Data Sufficiency Practice Questions — GMAT — Page 9

Question 161Question →

In a financial firm, a total of $100$ portfolio audits were conducted last year. Each audit was classified as High-Risk, International, or both. Exactly $60$ audits were classified as High-Risk, and exactly $50$ audits were classified as International. What was the average (arithmetic mean) duration, in days, of the $100$ portfolio audits?

(1) The average duration of the High-Risk audits was $14$ days, and the average duration of the International audits was $18$ days.
(2) The average duration of the audits classified as both High-Risk and International was $20$ 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 answer states that both statements together are sufficient, but neither statement alone is sufficient. By set theory, the number of audits in both categories is

60 + 50 - 100 = 10

. The total sum of days across all

100

audits satisfies

S_{\text{total}} = S_{\text{High-Risk}} + S_{\text{International}} - S_{\text{Both}}

. Statement (1) gives

S_{\text{High-Risk}} = 60 \times 14 = 840

and

S_{\text{International}} = 50 \times 18 = 900

, yielding

S_{\text{total}} = 1740 - S_{\text{Both}}

, which is insufficient alone. Statement (2) gives

S_{\text{Both}} = 10 \times 20 = 200

, which is insufficient alone. Combining both gives

S_{\text{total}} = 1740 - 200 = 1540

days, allowing us to find a unique overall mean of

15.4

days.

Step-by-Step Solution

1

Determine set sizes using Principle of Inclusion-Exclusion.

Total

N = 100

.

N(H) = 60

,

N(I) = 50

. Since every audit is in

H

,

I

, or both,

N(H \cup I) = 100 = 60 + 50 - N(H \cap I) \implies N(H \cap I) = 10

.

Identify the count of audits in the overlapping set (both High-Risk and International).

2

Formulate total duration sum in terms of set sums.

S_{\text{total}} = S_H + S_I - S_{H \cap I}

, where

S_H

is sum of High-Risk durations,

S_I

is sum of International durations, and

S_{H \cap I}

is sum of durations in both categories.

Double counting principle applies to the sum of values across sets.

3

Evaluate Statement (1) alone.

S_H = 60 \times 14 = 840

days,

S_I = 50 \times 18 = 900

days.

S_{\text{total}} = 840 + 900 - S_{H \cap I} = 1740 - S_{H \cap I}

.

Since

S_{H \cap I}

is unknown,

S_{\text{total}}

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

4

Evaluate Statement (2) alone.

S_{H \cap I} = 10 \times 20 = 200

days.

Without

S_H

or

S_I

,

S_{\text{total}}

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

5

Evaluate Statements (1) and (2) together.

S_{\text{total}} = 1740 - 200 = 1540

days. Mean duration =

1540 / 100 = 15.4

days.

A unique value for the overall mean is obtained. Both statements together are sufficient.

Key Concept

Overlapping Sets and Weighted Averages (Sum Double Counting)

Estimated Time:2m 0s

Question 162Question →

A financial advisory firm manages a portfolio of $90$ assets, each of which is classified as a Tech asset, a Biotech asset, or both. Exactly $50$ of the assets are Tech assets, and exactly $60$ of the assets are Biotech assets. Is the average (arithmetic mean) value of the Tech-only assets greater than the average value of the Biotech-only assets?

(1) The average value of all $50$ Tech assets is $\$12$ million, and the average value of all $60$ Biotech assets is $\$12$ million.
(2) The average value of the $20$ assets classified as both Tech and Biotech assets is $\$15$ million.

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 allows us to solve for the exact average value of the Tech-only assets (

10

million) and the Biotech-only assets (

10.5

million). Because we can answer the stem's question with a definitive 'No' (

10

is not greater than

10.5

), both statements together are sufficient.

Step-by-Step Solution

1

Determine the number of assets in each non-overlapping subgroup using set theory.

Total assets

N = 90

, Tech assets

|T| = 50

, Biotech assets

|B| = 60

. Using

|T \cup B| = |T| + |B| - |T \cap B|

, we get

90 = 50 + 60 - |T \cap B| \implies |T \cap B| = 20

. Therefore, Tech-only assets

= 50 - 20 = 30

, Biotech-only assets

= 60 - 20 = 40

, and Both

= 20

.

Rephrasing the question stem into distinct, non-overlapping subset counts simplifies the algebraic system.

2

Evaluate Statement (1) independently.

Let

M_{T}

,

M_{B}

, and

M_{both}

represent the mean values of Tech-only, Biotech-only, and Both subsets respectively. From Statement (1), total value of Tech

= 30 M_T + 20 M_{both} = 50 \times 12 = 600

, and total value of Biotech

= 40 M_B + 20 M_{both} = 60 \times 12 = 720

. Subtracting the equations yields

40 M_B - 30 M_T = 120 \implies 4 M_B - 3 M_T = 12

.

Since

M_T

could be

16

(making

M_B = 15

, so

M_T > M_B

is Yes) or

M_T

could be

8

(making

M_B = 9

, so

M_T > M_B

is No), Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

Statement (2) gives

M_{both} = 15

. Without knowing the total asset values or subgroup means, we cannot compare

M_T

and

M_B

.

Statement (2) alone is NOT sufficient.

4

Evaluate Statements (1) and (2) together.

Substituting

M_{both} = 15

into Statement (1) equations:

30 M_T + 20(15) = 600 \implies 30 M_T = 300 \implies M_T = 10

. Similarly,

40 M_B + 20(15) = 720 \implies 40 M_B = 420 \implies M_B = 10.5

. Since

10 > 10.5

is definitively FALSE, we get a clear 'No' response.

In Yes/No Data Sufficiency questions, a definitive 'No' means the combined statements are sufficient.

Key Concept

Data Sufficiency decision logic for overlapping sets combined with weighted averages

Estimated Time:2m 0s

Question 163Question →

For all positive real numbers $a$ and $b$ , the Data Sufficiency Yes/No target question "Is $\frac{a^2 + 3ab}{b^2 + 3ab} > 1$ ?" is algebraically equivalent to asking "Is $a > b$ ?"

True

False

Show answer & explanation

Answer: True

Answer

True. The target question simplifies directly to "Is

a > b

?"

