Overlapping Sets, Statistics, and Data Distributions Practice Questions — GMAT

Question 1Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct option states that both statements together are sufficient, but neither statement alone is sufficient. By applying the formula

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

, we see that knowing the total (

30

) along with both individual sets (

20

and

15

) allows us to solve for the overlap uniquely (

5

). Neither statement alone provides both set totals.

Step-by-Step Solution

1

Rephrase the question stem using the overlapping set formula.

The total number of tourists is given by

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

, where

\text{Total} = 30

. To find

N(X \cap Y)

, we need the sum

N(X) + N(Y)

.

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

2

Evaluate Statement (1) independently.

Statement (1) gives

N(X) = 20

. Substituting into the equation gives

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

. Since

N(Y)

is unknown,

N(X \cap Y)

cannot be determined.

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

3

Evaluate Statement (2) independently.

Statement (2) gives

N(Y) = 15

. Substituting into the equation gives

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

. Since

N(X)

is unknown,

N(X \cap Y)

cannot be determined.

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

4

Evaluate both statements together.

Combining Statement (1) and Statement (2) provides

N(X) = 20

and

N(Y) = 15

. Plugging both into the formula:

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

. This gives a single, unique value.

Determines if combining both statements provides sufficient information.

Key Concept

Two-Group Overlapping Sets Principle

Estimated Time:1m 0s

Question 2Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The mean of a dataset of

n

items is defined as the sum of all elements divided by

n

. Since

n = 5

, knowing that the sum of the

5

numbers is

45

immediately gives the mean as

45 / 5 = 9

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

Step-by-Step Solution

1

Rephrase the question stem

The arithmetic mean of

5

numbers is equal to

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

. Thus, knowing the sum of the

5

numbers is necessary and sufficient to find the mean.

By definition,

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

where

n = 5

.

2

Evaluate Statement (1)

Statement (1) gives

\text{Sum} = 45

. Therefore,

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

. This gives a unique value.

Statement (1) alone provides all required information.

3

Evaluate Statement (2)

Consider two possible sets with median

9

and range

8

: Set X =

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

(Sum =

45

, Mean =

9

) and Set Y =

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

(Sum =

45

, Mean =

9

), but also Set Z =

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

(Sum =

43

, Mean =

8.6

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

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

Key Concept

Arithmetic Mean Definition in Data Sufficiency

Question 3Question →

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

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

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

, where

x

is analysts tracking Equities only,

y

is analysts tracking Bonds only,

z

is analysts tracking both, and

\bar{m}

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

\bar{m} = 15

, simplifying the target to

y > x

, which is insufficient alone. Statement (2) gives

y > x

, which is insufficient alone because

\bar{m}

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

y > x

and Statement (2) confirms

y > x

, yielding a definitive 'Yes'.

Step-by-Step Solution

1

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

Let

x

be the number of analysts tracking Equities only,

y

be the number of analysts tracking Bonds only, and

z

be the number of analysts tracking both. Total analysts

N = x + y + z

.
Sum of experience for Equities group:

S_E = 12(x+z)

.
Sum of experience for Bonds group:

S_B = 18(y+z)

.
Let

\bar{m}

be the mean experience of the

z

analysts tracking both, so their total experience is

z\bar{m}

.
Total experience

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

.

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

2

Rephrase the target question stem inequality.

The overall mean is

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

.
The question asks if

\bar{M} > 15

:

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

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

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

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

3

Evaluate Statement (1) alone.

Statement (1) states

\bar{m} = 15

.
Substituting

\bar{m} = 15

into the rephrased inequality yields

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

.
Since we do not know whether

y > x

or

y \leq x

, Statement (1) alone is NOT sufficient.

Knowing the mean of the overlap eliminates the

z

term but leaves the relationship between

x

and

y

unknown.

4

Evaluate Statement (2) alone.

