = a

gives

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

. If

n > 1

, any prime factor

p

of

n

must divide

m(m + 2n)

, which implies

p

divides

m^2

and thus

p

divides

m

. This contradicts

\gcd(m, n) = 1

. Thus,

n

must equal

1

, proving

x

is an integer.

Key Concept

Using algebraic combination of Data Sufficiency statements and integer polynomial constraints to establish sufficiency without assuming variables are integers.

Question 152Question →

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

(1) $x^2 y - x y^2 = 2xy$
(2) $x^2 + y^2 = 5xy$

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 statement that both statements together are not sufficient is correct because combining Statement (1) (

x = y + 2

) and Statement (2) (

x^2 + y^2 = 5xy

) leads to a quadratic equation

3y^2 + 6y - 4 = 0

with two distinct real roots for

y

. Evaluating

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

for both roots yields two distinct possible values,

\frac{5 + \sqrt{21}}{2}

and

\frac{5 - \sqrt{21}}{2}

. Because a unique numerical value cannot be determined, the information provided is not sufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Simplifies to

x - y = 2 \implies x = y + 2

.

Since

x, y \neq 0

,

xy \neq 0

. Dividing

x^2 y - x y^2 = 2xy

by

xy

yields

x - y = 2

. Thus

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

, which varies with

y

. Not sufficient.

2

Evaluate Statement (2) independently.

Yields

(\frac{x}{y})^2 - 5(\frac{x}{y}) + 1 = 0

.

Dividing

x^2 + y^2 = 5xy

by

y^2

gives a quadratic in

k = \frac{x}{y}

:

k^2 - 5k + 1 = 0

. Solving for

k

gives

k = \frac{5 \pm \sqrt{21}}{2}

. Because there are two distinct real solutions for the ratio, Statement (2) alone is not sufficient.

3

Evaluate Statements (1) and (2) combined.

Two distinct pairs of real numbers satisfy both conditions, giving two distinct values for

\frac{x}{y}

.

Substitute

x = y + 2

into

x^2 + y^2 = 5xy

:

(y+2)^2 + y^2 = 5y(y+2) \implies 3y^2 + 6y - 4 = 0

. This gives two real roots for

y

:

y = \frac{-3 \pm \sqrt{21}}{3}

. Substituting each

y

back into

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

yields

\frac{x}{y} = \frac{5 + \sqrt{21}}{2}

and

\frac{x}{y} = \frac{5 - \sqrt{21}}{2}

. Since two distinct ratios remain possible, combined statements are not sufficient.

Key Concept

Non-linear systems in Data Sufficiency often result in multiple valid solutions, requiring explicit verification of solution uniqueness rather than assuming two equations with two variables yield a single solution.

Estimated Time:2m 15s

Question 153Question →

An agricultural research station evaluated a sample of $90$ fruit trees. Each tree was treated with Fertilizer X, Fertilizer Y, or both. Exactly $60$ trees were treated with Fertilizer X, and exactly $50$ trees were treated with Fertilizer Y. What was the average (arithmetic mean) yield, in kilograms, of all $90$ trees?

(1) The average yield of the trees treated with Fertilizer X was $45$ kg, and the average yield of the trees treated only with Fertilizer Y was $35$ kg.
(2) The average yield of the trees treated with Fertilizer Y was $42$ kg, and the average yield of the trees treated only with Fertilizer X was $48$ kg.

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.

Using the principal formula for overlapping sets

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

, we find that

90 = 60 + 50 - N(\text{X } \cap \text{ Y})

, meaning exactly

20

trees received both fertilizers. This partitions the

90

trees into three mutually exclusive groups:

40

trees receiving Only X,

20

trees receiving Both, and

30

trees receiving Only Y.

Statement (1) provides the average for all trees receiving X (which combines 'Only X' and 'Both', totaling

60

trees) as

45

kg, giving a subgroup total yield of

60 \times 45 = 2,700

kg. It also gives the average for the remaining

30

trees ('Only Y') as

35

kg, giving

30 \times 35 = 1,050

kg. Summing these gives the exact total yield of all

90

trees (

3,750

kg), which allows computing a unique overall mean. Hence Statement (1) alone is sufficient.