The statement is true because cross-multiplying the strictly positive denominator

b^2 + 3ab

and subtracting the shared term

3ab

reduces

\frac{a^2 + 3ab}{b^2 + 3ab} > 1

directly to

a^2 > b^2

. Since

a

and

b

are both positive,

a^2 > b^2

is logically equivalent to

a > b

.

Step-by-Step Solution

1

Determine the sign of the denominator.

Since

a > 0

and

b > 0

,

b^2 + 3ab > 0

.

Knowing the denominator is strictly positive ensures we can cross-multiply without reversing the inequality sign.

2

Cross-multiply the inequality by the denominator.

a^2 + 3ab > b^2 + 3ab

Multiplying both sides of

\frac{a^2 + 3ab}{b^2 + 3ab} > 1

by

(b^2 + 3ab)

clears the fraction.

3

Subtract

3ab

from both sides.

a^2 > b^2

Eliminating the common linear term isolates the quadratic terms.

4

Take the square root of both sides.

a > b

Because

a

and

b

are positive,

a^2 > b^2

holds if and only if

a > b

.

Key Concept

Data Sufficiency Question Stem Simplification

Question 164Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

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 determines the parity of

m

. However, combining Statement (1) (

m+n

is odd) and Statement (2) (

n=5

, an odd integer) proves that

m

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

Step-by-Step Solution

1

Analyze Statement (1) independently.

Statement (1) states that

m + n

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

n

is odd,

m

is even (Yes). If

n

is even,

m

is odd (No). Since the parity of

n

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

Determining sufficiency requires a unique Yes or No answer.

2

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

Statement (2) states that

n = 5

. This gives no information about

m

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

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

3

Combine Statement (1) and Statement (2).

From Statement (2),

n = 5

, which is an odd integer. Substituting

n = 5

into Statement (1),

m + 5

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

m

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

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

Key Concept

Statement Independence and Combination in Data Sufficiency

Question 165Question →

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

(1) $r^2 + 2r$ is an integer.
(2) $r^2 - 2r$ 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 irrational numbers like

\sqrt{2}-1

and

\sqrt{2}+1

can satisfy statements (1) and (2) respectively. However, combining both statements proves that

4r

is an integer

k

, and substituting

r = k/4

into the quadratic equation shows that

k

must be a multiple of 4, forcing

r

to be a positive integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is INSUFFICIENT.

If

r = 1

, then

r^2 + 2r = 3

(an integer), and

r

is an integer (YES). If

r = \sqrt{2} - 1

, then

r^2 + 2r = (3 - 2\sqrt{2}) + (2\sqrt{2} - 2) = 1

(an integer), but

r

is NOT an integer (NO). Since

r

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

2

Evaluate Statement (2) independently.

Statement (2) is INSUFFICIENT.

If

r = 3

, then

r^2 - 2r = 3

(an integer), and

r

is an integer (YES). If

r = \sqrt{2} + 1

, then

r^2 - 2r = (3 + 2\sqrt{2}) - (2\sqrt{2} + 2) = 1

(an integer), but

r

is NOT an integer (NO). Since

r

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

3

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

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

Let

r^2 + 2r = a

and

r^2 - 2r = b

, where

a

and

b

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

(r^2 + 2r) - (r^2 - 2r) = a - b \implies 4r = a - b

. Since

a

and

b

are integers,

a - b = k

for some integer

k

. Thus,

r = \frac{k}{4}

. Now substitute

r = \frac{k}{4}

back into Statement (1):

\left(\frac{k}{4}\right)^2 + 2\left(\frac{k}{4}\right) = a \implies \frac{k^2}{16} + \frac{k}{2} = a \implies \frac{k^2 + 8k}{16} = a

. This means

16

must divide

k^2 + 8k = k(k+8)

. If

k

were odd,

k(k+8)

would be odd and thus not divisible by 16. So

k

must be even. Let

k = 2m

. Then

k(k+8) = 2m(2m+8) = 4m(m+4)

. For

16

to divide

4m(m+4)

,

4

must divide

m(m+4)

. If

m

were odd,

m(m+4)

would be odd, which is not divisible by 4. Thus,

m

must be even. Let

m = 2p

. Then

k = 2m = 4p

, which means

k

is a multiple of 4. Therefore,

r = \frac{k}{4} = \frac{4p}{4} = p

, an integer. Thus,

r

MUST be an integer.

Key Concept

Testing integer constraints in Data Sufficiency requires evaluating whether non-integer real numbers satisfy given equations, and combining equations to determine if rational and integer bounds strictly hold.

Question 166Question →

If $a$ and $b$ are positive real numbers such that $ab$ is an integer, is $a$ an integer?

(1) $a + b$ is an integer.
(2) $a - b$ 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.

Combining both statements shows that

2a

and

2b

are integers. This restricts

a

to being either a whole integer or a half-integer. Evaluating the product of two half-integers reveals that their product has a denominator of 4 and can never be an integer. Since the question stem specifies that

ab

is an integer,

a

must be an integer.

Step-by-Step Solution

1

Analyze the question stem and given constraints

a

and

b

are positive real numbers and

ab = N

, where

N

is an integer. We must determine whether

a

is guaranteed to be an integer.

The variables are defined as real numbers, so we cannot assume they are integers unless forced by the mathematical statements.

2

Evaluate Statement (1):

a + b

is an integer

Statement (1) is NOT sufficient.

Let

a = 2 + \sqrt{3}

and

b = 2 - \sqrt{3}

. Both are positive real numbers. Here

a + b = 4

(an integer) and

ab = (2+\sqrt{3})(2-\sqrt{3}) = 1

(an integer). However,

a = 2 + \sqrt{3}

is not an integer. Conversely, if

a = 3

and

b = 1

, then

a + b = 4

and

ab = 3

, and

a

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

a

is an integer.

3

Evaluate Statement (2):

a - b

is an integer

Statement (2) is NOT sufficient.

Let

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

and

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

. Both are positive real numbers. Here

a - b = 1

(an integer) and

ab = \frac{(\sqrt{5}+1)(\sqrt{5}-1)}{4} = 1

(an integer). However,

a

is not an integer. Conversely, if

a = 4

and

b = 1

, then

a - b = 3

and

ab = 4

, and

a

is an integer. Thus, Statement (2) alone is not sufficient.

4

Evaluate Statement (1) and Statement (2) combined

Both statements together are SUFFICIENT.