Statement (2) states

y > x

.
Without knowing the value of

\bar{m}

(the mean of the overlapping group), if

\bar{m}

is very large (e.g.,

\bar{m} = 50

), then

(15 - \bar{m})z

could be negative enough to make

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

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

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

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

cannot be determined.

5

Evaluate Statements (1) and (2) together.

From Statement (1), the condition simplifies to

y > x

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

y > x

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

Combining both statements satisfies the rephrased target condition completely.

Key Concept

Overlapping Sets and Weighted Averages in Data Sufficiency

Estimated Time:2m 0s

Question 4Question →

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

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

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

500 and

600). Therefore, the statements together remain insufficient.

Step-by-Step Solution

1

Determine subgroup counts using the Principle of Inclusion-Exclusion.

Total workshops

N = 60

.

N(X) = 35

,

N(Y) = 40

. Since every workshop is in

X

or

Y

,

N(X \cup Y) = 60

. Thus,

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

. Workshops in ONLY

X = 35 - 15 = 20

. Workshops in ONLY

Y = 40 - 15 = 25

.

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

2

Evaluate Statement (1) alone.

The 20 ONLY

X

workshops cost

400 each, and the 25 ONLY

Y

workshops cost

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

400 (if all BOTH costs are low) or

600 (if all BOTH costs are high).

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

3

Evaluate Statement (2) alone.

Statement (2) provides the median (

500) and sum (

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

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

X

or ONLY

Y$ workshops.

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

4

Evaluate Statements (1) and (2) together.

From Statement (1), the sum of ONLY

X

and ONLY

Y

workshops is

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

. From Statement (2), total sum is

30,000

, so the 15 BOTH workshops sum to

30,000 - 23,000 = 7,000

, with a median of

500

.
Case A: If 1 BOTH workshop has cost

0

and 14 have cost

500

(median =

500

, sum =

7,000

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

400

, 14 values of

500

, and 25 values of

600

. The 30th and 31st values are both

500

, so overall median =

500

.
Case B: If 7 BOTH workshops have cost

0

, 1 has cost

500

, and 7 have cost

642.85

(median =

500

, sum =

7,000

), the sorted values put the 20 values of

400

in positions 8–27, the single

500

at position 28, and the 25 values of

600

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

600

, so overall median =

600

.

Since the overall median can be

500

or

600

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

Key Concept

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

Question 5Question →

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

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

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

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

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

45

Alpha-only,

60

Beta-only, and

45

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

Step-by-Step Solution

1

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

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

. Thus,

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

,

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

, and

N(\text{Both}) = 45

.

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

2

Evaluate Statement (1) alone.

Statement (1) provides

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

minutes and

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

minutes. However,

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

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

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

3

Evaluate Statement (2) alone.

Statement (2) provides

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

minutes. However,

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

and

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

are unknown. Statement (2) is NOT sufficient.

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

4

Evaluate Statements (1) and (2) combined.

Combining both statements gives all three subgroup means:

12

minutes for

45

assemblies,

18

minutes for

60

assemblies, and

25

minutes for

45

assemblies. Total testing duration

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

minutes. The overall average duration is

\frac{2745}{150} = 18.3

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

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

Key Concept

Overlapping Sets and Weighted Averages in Data Sufficiency

Estimated Time:2m 0s

Question 6Question →

At a technology firm, $60$ software engineers work on Project Alpha, Project Beta, or both. Exactly $40$ engineers work on Project Alpha, and exactly $35$ engineers work on Project Beta. Is the average (arithmetic mean) years of experience of all $60$ engineers combined greater than $7$ years?

(1) The average years of experience of the engineers who work ONLY on Project Alpha is $8$ years, and the average years of experience of the engineers who work ONLY on Project Beta is $5$ years.
(2) The average years of experience of all $40$ engineers on Project Alpha is $8.75$ 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 with a 'Yes', but neither statement alone is sufficient.