Statement (2) provides the average for all trees receiving Y (combining 'Only Y' and 'Both', totaling

50

trees) as

42

kg, giving a subgroup total yield of

50 \times 42 = 2,100

kg. It also gives the average for the remaining

40

trees ('Only X') as

48

kg, giving

40 \times 48 = 1,920

kg. Summing these gives the exact total yield of all

90

trees (

4,020

kg), which allows computing a unique overall mean. Hence Statement (2) alone is sufficient.

Since each statement alone is sufficient, the correct option is the one stating that each statement alone is sufficient.

Step-by-Step Solution

1

Determine the number of trees in each disjoint subset using overlapping set principles.

Number of trees receiving both fertilizers is

20

; 'Only X' is

40

; 'Only Y' is

30

.

By the inclusion-exclusion principle:

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

. Thus,

90 = 60 + 50 - N(\text{Both})

, which yields

N(\text{Both}) = 20

. Consequently,

N(\text{Only X}) = 60 - 20 = 40

and

N(\text{Only Y}) = 50 - 20 = 30

.

2

Evaluate Statement (1) independently.

Total yield

= 3,750

kg, yielding a unique overall average of

\frac{3,750}{90} = \frac{125}{3}

kg.

Statement (1) gives the average yield for all

60

trees treated with Fertilizer X (

45

kg) and for the

30

trees treated only with Fertilizer Y (

35

kg). The total yield of all

90

trees is

(60 \times 45) + (30 \times 35) = 2,700 + 1,050 = 3,750

kg. Dividing by

90

gives a single, deterministic value.

3

Evaluate Statement (2) independently.

Total yield

= 4,020

kg, yielding a unique overall average of

\frac{4,020}{90} = \frac{134}{3}

kg.

Statement (2) gives the average yield for all

50

trees treated with Fertilizer Y (

42

kg) and for the

40

trees treated only with Fertilizer X (

48

kg). The total yield of all

90

trees is

(50 \times 42) + (40 \times 48) = 2,100 + 1,920 = 4,020

kg. Dividing by

90

gives a single, deterministic value.

4

Synthesize data sufficiency evaluation.

Since each statement independently allows us to calculate the exact average yield, each statement alone is sufficient.

Data Sufficiency requires identifying whether each statement alone yields a unique solution to the question asked.

Key Concept

Overlapping Sets and Weighted Averages

Question 154Question →

In a graduating class of $120$ students, each student participated in at least one of two extracurricular activities: the Science Club or the Debate Team. The mean score on a national mathematics exam for all students who participated in the Science Club was $85$ , and the mean score for all students who participated in the Debate Team was $80$ . What was the mean mathematics score for all $120$ students in the graduating class?

(1) Exactly $40$ students participated in both the Science Club and the Debate Team, and their mean mathematics score on the exam was $90$ .
(2) The total number of students who participated in the Science Club was equal to the total number of students who participated in the Debate Team.

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.

Both statements together are sufficient. Statement (1) establishes that

40

students are in both activities with a mean score of

90

, leaving

80

students in only one activity, but does not specify how those

80

students are divided between the two clubs. Statement (2) specifies that the two club sizes are equal, which implies that the number of students participating only in Science equals the number participating only in Debate. Combining these facts determines that exactly

40

students are in Science only,

40

in Debate only, and

40

in both, allowing the total score sum (

9600

) and overall mean (

80

) to be uniquely calculated.

Step-by-Step Solution

1

Set up the algebraic model for overlapping set counts and statistics sums.

Let

a

be the number of students in Science Club only,

b

be the number of students in Debate Team only, and

c

be the number of students in both.

a + b + c = 120

. Total sum of scores =

85(a+c) + 80(b+c) - \text{Sum}(S \cap D)

.

Scores of students in the intersection are counted in both club averages, so subtracting the overlap sum prevents double counting.

2

Evaluate Statement (1) independently.