Let

a + b = k

and

a - b = m

, where

k

and

m

are integers. Adding the two equations gives

2a = k + m \implies a = \frac{k+m}{2}

. Subtracting gives

2b = k - m \implies b = \frac{k-m}{2}

. Since

k

and

m

are integers,

k+m

and

k-m

are integers with the same parity. If

k+m

is even, both

a

and

b

are integers. If

k+m

is odd,

a = \frac{p}{2}

and

b = \frac{q}{2}

for odd integers

p

and

q

. But then

ab = \frac{pq}{4}

, which cannot be an integer because the product of two odd integers is odd and thus not divisible by 4. This contradicts the stem constraint that

ab

is an integer. Therefore,

k+m

must be even, forcing

a

to be an integer.

Key Concept

Real number vs. integer constraints and parity properties in quadratic system equations

Question 167Question →

If $k$ is a positive integer, is $\frac{k^3 - 4k}{k + 2}$ an odd integer?

(1) $k^2 + 3k$ is an even integer.
(2) $3k + 1$ 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 (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Answer

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

Rephrasing the target question

\frac{k^3 - 4k}{k + 2}

by factoring the numerator gives

\frac{k(k-2)(k+2)}{k+2} = k(k-2)

. Because

k

and

k-2

share the same parity, their product

k(k-2)

is odd if and only if

k

is odd. Thus, the question simplifies to: 'Is

k

an odd integer?' Statement (1) tells us

k^2 + 3k = k(k+3)

is even, which is true for all integers

k

, so it provides no information about whether

k

is odd or even (NOT sufficient). Statement (2) tells us

3k + 1

is odd, which implies

3k

is even, and therefore

k

must be even. Since

k

is definitively even, the answer to 'Is

k

an odd integer?' is a definitive 'No'. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

1

Rephrase and simplify the question stem algebraically.

Since

k

is a positive integer,

k + 2 > 0

. Factoring the numerator gives

k^3 - 4k = k(k^2 - 4) = k(k - 2)(k + 2)

. Dividing by

k + 2

yields

\frac{k(k - 2)(k + 2)}{k + 2} = k(k - 2)

.

Simplifying the target expression reduces cognitive complexity before evaluating statements.

2

Determine the parity condition for

k(k - 2)

to be an odd integer.

The factors

k

and

k - 2

differ by 2, so they have the same parity. If

k

is even,

k(k - 2)

is (even)(even) = even. If

k

is odd,

k(k - 2)

is (odd)(odd) = odd. Thus, the target question 'Is

k(k - 2)

an odd integer?' simplifies to 'Is

k

an odd integer?'

Rephrasing the target question to 'Is

k

odd?' establishes a clear, minimal sufficiency criterion.

3

Evaluate Statement (1):

k^2 + 3k

is an even integer.

Factor

k^2 + 3k

as

k(k + 3)

. The terms

k

and

k + 3

differ by 3, meaning one is always even and the other is odd. Their product is ALWAYS even regardless of whether

k

is odd or even. Therefore, Statement (1) provides no information about whether

k

is odd. NOT sufficient.

A statement that holds true for all integers provides no distinguishing information about

k

.

4

Evaluate Statement (2):

3k + 1

is an odd integer.

If

3k + 1

is odd, then

3k

must be even. Since 3 is odd,

3k

is even if and only if

k

is even. Hence,

k

is definitely even. This gives a definitive 'No' to the rephrased question 'Is

k

an odd integer?'. A definitive 'No' means the statement IS sufficient.

In Data Sufficiency Yes/No questions, a statement that yields a conclusive 'No' is sufficient.

Key Concept

Data Sufficiency Question Stem Simplification and Parity Analysis

Estimated Time:1m 30s

Question 168Question →

If $p$ and $q$ are real numbers, what is the value of $p + q$ ?

(1) $p^2 q + p q^2 = 36$ and $p^2 + 2pq + q^2 = 9$
(2) $p^2 - q^2 = 15$

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 simplifying

(p+q)^2 = 9

yields two potential values for the sum:

p+q = 3

or

p+q = -3

. Combining each sum with

pq(p+q) = 36

gives corresponding products

pq = 12

and

pq = -12

. For real numbers

p

and

q

, the discriminant

S^2 - 4P

must be non-negative. The sum

p+q = 3

gives a negative discriminant

3^2 - 4(12) = -39

, which yields no real solutions for

p

and

q

. The sum

p+q = -3

gives a positive discriminant

(-3)^2 - 4(-12) = 57

, which yields valid real numbers. Thus,

p+q

uniquely equals

-3

.

Statement (2) alone is not sufficient because

p^2 - q^2 = (p-q)(p+q) = 15

allows infinitely many different values for

p+q

depending on

p-q

. Hence, the correct statement selection is that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase and analyze Statement (1) algebraically.

From

p^2 + 2pq + q^2 = 9

, we have

(p + q)^2 = 9

, which implies

p + q = 3

or

p + q = -3

. Factor the second equation:

p^2 q + p q^2 = pq(p + q) = 36

.

Factoring exposes the sum

S = p + q

and product

P = pq

structure of the algebraic system.

2

Evaluate the real number constraint for Case 1 (

p + q = 3

) in Statement (1).

If

p + q = 3

, then

pq(3) = 36 \implies pq = 12

. For real numbers

p

and

q

with sum

3

and product

12

, they must be real roots of

t^2 - 3t + 12 = 0

. The discriminant is

\Delta = (-3)^2 - 4(1)(12) = 9 - 48 = -39 < 0

.

A negative discriminant means no real numbers

p

and

q

exist for which

p + q = 3

and

pq = 12

. Thus,

p + q = 3

is impossible.

3

Evaluate the real number constraint for Case 2 (

p + q = -3

) in Statement (1).

If

p + q = -3

, then

pq(-3) = 36 \implies pq = -12

. The corresponding quadratic equation for

p

and

q

is

t^2 + 3t - 12 = 0

, with discriminant

\Delta = 3^2 - 4(1)(-12) = 57 > 0

. Real solutions exist.

Since a positive discriminant guarantees real solutions for

p

and

q

,

p + q

must uniquely equal

-3

. Therefore, Statement (1) alone is sufficient.

4

Evaluate Statement (2) independently.

p^2 - q^2 = 15 \implies (p - q)(p + q) = 15

. For example, if

p = 4, q = 1

, then

p + q = 5

. If

p = 8, q = 7

, then

p + q = 15

.