Combining both statements provides the exact sum of values across all three mutually exclusive sub-groups (Only Alpha, Only Beta, and Both), yielding a unique overall average of 7.5 years, which yields a definitive 'Yes' answer.

Step-by-Step Solution

1

Determine the sizes of the non-overlapping and overlapping groups.

Group sizes: Only Alpha = 25, Only Beta = 20, Both = 15.

Using the overlapping sets formula

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

, we get

60 = 40 + 35 - N(A \cap B)

, which yields

N(A \cap B) = 15

. Thus,

N(\text{Only Alpha}) = 40 - 15 = 25

and

N(\text{Only Beta}) = 35 - 15 = 20

.

2

Evaluate Statement (1) alone.

Statement (1) is NOT sufficient.

Statement (1) gives

S_{\text{Only A}} = 25 \times 8 = 200

and

S_{\text{Only B}} = 20 \times 5 = 100

. The sum of experience for the

15

overlap engineers (

S_{\text{Both}}

) remains unknown. Total mean

= \frac{300 + S_{\text{Both}}}{60}

. If

S_{\text{Both}} = 150

(mean

10

), total mean

= 7.5 > 7

(Yes). If

S_{\text{Both}} = 30

(mean

2

), total mean

= 5.5 \le 7

(No). Hence, Statement (1) alone is insufficient.

3

Evaluate Statement (2) alone.

Statement (2) is NOT sufficient.

Statement (2) gives the total experience of all

40

engineers on Project Alpha:

S_{\text{Alpha}} = 40 \times 8.75 = 350

. This means

S_{\text{Only A}} + S_{\text{Both}} = 350

. Total mean

= \frac{350 + S_{\text{Only B}}}{60}

. Since

S_{\text{Only B}}

is unknown, the total mean could be

7.5

(if

S_{\text{Only B}} = 100

) or

6.17

(if

S_{\text{Only B}} = 20

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

4

Evaluate Statements (1) and (2) together.

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

From Statement (1),

S_{\text{Only A}} = 200

and

S_{\text{Only B}} = 100

. From Statement (2),

S_{\text{Only A}} + S_{\text{Both}} = 350

, which implies

200 + S_{\text{Both}} = 350 \implies S_{\text{Both}} = 150

. The combined total experience is

S_{\text{Total}} = 200 + 100 + 150 = 450

. The overall mean is

\frac{450}{60} = 7.5

years. Since

7.5 > 7

, we get a definitive 'Yes'.

Key Concept

Data Sufficiency evaluation for combined weighted averages and overlapping set partitions

Question 7Question →

A department has $40$ employees, and each employee speaks at least one of two languages: French or Spanish. How many employees in the department speak Spanish?

(1) $25$ employees speak French.
(2) $10$ employees speak both French and Spanish.

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 total number of employees who speak at least one language is given by

\text{Total} = \text{French} + \text{Spanish} - \text{Both}

. Neither statement alone provides both the number of French speakers and the number of employees who speak both languages. However, combining both statements gives

40 = 25 + \text{Spanish} - 10

, which uniquely solves to

\text{Spanish} = 25

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

Step-by-Step Solution

1

Rephrase the question stem using the overlapping sets formula.

Since every employee speaks at least one language,

\text{Neither} = 0

. The relationship is

\text{Total} = \text{French} + \text{Spanish} - \text{Both}

, which simplifies to

40 = \text{French} + \text{Spanish} - \text{Both}

.

Establishing the mathematical relationship before evaluating statements clarifies what data is missing.

2

Evaluate Statement (1) independently.

Statement (1) gives

\text{French} = 25

. Substituting this gives

40 = 25 + \text{Spanish} - \text{Both}

, or

\text{Spanish} - \text{Both} = 15

.

Since

\text{Both}

is unknown,

\text{Spanish}

could take multiple values. Statement (1) is NOT sufficient.

3

Evaluate Statement (2) independently.

Statement (2) gives

\text{Both} = 10

. Substituting this gives

40 = \text{French} + \text{Spanish} - 10

, or

\text{French} + \text{Spanish} = 50

.