Statement (1) gives

c = 40

and

\text{Sum}(S \cap D) = 40 \times 90 = 3600

. Then

a + b = 80

, and Total Sum =

85a + 80b + 3000 = 5a + 9400

. Since

a

can vary from

0

to

80

, Total Sum is not unique.

Statement (1) alone is insufficient because

a

remains a free variable.

3

Evaluate Statement (2) independently.

Statement (2) gives

a + c = b + c \implies a = b

. Without

c

or intersection scores, Total Sum cannot be computed.

Statement (2) alone is insufficient.

4

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

From (1),

a + b = 80

and

c = 40

. From (2),

a = b

. Thus

2a = 80 \implies a = 40

and

b = 40

. Substituting

a = 40

gives Total Sum =

5(40) + 9400 = 9600

. Overall mean =

9600 / 120 = 80

.

The combined system yields a single unique overall average score.

Key Concept

Weighted averages in overlapping sets using principle of inclusion-exclusion for statistical sums.

Estimated Time:2m 0s

Question 155Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

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 stem by squaring both sides gives

|a + b|^2 > |a - b|^2 \iff (a + b)^2 > (a - b)^2 \iff 4ab > 0 \iff ab > 0

. Statement (1) simplifies similarly by squaring both sides to

(a + 2b)^2 < (a - 2b)^2 \iff 8ab < 0 \iff ab < 0

. Since

ab < 0

guarantees that

ab > 0

is false, Statement (1) produces a definitive 'No' and is therefore sufficient. Statement (2) is satisfied when both

a

and

b

are negative (giving

ab > 0

, a 'Yes' answer) as well as when

a

and

b

have opposite signs with

|a| > |b|

(giving

ab < 0

, a 'No' answer). Because Statement (2) permits both outcomes, it is insufficient.

Step-by-Step Solution

1

Rephrase the question stem target using algebraic properties of absolute values.

Squaring both non-negative sides of

|a + b| > |a - b|

yields

(a + b)^2 > (a - b)^2 \implies a^2 + 2ab + b^2 > a^2 - 2ab + b^2 \implies 4ab > 0 \implies ab > 0

. The question asks: 'Is

ab > 0

?'

Simplifying the stem target into a fundamental sign constraint (

ab > 0

) prevents algebraic errors during statement evaluation.

2

Evaluate Statement (1):

|a + 2b| < |a - 2b|

.

Squaring both sides yields

(a + 2b)^2 < (a - 2b)^2 \implies a^2 + 4ab + 4b^2 < a^2 - 4ab + 4b^2 \implies 8ab < 0 \implies ab < 0

.

Since

ab < 0

,

ab

cannot be greater than

0

. This yields a definitive 'No' to the question 'Is

ab > 0

?'. A definitive 'No' is sufficient.

3

Evaluate Statement (2):

\frac{|a|}{b} + \frac{|b|}{a} < 0

.

Combine the fractions:

\frac{a|a| + b|b|}{ab} < 0

. Test specific cases:
- Case A:

a = -2, b = -2 \implies \frac{|-2|}{-2} + \frac{|-2|}{-2} = -2 < 0

. Here

ab = 4 > 0

(Answer: YES).
- Case B:

a = 3, b = -1 \implies \frac{|3|}{-1} + \frac{|-1|}{3} = -3 + \frac{1}{3} = -\frac{8}{3} < 0

. Here

ab = -3 < 0

(Answer: NO).

Since Statement (2) allows both a 'Yes' and a 'No' response to whether

ab > 0

, Statement (2) alone is not sufficient.

Key Concept

Data Sufficiency Yes/No decision logic combined with distance and squaring properties of absolute value inequalities.

Question 156Question →

At a technical conference, a total of $150$ software engineers attended at least one of two technical sessions: System Architecture or Distributed Systems. Exactly $90$ engineers attended the System Architecture session. If the mean years of experience for all $150$ engineers combined was $8$ years, what was the mean years of experience for the engineers who attended ONLY the Distributed Systems session?