Without knowing

p - q

, infinitely many pairs of real numbers satisfy

p^2 - q^2 = 15

with different sums

p + q

. Statement (2) alone is not sufficient.

Key Concept

Evaluating non-linear algebraic systems in Data Sufficiency using real variable discriminant constraints

Question 169Question →

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

(1) $m - n$ is an integer.
(2) $m^2 - n^2$ 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.

Evaluating Statement (1) alone: If

m = \sqrt{2}

and

n = \sqrt{2}

, then

m - n = 0

(an integer), but

m + n = 2\sqrt{2}

is not an integer. If

m = 2

and

n = 1

, then

m - n = 1

(an integer) and

m + n = 3

is an integer. Thus, Statement (1) alone is not sufficient.

Evaluating Statement (2) alone: If

m = \sqrt{2}

and

n = 1

, then

m^2 - n^2 = 1

(an integer), but

m + n = \sqrt{2} + 1

is not an integer. If

m = 2

and

n = 1

, then

m^2 - n^2 = 3

(an integer) and

m + n = 3

is an integer. Thus, Statement (2) alone is not sufficient.

Evaluating Statements (1) and (2) together: Factoring gives

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

. If

m - n = k \neq 0

and

m^2 - n^2 = p

for integers

k

and

p

, then

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

. This establishes that

m + n

must be rational, but a rational number is not necessarily an integer. For example, if

m = \frac{5}{4}

and

n = -\frac{3}{4}

, then

m - n = 2

(an integer) and

m^2 - n^2 = 1

(an integer), but

m + n = \frac{1}{2}

is not an integer. If

m = 2

and

n = 0

, both statements hold and

m + n = 2

is an integer. Since

m + n

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

Step-by-Step Solution

1

Analyze the question stem and identify variable constraints.

The variables

m

and

n

are defined as real numbers, NOT necessarily integers.

Failing to account for non-integer real values is a primary trap in GMAT Data Sufficiency number property questions.

2

Evaluate Statement (1) independently.

Statement (1) states

m - n = k

for some integer

k

.
If

m = 2

and

n = 1

,

m - n = 1

(integer) and

m + n = 3

(integer).
If

m = \sqrt{2}

and

n = \sqrt{2}

,

m - n = 0

(integer) and

m + n = 2\sqrt{2}

(not an integer).
Since

m + n

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

Testing both integer and irrational values reveals that a difference being an integer does not constrain the sum to be an integer.

3

Evaluate Statement (2) independently.

Statement (2) states

m^2 - n^2 = p

for some integer

p

.
If

m = 2

and

n = 1

,

m^2 - n^2 = 3

(integer) and

m + n = 3

(integer).
If

m = \sqrt{2}

and

n = 1

,

m^2 - n^2 = 1

(integer) and

m + n = \sqrt{2} + 1

(not an integer).
Since

m + n

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

Testing non-integer values shows that the difference of squares being an integer does not guarantee that the sum is an integer.

4

Evaluate Statements (1) and (2) together.

We have

m - n = k

and

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

for integers

k

and

p

.
If

k = 0

,

m = n

, so

m - n = 0

and

m^2 - n^2 = 0

. In this case

m + n = 2m

, which can be non-integer if

m = \sqrt{2}

.
If

k \neq 0

, then

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

. This proves that

m + n

is a rational number, but not necessarily an integer.
For instance, let

m = \frac{5}{4}

and

n = -\frac{3}{4}

:
-

m - n = \frac{5}{4} - \left(-\frac{3}{4}\right) = 2

(integer)
-

m^2 - n^2 = \frac{25}{16} - \frac{9}{16} = 1

(integer)
-

m + n = \frac{5}{4} + \left(-\frac{3}{4}\right) = \frac{1}{2}

(not an integer).
Conversely, if

m = 2

and

n = 0

,

m - n = 2

(integer),

m^2 - n^2 = 4

(integer), and

m + n = 2

(integer).
Because

m + n

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

Algebraic division demonstrates that the sum is guaranteed to be rational, but rational numbers include non-integer fractions.

Key Concept

Real Number Constraints vs. Integer Constraints in Data Sufficiency

Estimated Time:2m 0s

Question 170Question →

In a commercial survey of $120$ retail stores, each store sells products from Brand P, Brand Q, or both. The average (arithmetic mean) monthly revenue of the stores that sell Brand P is $\$40,000$ , and the average monthly revenue of the stores that sell Brand Q is $\$50,000$ . What is the average monthly revenue of all $120$ stores combined?

(1) Exactly $30$ stores sell both Brand P and Brand Q, and their average monthly revenue is $\$60,000$ .
(2) Exactly $40$ stores sell only Brand P.

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 neither statement alone provides enough information to determine both the exact store counts in each category and the overlap revenue sum, but combining both statements gives a unique system of linear equations that yields an exact combined average revenue of

\frac{\$5,000,000}{120} \approx \$41,666.67

.

Step-by-Step Solution

1

Formulate the total revenue expression in terms of overlapping sets and subgroup means.

Let

n(P)

be the number of stores selling Brand P and

n(Q)

be the number of stores selling Brand Q. Total stores

N = n(P \text{ only}) + n(Q \text{ only}) + n(P \cap Q) = 120

. Total revenue

S_{total} = S(P) + S(Q) - S(P \cap Q) = 40,000 n(P) + 50,000 n(Q) - S(P \cap Q)

.

Stores selling both brands are counted twice when summing

S(P)

and

S(Q)

, so their revenue sum

S(P \cap Q)

must be subtracted once.

2

Evaluate Statement (1) alone.

Statement (1) gives

n(P \cap Q) = 30

and average overlap revenue

= \$60,000 \implies S(P \cap Q) = 30 \times 60,000 = 1,800,000

. Also

n(P) + n(Q) = 120 + 30 = 150

.

Since

n(P)

and

n(Q)

can still vary while their sum is

150

(e.g.,

n(P)=70, n(Q)=80

vs

n(P)=90, n(Q)=60

),

S_{total} = 40,000 n(P) + 50,000 n(Q) - 1,800,000

yields multiple possible total values. Statement (1) alone is INSUFFICIENT.

3

Evaluate Statement (2) alone.

Statement (2) gives

n(P \text{ only}) = 40

. Thus

n(Q) = 120 - 40 = 80

.