Since

\text{French}

is unknown,

\text{Spanish}

cannot be uniquely determined. Statement (2) is NOT sufficient.

4

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

Substitute both values into the equation:

40 = 25 + \text{Spanish} - 10 \implies 40 = 15 + \text{Spanish} \implies \text{Spanish} = 25

.

The equation yields a single, unique value for the target variable. Both statements together are sufficient.

Key Concept

Overlapping Sets (Two Groups)

Estimated Time:1m 0s

Question 8Question →

A class of 50 students took tests in both Mathematics and Science. How many students passed both tests?

(1) 35 students passed Mathematics and 30 students passed Science.
(2) 10 students failed both tests.

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

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

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

EACH statement ALONE is sufficient.

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

Show answer & explanation

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

Answer

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

The correct option is the choice stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) leaves the number of students failing both tests unknown, while Statement (2) leaves the individual subject pass numbers unknown. Combined, the standard set equation

\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither}

yields a unique value of

25

for students passing both tests.

Step-by-Step Solution

1

Set up the overlapping sets formula for two groups

\text{Total} = \text{Math} + \text{Science} - \text{Both} + \text{Neither}

, which becomes

50 = \text{Math} + \text{Science} - \text{Both} + \text{Neither}

.

This formula connects all four components of a two-group overlapping set.

2

Evaluate Statement (1) independently

Substituting

\text{Math} = 35

and

\text{Science} = 30

into the formula yields

50 = 35 + 30 - \text{Both} + \text{Neither}

, or

\text{Both} - \text{Neither} = 15

.

Since

\text{Neither}

is unknown,

\text{Both}

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

3

Evaluate Statement (2) independently

Substituting

\text{Neither} = 10

yields

50 = \text{Math} + \text{Science} - \text{Both} + 10

.

Since

\text{Math}

and

\text{Science}

are unknown,

\text{Both}

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

4

Evaluate Statements (1) and (2) combined

Combining all given values gives

50 = 35 + 30 - \text{Both} + 10 \Rightarrow 50 = 75 - \text{Both} \Rightarrow \text{Both} = 25

.

A single unique value of 25 is obtained for the number of students who passed both tests.

Key Concept

Overlapping Sets (Two-Group Venn Diagram Formula)

Question 9Question →

In a technology consulting firm of 100 employees, every employee works in either the Analytics department, the Engineering department, or both. How many employees work in both departments?

(1) Exactly 70 employees work in the Analytics department, and 60 employees work in the Engineering department.
(2) Exactly 40 employees work ONLY in the Analytics department.

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 states that Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient. Using the standard formula for overlapping sets, Total = Group A + Group B - Both, Statement (1) provides values for Total (100), Group A (70), and Group B (60), allowing us to solve directly for Both = 30. Statement (2) only specifies the number of members belonging strictly to Group A, which leaves the overlap dependent on the unstated size of Group B.

Step-by-Step Solution

1

Set up the overlapping sets formula for two groups.

Total = N(Analytics) + N(Engineering) - N(Both)

Since every employee belongs to at least one of the two departments, the union of the two sets equals the total number of employees, 100.

2

Evaluate Statement (1) independently.

100 = 70 + 60 - N(Both) => N(Both) = 30.

Statement (1) provides N(Analytics) = 70 and N(Engineering) = 60. Substituting these values into the formula yields a unique value of 30 for N(Both). Thus, Statement (1) alone is sufficient.

3

Evaluate Statement (2) independently.

100 = N(Only Analytics) + N(Only Engineering) + N(Both) => 100 = 40 + N(Only Engineering) + N(Both).

We have two unknown variables: N(Only Engineering) and N(Both). Multiple non-negative integer pairs satisfy this equation (e.g., N(Both) could be 0, 10, 20, etc.). Thus, Statement (2) alone is not sufficient.