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 answer is Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Rephrasing the question stem reveals that the total group of 150 engineers is divided into two disjoint subsets: the 90 engineers in the System Architecture session and the 60 engineers who attended ONLY the Distributed Systems session. Because the total experience of all 150 engineers is fixed at 1200 years (150 × 8), knowing the mean experience of the 90 System Architecture engineers in Statement (1) allows calculation of their total experience sum (90 × 7.5 = 675), which directly yields the remaining experience sum (1200 - 675 = 525) and mean (525 / 60 = 8.75) for the engineers in ONLY the Distributed Systems session. Statement (2) gives information about the overlap group but leaves the experience sum of the engineers in ONLY the System Architecture session unknown.

Step-by-Step Solution

1

Rephrase the question stem using set relationships and statistics formulas.

Let

S

be System Architecture attendees (

|S| = 90

) and

D

be Distributed Systems attendees. The total combined group is

S \cup D

with

|S \cup D| = 150

. The number of engineers attending ONLY Distributed Systems is

|D \setminus S| = |S \cup D| - |S| = 150 - 90 = 60

.

Since every attendee is in at least one session, the total group consists of all attendees in

S

plus those in

D \setminus S

, which are mutually disjoint sets.

2

Express the total sum of experience and set up the target equation.

Total combined experience sum

T_{\text{total}} = 150 \times 8 = 1200

. Since

S

and

D \setminus S

partition the entire population,

T_{\text{total}} = T_S + T_{D \setminus S}

, where

T_S

is the sum of experience of all

90

engineers in

S

. Therefore,

T_{D \setminus S} = 1200 - T_S

, and the target mean is

\frac{T_{D \setminus S}}{60} = \frac{1200 - T_S}{60}

.

Finding the mean experience for

D \setminus S

depends entirely on finding the total experience sum

T_S

of the

90

engineers in System Architecture.

3

Evaluate Statement (1): The mean years of experience for the engineers who attended the System Architecture session was

7.5

years.

T_S = 90 \times 7.5 = 675

. Then

T_{D \setminus S} = 1200 - 675 = 525

. Target mean

= \frac{525}{60} = 8.75

years. Statement (1) is SUFFICIENT.

Statement (1) directly gives the mean of set

S

, allowing exact computation of

T_S

and thus the target mean.

4

Evaluate Statement (2): Exactly

40

engineers attended BOTH sessions, and their mean years of experience was

9

years.

This gives

|S \cap D| = 40

and sum

T_{S \cap D} = 40 \times 9 = 360

. Set

S

is divided into

S \setminus D

(size

50

) and

S \cap D

(size

40

).

T_S = T_{S \setminus D} + 360

. Since

T_{S \setminus D}

remains unknown,

T_S

cannot be determined. Statement (2) is INSUFFICIENT.

Without the experience sum or mean of the engineers who attended ONLY System Architecture, we cannot determine

T_S

.

Key Concept

Partitioning combined sets in weighted averages and Data Sufficiency rephrasing

Question 157Question →

A logistics company analyzed the operational downtime of a fleet of $80$ delivery trucks over a one-month period. Each truck in the fleet completed at least one of two specialized maintenance programs: Program A or Program B. The arithmetic mean downtime for trucks that completed only Program A was $12$ hours, and the arithmetic mean downtime for trucks that completed only Program B was $18$ hours. What was the average downtime per truck, in hours, for all $80$ trucks in the fleet?

(1) Exactly $50$ trucks completed Program A, and exactly $45$ trucks completed Program B.
(2) The total downtime of all trucks in the fleet that completed Program B was $765$ 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: 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 response identifies that both statements together provide complementary pieces of required information: the first statement determines the number of trucks in the 'only A' group (

35

trucks), while the second statement provides the combined downtime of all trucks in the 'Program B' group (

765

hours). Together, they uniquely specify the fleet's total downtime as

12(35) + 765 = 1185

hours, giving a single average of

14.8125

hours.

Step-by-Step Solution

1

Define set variables and express the target fleet average in terms of known and unknown quantities.