No revenue information or overlap count is given. Statement (2) alone is INSUFFICIENT.

4

Evaluate Statements (1) and (2) together.

From Statement (2),

n(P \text{ only}) = 40

. From Statement (1),

n(P \cap Q) = 30

. Thus

n(P) = 40 + 30 = 70

. Since

n(P) + n(Q) = 150

,

n(Q) = 80

.

We now have exact values:

n(P) = 70

,

n(Q) = 80

,

S(P \cap Q) = 1,800,000

.

S_{total} = 40,000(70) + 50,000(80) - 1,800,000 = 2,800,000 + 4,000,000 - 1,800,000 = 5,000,000

. Combined mean revenue

= \frac{5,000,000}{120} = \$41,666.67

. Both statements together are SUFFICIENT.

Key Concept

Weighted averages in overlapping sets requiring exact subgroup counts and sum reconciliations.

Question 171Question →

In a laboratory, $80$ experimental trials were conducted. Each trial was monitored by Sensor P, Sensor Q, or both. The average (arithmetic mean) duration of the trials monitored by Sensor P only was $45$ minutes, and the average duration of the trials monitored by Sensor Q only was $60$ minutes. What was the average duration of all $80$ trials?

(1) Exactly $20$ trials were monitored by Sensor P only, and $30$ trials were monitored by Sensor Q only.
(2) The average duration of all trials monitored by Sensor Q was $55$ minutes, and the average duration of the trials monitored by both sensors was $50$ minutes.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Evaluating Statement (1) gives the counts for all three disjoint regions (

20

P-only,

30

Q-only, and

30

Both), but does not provide the mean duration of the

30

trials in the 'Both' category. Evaluating Statement (2) gives the mean duration of the 'Both' category (

50

minutes) and establishes that

n_{B} = n_{Q}

, but does not give absolute counts. Combining both statements provides all subgroup sizes and their respective means, allowing us to calculate the exact weighted average of

52.5

minutes.

Step-by-Step Solution

1

Formulate the algebraic expressions for total trials and total duration.

Let

n_{P}

,

n_{Q}

, and

n_{B}

be the counts for Sensor P only, Sensor Q only, and Both sensors respectively.

n_{P} + n_{Q} + n_{B} = 80

. The total duration is

45 n_{P} + 60 n_{Q} + M_{B} n_{B}

, where

M_{B}

is the mean duration of the trials monitored by both sensors.

Deconstruct the overlapping sets into three mutually exclusive subgroups: P only, Q only, and Both.

2

Evaluate Statement (1) alone.

n_{P} = 20

and

n_{Q} = 30 \implies n_{B} = 80 - 20 - 30 = 30

. Total duration

= 45(20) + 60(30) + M_{B}(30) = 2700 + 30 M_{B}

.

Since the mean duration

M_{B}

for trials monitored by both sensors is unknown, Statement (1) is insufficient.

3

Evaluate Statement (2) alone.

The total group Q consists of Q only (

n_{Q}

, mean

60

) and Both (

n_{B}

, mean

50

). Average for Q is

55 \implies \frac{60 n_{Q} + 50 n_{B}}{n_{Q} + n_{B}} = 55 \implies 5 n_{B} = 5 n_{Q} \implies n_{B} = n_{Q}

.

Statement (2) proves

n_{B} = n_{Q}

, which gives

n_{P} + 2 n_{Q} = 80

, but without specific numbers for

n_{Q}

or

n_{P}

, the overall average cannot be uniquely determined. Thus Statement (2) is insufficient.

4

Evaluate Statements (1) and (2) together.

From (1),

n_{P} = 20

and

n_{Q} = 30

, so

n_{B} = 30

. From (2),

M_{B} = 50

. Total duration

= 45(20) + 60(30) + 50(30) = 900 + 1800 + 1500 = 4200

. Overall average

= \frac{4200}{80} = 52.5

minutes.

Combining both statements yields a unique numeric value for the overall average duration.

Key Concept

Weighted Averages and Overlapping Sets Partitioning

Question 172Question →

If $m$ and $n$ are positive integers, is $m^2 - n^2$ a prime number?

(1) $m - n = 1$
(2) $m + n = 7$

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.

Rephrasing the question stem shows

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

. Statement (1) alone allows

2n + 1

to be prime (for

n=1

) or composite (for

n=4

), so it is insufficient. Statement (2) alone allows pairs like

(4,3)

which gives

7

(prime) and

(5,2)

which gives

21

(composite), so it is insufficient. Combining both statements yields

m = 4

and

n = 3

, giving

m^2 - n^2 = 7

, which definitively answers 'Yes'.

Step-by-Step Solution

1

Rephrase the question stem using algebraic factorization.

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

. For

m^2 - n^2

to be a prime number, since

m

and

n

are positive integers, the smaller factor

(m - n)

must equal 1 and the larger factor

(m + n)

must be a prime number.

Factoring highlights the conditions required for a product of integers to yield a prime number.

2

Evaluate Statement (1):

m - n = 1

.

Substitute

m = n + 1

into

m^2 - n^2

:

(1)(2n + 1) = 2n + 1

. If

n = 1

, then

2(1) + 1 = 3

(prime, Yes). If

n = 4

, then

2(4) + 1 = 9

(composite, No).

Since Statement (1) produces both Yes and No outcomes, it is NOT sufficient.

3

Evaluate Statement (2):

m + n = 7

.

Since

m

and

n

are positive integers, test pairs with

m > n

: If

(m, n) = (4, 3)

, then

m^2 - n^2 = 16 - 9 = 7

(prime, Yes). If

(m, n) = (5, 2)

, then

m^2 - n^2 = 25 - 4 = 21

(composite, No).

Since Statement (2) produces both Yes and No outcomes, it is NOT sufficient.

4

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

System of equations:

m - n = 1

and

m + n = 7

. Adding the equations gives

2m = 8 \implies m = 4

, which leads to

n = 3

. Then

m^2 - n^2 = 4^2 - 3^2 = 7

, which is prime (Definitive Yes).

Combining the statements yields a single unique pair of values and a definitive Yes answer.

Key Concept

Difference of squares factorization and prime number properties in Data Sufficiency

Estimated Time:2m 0s

Question 173Question →

If $p$ and $q$ are positive integers, is $p^2 + 3q$ divisible by 6?