Key Concept

Overlapping Sets Formula for Two Groups

Estimated Time:1m 30s

Question 10Question →

A community library cataloged a collection of $120$ historical manuscripts. Each manuscript is written in either Latin, Ancient Greek, or both. How many of the manuscripts are written in both Latin and Ancient Greek?

(1) Exactly $80$ manuscripts are written in Latin.
(2) The number of manuscripts written only in Ancient Greek is twice the number of manuscripts written in both languages.

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 enough independent relationships to solve for the intersection. Statement (1) determines the count of manuscripts written exclusively in Ancient Greek (

120 - 80 = 40

), while Statement (2) establishes a proportional link between manuscripts exclusively in Ancient Greek and those in both languages (

2x

). Combining both facts produces the linear equation

2x = 40

, which uniquely solves to

x = 20

.

Step-by-Step Solution

1

Rephrase the question stem using overlapping set principles.

Let

L

be the set of Latin manuscripts,

G

be the set of Ancient Greek manuscripts, and

x

be the number of manuscripts in both (

L \cap G

). Since every manuscript is in at least one set, Total =

(\text{Latin only}) + (\text{Greek only}) + x = 120

. Also, Total =

|L| + (\text{Greek only}) = 120

. We need to find the unique value of

x

.

Formulating the set relationships establishes the exact algebraic system needed to evaluate sufficiency.

2

Evaluate Statement (1) independently.

Statement (1) gives

|L| = 80

. Substituting into the total formula gives

80 + (\text{Greek only}) = 120 \implies \text{Greek only} = 40

. However,

|L| = (\text{Latin only}) + x = 80

. The overlap

x

can range anywhere from

0

to

80

.

Statement (1) does not provide enough information to isolate

x

from Latin only.

3

Evaluate Statement (2) independently.

Statement (2) states that

\text{Greek only} = 2x

. Substituting into the total equation yields

(\text{Latin only}) + 2x + x = (\text{Latin only}) + 3x = 120

. Multiple integer solutions exist for

x

(e.g., if

\text{Latin only} = 90

,

x = 10

; if

\text{Latin only} = 60

,

x = 20

).

Statement (2) presents one equation with two unknown variables.

4

Evaluate Statements (1) and (2) together.

From Statement (1),

\text{Greek only} = 120 - 80 = 40

. From Statement (2),

\text{Greek only} = 2x

. Therefore,

2x = 40 \implies x = 20

. This gives a single, unique answer.

Combining both statements provides two independent linear equations, uniquely solving for

x

.

Key Concept

Overlapping Sets (Two-Group Venn Diagram) Data Sufficiency

Estimated Time:2m 0s

Question 11Question →

Among a panel of $150$ medical research trials, each trial evaluates at least one of three experimental drugs: Drug $X$ , Drug $Y$ , or Drug $Z$ . If $80$ trials evaluate Drug $X$ and $70$ trials evaluate Drug $Y$ , how many trials evaluate Drug $Z$ only?

(1) Exactly $25$ trials evaluate both Drug $X$ and Drug $Y$ .
(2) Exactly $40$ trials evaluate at least two of the three drugs, and no trial evaluates all three drugs.

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 choice is the option stating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient. By rephrasing the question, the number of trials evaluating Drug Z only equals the total number of trials (

150

) minus the number of trials evaluating Drug X or Drug Y (

|X \cup Y|

). Statement (1) gives the intersection

|X \cap Y| = 25

, which immediately yields

|X \cup Y| = 80 + 70 - 25 = 125

, so Drug Z only

= 150 - 125 = 25

. Statement (2) only establishes that the double overlap equals

40

, leaving the specific overlap between X and Y unknown.

Step-by-Step Solution

1

Rephrase the target question using set notation.

Let

U

be the total set of trials (

|U| = 150

). Since every trial evaluates at least one drug,

|X \cup Y \cup Z| = 150

. The number of trials evaluating Drug

Z

only is given by

|Z \text{ only}| = |X \cup Y \cup Z| - |X \cup Y| = 150 - |X \cup Y|

.