Let

n_A

be the number of trucks completing only Program A,

n_B

be the number of trucks completing only Program B, and

n_{AB}

be the number of trucks completing both programs. The total number of trucks is

n_A + n_B + n_{AB} = 80

. The total downtime of the fleet is

T_{total} = (12 \cdot n_A) + T_B

, where

T_B

is the total downtime of all trucks that completed Program B (which comprises trucks completing only B and trucks completing both A and B).

Partitioning the fleet into trucks completing only Program A and trucks completing Program B allows us to rephrase the total downtime simply as

12 n_A + T_B

.

2

Evaluate Statement (1) independently.

Statement (1) states that

n_A + n_{AB} = 50

and

n_B + n_{AB} = 45

. Since

(n_A + n_{AB}) + (n_B + n_{AB}) - n_{AB} = 80

, we get

50 + 45 - n_{AB} = 80 \Rightarrow n_{AB} = 15

. This yields

n_A = 35

and

n_B = 30

. However, we do not know

T_B

or the downtime of the

15

trucks in both programs, so the fleet average cannot be calculated. Statement (1) alone is NOT sufficient.

Knowing set counts alone does not supply the necessary downtime data for the overlapping region.

3

Evaluate Statement (2) independently.

Statement (2) gives

T_B = 765

hours. Substituting this into our total downtime expression yields

T_{total} = 12 n_A + 765

. Since

n_A

(the number of trucks completing only Program A) is unknown,

T_{total}

cannot be uniquely calculated. Statement (2) alone is NOT sufficient.

Without knowing

n_A

, the contribution of trucks completing only Program A to total downtime cannot be determined.

4

Evaluate Statements (1) and (2) together.

From Statement (1),

n_A = 35

. From Statement (2),

T_B = 765

. Thus,

T_{total} = 12(35) + 765 = 420 + 765 = 1185

hours. The average downtime per truck is

\frac{1185}{80} = 14.8125

hours, which is a unique numerical value. Both statements together are SUFFICIENT.

Combining

n_A = 35

with

T_B = 765

uniquely determines the fleet's total downtime and overall average.

Key Concept

Overlapping set partitioning and weighted averages in Data Sufficiency

Question 158Question →

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

(1) $n + \frac{6}{n}$ is an integer.
(2) $n^2 + \frac{36}{n^2}$ is an integer.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

n

is an integer.

The correct answer identifies that neither statement alone is sufficient, and combining them offers no additional constraint because the second statement is algebraically derived by squaring the first. Since non-integer real numbers such as

n = 3 + \sqrt{3}

satisfy both statements, the statements together remain insufficient.

Step-by-Step Solution

1

Evaluate Statement (1) by setting

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

, where

k

is an integer.

Multiplying by

n

gives

n^2 - kn + 6 = 0

, which yields

n = \frac{k \pm \sqrt{k^2 - 24}}{2}

. For

n

to be a real number, we need

k^2 \ge 24

, so

k \ge 5

.

We must test whether all positive real solutions for

n

under this condition are integers.

2

Test specific integer values for

k

in Statement (1).

If

k = 5

, then

n = \frac{5 \pm 1}{2}

, giving

n = 3

or

n = 2

(both are integers, yielding a YES answer). However, if

k = 6

, then

n = \frac{6 \pm \sqrt{12}}{2} = 3 \pm \sqrt{3}

, which is a positive real number but NOT an integer (yielding a NO answer).

Since Statement (1) permits both integer and non-integer values for

n

, Statement (1) ALONE is NOT sufficient.

3

Evaluate Statement (2) independently and analyze its algebraic relationship to Statement (1).

If

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

, then squaring both sides gives

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

, which implies

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

. If

k

is an integer,

k^2 - 12

is automatically an integer.

This shows Statement (2) is logically and algebraically implied by Statement (1).

4

Evaluate Statements (1) and (2) together.

Because Statement (2) provides zero new information beyond Statement (1), combining them does not eliminate the non-integer solution

n = 3 + \sqrt{3}

. Thus, the question cannot be answered definitively.