(1) $p + q$ is an odd integer.
(2) $p^2 - q^2$ is divisible by 3.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The question requires determining whether

p^2 + 3q

is divisible by 6, which demands divisibility by both 2 and 3. By parity rules,

p^2 \equiv p \pmod 2

and

3q \equiv q \pmod 2

, meaning

p^2 + 3q

has the same parity as

p + q

. Statement (1) specifies that

p + q

is odd, so

p^2 + 3q

is odd and cannot be divisible by 6. This gives a definitive 'No' answer, making Statement (1) alone sufficient. Statement (2) permits cases where

p^2 + 3q

is divisible by 6 (e.g.,

p=3, q=3

) and cases where it is not (e.g.,

p=4, q=1

), making Statement (2) alone insufficient.

Step-by-Step Solution

1

Rephrase the question target using number properties and parity rules.

For

p^2 + 3q

to be divisible by 6, it must be divisible by both 2 (even) and 3. For parity,

p^2 \equiv p \pmod 2

and

3q \equiv q \pmod 2

, so

p^2 + 3q \equiv p + q \pmod 2

. Thus,

p^2 + 3q

is even if and only if

p + q

is even.

Simplifying the parity constraint simplifies evaluating statements regarding divisibility by 2.

2

Evaluate Statement (1):

p + q

is an odd integer.

Since

p + q

is odd,

p^2 + 3q

must be odd. An odd integer cannot be divisible by 2, so

p^2 + 3q

CANNOT be divisible by 6. This yields a definitive 'No'.

In Data Sufficiency Yes/No questions, a definitive 'No' answer means the statement is sufficient.

3

Evaluate Statement (2):

p^2 - q^2

is divisible by 3.

Test cases:
- If

p = 3, q = 3

:

p^2 - q^2 = 0

(divisible by 3), and

p^2 + 3q = 18

(divisible by 6 -> YES).
- If

p = 4, q = 1

:

p^2 - q^2 = 15

(divisible by 3), and

p^2 + 3q = 19

(not divisible by 6 -> NO).

Since Statement (2) yields both 'Yes' and 'No' outcomes, it is insufficient.

Key Concept

Parity rephrasing and Yes/No Data Sufficiency decision logic

Question 174Question →

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

(1) $a^4 - b^4 = 5(a^2 - b^2)$
(2) $ab = 6$ and $a + b = 5$

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

Answer

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

Statement (1) simplifies to

(a^2 - b^2)(a^2 + b^2 - 5) = 0

, which allows

a^2 + b^2 = 5

when

a \neq b

, but allows infinitely many values for

a^2 + b^2

when

a = b

. Hence, Statement (1) alone is insufficient. Statement (2) provides

a + b = 5

and

ab = 6

, allowing us to use

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

to find

a^2 + b^2 = 25 - 12 = 13

uniquely. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

1

Analyze Statement (1) algebraically.

Factor the left side as

(a^2 - b^2)(a^2 + b^2) = 5(a^2 - b^2)

. Rearranging yields

(a^2 - b^2)(a^2 + b^2 - 5) = 0

.

Dividing by

(a^2 - b^2)

is valid only if

a \neq b

.

2

Evaluate sufficiency of Statement (1).

If

a \neq b

, then

a^2 + b^2 = 5

. However, if

a = b > 0

, then

a^2 - b^2 = 0

, which makes the equation true for any positive value of

a

, meaning

a^2 + b^2 = 2a^2

can take infinitely many values (e.g., if

a=b=1

,

a^2+b^2=2

; if

a=b=3

,

a^2+b^2=18

). Thus, a unique value cannot be determined.

Multiple possible values mean Statement (1) is not sufficient.

3

Evaluate sufficiency of Statement (2).

Use the algebraic identity

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

. Substituting

a + b = 5

and

ab = 6

gives

5^2 = a^2 + 2(6) + b^2

, so

25 = a^2 + b^2 + 12 \implies a^2 + b^2 = 13

.

This determines a single, unique numerical value for

a^2 + b^2

.

4

Conclude overall sufficiency.

Statement (1) is NOT sufficient, but Statement (2) IS sufficient.

Statement (2) alone resolves the target question.

Key Concept

Algebraic Equations and Systems in Data Sufficiency

Question 175Question →

If $n$ is a positive integer, is $n$ a multiple of 18?

(1) $n^2$ is divisible by 108.
(2) $n^3$ is divisible by 243.

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 prime factor analysis of

n^2

being divisible by

108 = 2^2 \times 3^3

requires the exponent of 3 in

n^2

to be at least 4 (since

n

is an integer and exponents in

n^2

must be even). This guarantees

n

is divisible by

2^1 \times 3^2 = 18

. Statement (2) alone is not sufficient because

n^3

being divisible by

243 = 3^5

only guarantees

n

is a multiple of 9, allowing

n = 9

(not a multiple of 18) or

n = 18

(a multiple of 18).

Step-by-Step Solution

1

Rephrase the target question using prime factorizations.

The target asks if

n

is a multiple of

18 = 2^1 \times 3^2

. For

n

to be a multiple of 18, its prime factorization must contain at least one factor of 2 and at least two factors of 3.

Decomposing into prime factors simplifies the condition required for sufficiency.

2

Evaluate Statement (1):

n^2

is divisible by

108 = 2^2 \times 3^3

.

Since

n

is a positive integer, any prime factor in the prime factorization of

n^2

must have an even exponent. For

n^2

to be divisible by

2^2 \times 3^3

, the power of 2 in

n^2

must be at least 2, so

n

has at least one factor of 2. The power of 3 in

n^2

must be an even integer greater than or equal to 3, meaning it must be at least 4. Thus,

n

must contain at least two factors of 3 (

3^2 = 9

). Consequently,

n

must be divisible by

2^1 \times 3^2 = 18

. Statement (1) alone yields a definitive 'Yes'.

An integer's square must have even exponents for all its prime factors.

3

Evaluate Statement (2):

n^3

is divisible by

243 = 3^5

.

For

n^3

to be divisible by

3^5

, the exponent of 3 in

n^3

must be a multiple of 3 greater than or equal to 5, which means it must be at least 6. Therefore,

n

must contain at least two factors of 3 (

3^2 = 9

). However, statement (2) places no restriction on whether

n

is even. If

n = 9

,

n^3 = 729 = 243 \times 3

(divisible by 243), but 9 is NOT a multiple of 18. If

n = 18

,

n^3 = 5832 = 243 \times 24

(divisible by 243), and 18 IS a multiple of 18. Since

n

can yield both 'No' and 'Yes', Statement (2) alone is insufficient.