Simplifying the target target shows that finding

|X \cup Y|

is both necessary and sufficient to answer the question.

2

Evaluate Statement (1) independently.

Statement (1) gives

|X \cap Y| = 25

. Using the standard principle of inclusion-exclusion for two sets:

|X \cup Y| = |X| + |Y| - |X \cap Y| = 80 + 70 - 25 = 125

. Then

|Z \text{ only}| = 150 - 125 = 25

.

Statement (1) yields a single, unique numerical answer (

25

), so Statement (1) alone is sufficient.

3

Evaluate Statement (2) independently.

Let

a, b, c

be the double overlaps

|X \cap Y|

,

|Y \cap Z|

, and

|X \cap Z|

, and

d = |X \cap Y \cap Z| = 0

. Statement (2) states

a + b + c = 40

. Total set formula gives

150 = 80 + 70 + |Z| - 40 \implies |Z| = 40

. However,

|Z \text{ only}| = |Z| - (b + c) = 40 - (40 - a) = a = |X \cap Y|

, which is unknown.

Since the value of

|X \cap Y|

can vary, Statement (2) alone does not yield a unique numerical value and is insufficient.

Key Concept

Overlapping Sets and Data Sufficiency Target Rephrasing

Question 12Question →

In a technology firm of $100$ software engineers, each engineer works on at least one of two projects: Project Alpha or Project Beta. If $70$ engineers work on Project Alpha, what is the average (arithmetic mean) monthly salary of all $100$ engineers at the firm?

(1) The average monthly salary of the engineers who work ONLY on Project Alpha is $\$6,000$ , and the average monthly salary of all engineers who work on Project Beta is $\$10,800$ .
(2) The average monthly salary of the engineers who work ONLY on Project Beta is $\$12,000$ , and the average monthly salary of the engineers who work on BOTH Project Alpha and Project Beta is $\$9,000$ .

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

Answer

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

The option stating that BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient is correct. Evaluating either statement independently leaves the overlap count or salary undefined. However, combining both statements allows equating two expressions for the total salary of Project Beta, which uniquely yields

20

engineers working on both projects, leading to a single deterministic overall average salary of

\$8,400

.

Step-by-Step Solution

1

Define variables from the question stem.

Let

n_1

be the number of engineers working ONLY on Project Alpha,

n_2

be the number working ONLY on Project Beta, and

n_3

be the number working on BOTH. Total engineers =

n_1 + n_2 + n_3 = 100

. Given

70

work on Alpha (

n_1 + n_3 = 70

), we find

n_2 = 100 - 70 = 30

.

Establishing exact set counts reduces unknowns.

2

Evaluate Statement (1) alone.

Total salary =

(n_1 \times 6000) + ((n_2 + n_3) \times 10800) = (70 - n_3)(6000) + (30 + n_3)(10800) = 744,000 + 4800 n_3

. Since

n_3

is unknown, overall average cannot be determined.

Statement (1) depends on the unknown overlap

n_3

, so Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) alone.

We know the average salary for

n_2 = 30

is

\$12,000

and for

n_3

is

\$9,000

, but we have no salary information for the

n_1

engineers working ONLY on Project Alpha.

Without salary information for Project Alpha only, Statement (2) alone is NOT sufficient.

4

Evaluate Statements (1) and (2) together.

From Statement (2), total salary of Project Beta =

(30 \times 12000) + (n_3 \times 9000) = 360,000 + 9000 n_3

. From Statement (1), total salary of Project Beta =

(30 + n_3) \times 10800 = 324,000 + 10800 n_3

. Equating both:

360,000 + 9000 n_3 = 324,000 + 10800 n_3 \implies 1800 n_3 = 36,000 \implies n_3 = 20

.

Finding

n_3 = 20

gives

n_1 = 50

, allowing exact calculation of the overall average salary (

8,400

).