Redundant statements cannot resolve insufficiency present in the individual statements.

Key Concept

Number Properties and Integer Constraints in Data Sufficiency

Estimated Time:2m 30s

Question 159Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The target question asks whether

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

. Factoring the numerator as

ab(a+b)

and the denominator as

ab(a+b)(a-b)

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

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

?', which is equivalent to 'Is

a > b

?'. Statement (1) directly states that

a > b

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

a+b > 0

, which gives no information regarding whether

a > b

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

Step-by-Step Solution

1

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

Numerator:

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

. Denominator:

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

. The expression becomes

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

.

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

2

Cancel non-zero common terms in the simplified fraction.

Since

a b \neq 0

and

a + b \neq 0

(given

a^2 \neq b^2

), common terms cancel out to yield

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

.

The sign of

\frac{1}{a - b}

is strictly positive if and only if

a - b > 0

, which is equivalent to

a > b

.

3

Evaluate Statement (1):

a > b

.

This directly yields

a - b > 0

, so

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

. This provides a definitive 'Yes' answer.

Statement (1) alone provides sufficient information.

4

Evaluate Statement (2):

a + b > 0

.

Knowing

a + b > 0

gives no information about whether

a > b

or

a < b

(e.g., if

a=5, b=2

,

a > b

; if

a=2, b=5

,

a < b

).

Statement (2) alone is not sufficient to determine if

a > b

.

Key Concept

Question Stem Simplification and Target Rephrasing

Estimated Time:1m 40s

Question 160Question →

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

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

Neither statement alone is sufficient because non-integer real numbers can produce integer outputs for

k^2 - k

or

k^3 - k

. However, when both statements are combined, expressing

k^3 - k

as

m(k + 1)

proves that

k

must be a rational number. For any rational number

\frac{a}{b}

in lowest terms,

\left(\frac{a}{b}\right)^2 - \frac{a}{b}

can only be an integer if the denominator

b = 1

, which proves definitively that

k

must be an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is NOT sufficient.

We are given that

k^2 - k = m

for some integer

m

. If

m = 1

, then

k^2 - k - 1 = 0

, which gives

k = \frac{1 + \sqrt{5}}{2}

. Here,

k

is not an integer, but

k^2 - k = 1

is an integer. Thus, Statement (1) alone does not determine whether

k

is an integer.

2

Evaluate Statement (2) independently.

Statement (2) is NOT sufficient.

We are given that

k^3 - k = n

for some integer

n

. If

n = 1

, then

k^3 - k - 1 = 0

. By the Intermediate Value Theorem, there exists a real number

k \in (1, 2)

satisfying this equation. Since

1 < k < 2

,

k

cannot be an integer, yet

k^3 - k = 1

is an integer. Thus, Statement (2) alone does not determine whether

k

is an integer.

3

Evaluate Statements (1) and (2) combined.

The combined statements are SUFFICIENT.

Let

k^2 - k = m \in \mathbb{Z}

and

k^3 - k = n \in \mathbb{Z}

. Note that

k^3 - k = k(k^2 - 1) = k(k^2 - k + k - 1) = k(m + k - 1) = mk + k^2 - k = mk + m = m(k + 1)

. Thus,

n = m(k + 1)

. If

m = 0

, then

k^2 - k = 0 \implies k = 0

or

k = 1

, both of which are integers. If

m \neq 0

, then

mk = n - m \implies k = \frac{n - m}{m}

. Since

m

and

n

are integers,

k

must be a rational number. Let

k = \frac{a}{b}

in reduced form with

\gcd(a, b) = 1

and

b \ge 1

. Substituting into

k^2 - k = m

yields

\frac{a^2 - ab}{b^2} = m \implies a(a - b) = m b^2

. This means

b^2

must divide

a(a - b)

. Since

\gcd(a, b^2) = 1

,

b^2

must divide

a - b

, which implies

b = 1

. Therefore,

k = a

, so

k

MUST be an integer.

Key Concept

Testing implicit real number constraints vs integer constraints using polynomial reduction and rational root properties.

Data Sufficiency

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