Testing specific values demonstrates that

n

may or may not be even.

Key Concept

Prime Factor Exponents in Integer Powers and Data Sufficiency Sufficiency Logic

Estimated Time:2m 0s

Question 176Question →

A summer academic camp has a total of $120$ enrolled students. Every student participates in at least one of two workshops: Data Analysis or Public Speaking. Exactly $70$ students participate in Data Analysis and exactly $80$ students participate in Public Speaking. What is the average (arithmetic mean) test score of the students who participate ONLY in Public Speaking?

Statement (1): The average test score of all $120$ students in the camp is $84$ .
Statement (2): The average test score of the students who participate in BOTH workshops is $90$ , and the average test score of the students who participate ONLY in Data Analysis is $78$ .

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 uniquely determine the average score of the students participating only in Public Speaking, but neither statement alone is sufficient.

The correct response identifies that both statements together provide enough information to solve for the target average, whereas neither statement alone is sufficient. By deconstructing the overlapping sets into three distinct groups (Data Analysis only = 40, Both = 30, Public Speaking only = 50), the weighted average equation connects the overall average to the three subgroup averages. Statement (1) supplies only the overall average, leaving two unknown subgroup averages. Statement (2) supplies two subgroup averages, leaving the overall average unknown. Combining both statements yields a single equation with only one unknown (

A_{P\text{-only}} = 85.2

), producing a unique numerical solution.

Step-by-Step Solution

1

Rephrase the question stem by calculating the exact number of students in each mutually exclusive subset.

Using the overlapping sets formula

N(D \cup P) = N(D) + N(P) - N(D \cap P)

, we get

120 = 70 + 80 - N(D \cap P)

, so

N(D \cap P) = 30

students in both workshops. Therefore, students in Data Analysis only =

70 - 30 = 40

, students in Public Speaking only =

80 - 30 = 50

, and students in both =

30

.

Deconstructing the overlapping sets into three distinct, non-overlapping groups (Data Analysis only, Both, Public Speaking only) is essential for formulating a weighted average equation.

2

Formulate the weighted average equation relating all subset averages to the total average.

Total Score Sum =

120 \cdot A_{total} = 40 \cdot A_{D\text{-only}} + 30 \cdot A_{both} + 50 \cdot A_{P\text{-only}}

, where

A_{P\text{-only}}

is the target value.

The total sum of all scores is the weighted sum of scores from the three mutually exclusive subsets.

3

Evaluate Statement (1) independently.

Statement (1) gives

A_{total} = 84

. Substituting gives

120(84) = 10080 = 40 \cdot A_{D\text{-only}} + 30 \cdot A_{both} + 50 \cdot A_{P\text{-only}}

. Since

A_{D\text{-only}}

and

A_{both}

remain unknown,

A_{P\text{-only}}

cannot be uniquely solved.

One linear equation with three unknown variables does not yield a unique solution. Statement (1) alone is NOT sufficient.

4

Evaluate Statement (2) independently.

Statement (2) gives

A_{both} = 90

and

A_{D\text{-only}} = 78

. Substituting gives

120 \cdot A_{total} = 40(78) + 30(90) + 50 \cdot A_{P\text{-only}} = 5820 + 50 \cdot A_{P\text{-only}}

. Since

A_{total}

is unknown,

A_{P\text{-only}}

cannot be uniquely solved.

One linear equation with two unknown variables does not yield a unique solution. Statement (2) alone is NOT sufficient.

5

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

Combining both statements provides

A_{total} = 84

,

A_{both} = 90

, and

A_{D\text{-only}} = 78

. The equation becomes

120(84) = 40(78) + 30(90) + 50 \cdot A_{P\text{-only}}

, which simplifies to

10080 = 5820 + 50 \cdot A_{P\text{-only}} \implies 4260 = 50 \cdot A_{P\text{-only}} \implies A_{P\text{-only}} = 85.2

.

We have one linear equation with only one unknown variable, giving a single unique numerical value. Both statements together are SUFFICIENT.

Key Concept

Combining overlapping set cardinalities with group weighted averages in Data Sufficiency

Estimated Time:2m 0s

Question 177Question →

If $n$ is a positive integer, is $n^2 + 3n + 2$ divisible by 12?

(1) $n$ is a multiple of 3.
(2) $n + 1$ is a prime number.

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

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

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

EACH statement ALONE is sufficient.

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

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.

Factoring the stem expression gives

n^2 + 3n + 2 = (n + 1)(n + 2)

. Statement (1) tells us

n

is a multiple of 3, which implies neither

n + 1

nor

n + 2

is a multiple of 3. Consequently, their product can never be divisible by 3, and thus can never be divisible by 12. Because Statement (1) conclusively answers 'No' to the question stem, Statement (1) ALONE is sufficient. Statement (2) allows

n = 2

(giving 12, divisible by 12) and

n = 4

(giving 30, not divisible by 12), yielding both 'Yes' and 'No' responses, making Statement (2) alone insufficient.

Step-by-Step Solution

1

Rephrase the question stem target using factoring.

The expression

n^2 + 3n + 2

factors into

(n + 1)(n + 2)

. The question asks whether

(n + 1)(n + 2)

is divisible by

12 = 2^2 \times 3

.

Factoring quadratic expressions helps reveal divisibility properties of consecutive integers.

2

Evaluate Statement (1):

n

is a multiple of 3.

If

n = 3k

for some positive integer

k

, then

n + 1 = 3k + 1

and

n + 2 = 3k + 2

. Neither factor contains 3 as a prime factor, so their product

(3k + 1)(3k + 2)

leaves a remainder of

1 \times 2 = 2

when divided by 3. Since the product is never divisible by 3, it can NEVER be divisible by 12.

In a Yes/No Data Sufficiency question, a statement that yields a definitive 'No' to the question is SUFFICIENT.

3

Evaluate Statement (2):

n + 1

is a prime number.

If

n = 2

, then

n + 1 = 3

(prime). The expression

(2+1)(2+2) = 12

, which is divisible by 12 (Answer: YES). If

n = 4

, then

n + 1 = 5

(prime). The expression

(4+1)(4+2) = 30

, which is not divisible by 12 (Answer: NO). Since both 'Yes' and 'No' are possible, Statement (2) is NOT sufficient.

Testing specific values shows that Statement (2) does not yield a consistent answer.

Key Concept

Definitive Yes/No decision logic in Data Sufficiency combined with divisibility and prime factor properties of consecutive integers.

Question 178Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

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

ab(a - b) < 0

, which is insufficient by itself because the sign of

(a - b)

depends on whether

ab

is positive or negative. Statement (2) simplifies to

ab < 0

(since

a^2 + b^2 > 0

), which alone does not reveal whether

a

or

b

is the larger number. When evaluated together, knowing

ab < 0

allows us to divide

ab(a - b) < 0

by the negative value

ab

and flip the inequality sign, yielding

a - b > 0

(i.e.,

a > b

) definitively.

Step-by-Step Solution

1

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

Statement (1) can be rewritten as

a^2 b - a b^2 < 0

, which factors into

ab(a - b) < 0

.

Factoring isolates the relationship between

ab

and

(a - b)

.

2

Test Statement (1) independently.

If

a = 1

and

b = 2

, then

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

, giving

a < b

(No). If

a = 2

and

b = -1

, then

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

, giving

a > b

(Yes).

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

a

and

b

, it is NOT sufficient alone.

3

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

Combining terms gives

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

. Because

a^2 + b^2 > 0

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

ab < 0

.

Knowing

ab < 0

implies

a

and

b

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

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

, whereas

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

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

4

Combine Statement (1) and Statement (2).

From Statement (2),

ab < 0

. From Statement (1),

ab(a - b) < 0

. Dividing

ab(a - b) < 0

by the negative quantity

ab

requires flipping the inequality sign, yielding

a - b > 0

, or

a > b

.

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

Key Concept

Statement Independence Evaluation and Statement Combination in Data Sufficiency

Question 179Question →

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

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

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, but statement (1) alone is not sufficient.

The question stem explicitly states that

x \neq y

, establishing that

x - y \neq 0

. In Statement (2), factoring

x^2 - y^2

into

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

and dividing by non-zero

(x - y)

directly yields the unique value

x + y = 3

. In Statement (1), factoring gives

x^2 + xy + y^2 = 7

, which leaves

x + y

undetermined because

xy

can vary. Consequently, Statement (2) alone is sufficient, while Statement (1) alone is not.

Step-by-Step Solution

1

Analyze the question stem constraints and target expression.

We are given that

x

and

y

are real numbers with

x \neq y

, which implies

x - y \neq 0

. The goal is to determine a single unique value for

x + y

.

Establishing

x - y \neq 0

allows valid algebraic division by

(x - y)

in the given statements.

2

Evaluate Statement (1) independently:

x^3 - y^3 = 7(x - y)

.

Factor the left side using the difference of cubes identity:

(x - y)(x^2 + xy + y^2) = 7(x - y)

. Since

x - y \neq 0

, divide both sides by

(x - y)

to get

x^2 + xy + y^2 = 7

, which rewrites to

(x + y)^2 - xy = 7

.

Because

xy

is unknown and variable,

(x + y)

can take multiple values. For instance,

(x, y) = (2, 1)

gives

x+y=3

and

4+2+1=7

, whereas

(x, y) = (\sqrt{7}, -\sqrt{7})

gives

x+y=0

and

7-7+7=7

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

3

Evaluate Statement (2) independently:

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

.

Factor the left side using the difference of squares identity:

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

. Since

x - y \neq 0

, divide both sides by

(x - y)

to yield

x + y = 3

.

This provides a single, unambiguous numerical value for

x + y

. Therefore, Statement (2) alone IS sufficient.

Key Concept

Algebraic expression simplification using stem constraints in Data Sufficiency

Question 180Question →

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

(1) $n^2 + 5n$ is an integer.

(2) $n^2 - n$ is an integer.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

Evaluating each statement alone shows that non-integer real numbers can produce integer values for

n^2 + 5n

or

n^2 - n

. However, combining both statements gives

6n = (n^2 + 5n) - (n^2 - n)

, proving

n = \frac{k}{6}

for some integer

k

. Substituting

n = \frac{k}{6}

back into

n^2 - n

requires

\frac{k^2 - 6k}{36}

to be an integer, which forces

k(k-6)

to be divisible by 36. Examining remainders modulo 6 shows

k

must be a multiple of 6, ensuring

n

is an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is NOT sufficient.

If

n^2 + 5n = p

where

p

is an integer,

n

can be an integer (e.g.,

n=1 \implies n^2+5n=6

) or a non-integer real number (e.g.,

n^2+5n-1=0 \implies n = \frac{-5 + \sqrt{29}}{2}

, which gives

n^2+5n=1

).

2

Evaluate Statement (2) independently.

Statement (2) is NOT sufficient.

If

n^2 - n = q

where

q

is an integer,

n

can be an integer (e.g.,

n=2 \implies n^2-n=2

) or a non-integer real number (e.g.,

n^2-n-1=0 \implies n = \frac{1 + \sqrt{5}}{2}

, which gives

n^2-n=1

).

3

Combine Statement (1) and Statement (2).

6n

is equal to an integer

k = p - q

, so

n = \frac{k}{6}

.

Subtracting

(n^2 - n = q)

from

(n^2 + 5n = p)

yields

6n = p - q

. Since

p

and

q

are integers,

k = p - q

must be an integer.

4

Substitute

n = \frac{k}{6}

back into Statement (2) to check integer constraints on

k

.

k

must be a multiple of 6, which implies

n = \frac{k}{6}

is an integer.

Substituting

n = \frac{k}{6}

into

n^2 - n = q

gives

\left(\frac{k}{6}\right)^2 - \frac{k}{6} = \frac{k^2 - 6k}{36} = q

. Thus,

36

must divide

k(k-6)

. Consequently,

6

divides

k(k-6)

. Since

k

and

k-6

have the same remainder

r

modulo 6,

k(k-6) \equiv r^2 \pmod 6

. For

r^2

to be divisible by 6 where

r \in \{0, 1, 2, 3, 4, 5\}

,

r

must be

0

. Hence,

k

is a multiple of 6, making

n

an integer.

Key Concept

Number properties of non-integer real variables vs. integer constraints when combining polynomial equations.

Data Sufficiency

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