Key Concept

Weighted Averages and Overlapping Sets in Data Sufficiency

Question 13Question →

A university research institute evaluated $N$ technology projects completed last year. Each project received grant funding from at least one of two foundations: Foundation $X$ or Foundation $Y$ . Exactly $30\%$ of the $N$ projects received funding from both foundations. If the average (arithmetic mean) grant amount per project among projects funded by Foundation $X$ was $\$50,000$ and the average grant amount per project among projects funded by Foundation $Y$ was $\$60,000$ , what was the average total grant funding per project across all $N$ projects?

(1) The number of projects that received funding from Foundation $X$ only was equal to the number of projects that received funding from Foundation $Y$ only.
(2) The total dollar amount disbursed by Foundation $Y$ exceeded the total dollar amount disbursed by Foundation $X$ by $\$1,800,000$ .

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

Answer

Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not sufficient.

Statement (1) alone is sufficient because equating the counts of projects funded exclusively by Foundation X and exclusively by Foundation Y implies that the total number of projects funded by Foundation X equals the total number funded by Foundation Y. Combined with the principle of overlapping sets where thirty percent of projects received funding from both, this uniquely determines that sixty-five percent of projects were funded by Foundation X and sixty-five percent were funded by Foundation Y, yielding a unique overall average grant of $71,500. Statement (2) alone is insufficient because the equation relating total disbursements depends on the unknown total number of projects N.

Step-by-Step Solution

1

Express the total funding and set proportions in terms of the total number of projects

N

.

Let

n_X

be the number of projects funded by Foundation

X

and

n_Y

be the number of projects funded by Foundation

Y

. The number of projects funded by both is

0.30N

. Since every project is funded by at least one foundation,

N = n_X + n_Y - 0.30N \implies n_X + n_Y = 1.30N

. Dividing by

N

gives the ratio sum

\frac{n_X}{N} + \frac{n_Y}{N} = 1.30

.

Establishing the relationship between the subset counts and total projects simplifies the target average formula.

2

Formulate the expression for the average total grant funding per project.

Total Funding

= 50,000 n_X + 60,000 n_Y

. Therefore, the average funding per project is

\text{Average} = \frac{50,000 n_X + 60,000 n_Y}{N} = 50,000\left(\frac{n_X}{N}\right) + 60,000\left(\frac{n_Y}{N}\right)

.

Since total grant funding across all projects is the sum of all money disbursed by Foundation

X

and Foundation

Y

, the overall average depends strictly on the ratios

\frac{n_X}{N}

and

\frac{n_Y}{N}

.

3

Evaluate Statement (1) independently.

Statement (1) states that the number of projects funded by

X

only equals the number funded by

Y

only:

n_X - 0.30N = n_Y - 0.30N \implies n_X = n_Y

. Since

\frac{n_X}{N} + \frac{n_Y}{N} = 1.30

and

n_X = n_Y

, we get

2\left(\frac{n_X}{N}\right) = 1.30 \implies \frac{n_X}{N} = 0.65

and

\frac{n_Y}{N} = 0.65

. Substituting these into the average formula yields

\text{Average} = 50,000(0.65) + 60,000(0.65) = 71,500

. Statement (1) ALONE is sufficient.

Knowing that the two single-foundation set sizes are equal determines the exact proportions of

N

funded by each foundation.

4

Evaluate Statement (2) independently.

Statement (2) states that

60,000 n_Y - 50,000 n_X = 1,800,000

. Dividing by

N

gives

60,000\left(\frac{n_Y}{N}\right) - 50,000\left(\frac{n_X}{N}\right) = \frac{1,800,000}{N}

. Because

N

is unknown, the right-hand side is not fixed, so

\frac{n_X}{N}

and

\frac{n_Y}{N}

cannot be uniquely determined. Statement (2) ALONE is not sufficient.

An absolute dollar equation introduces a dependency on the total count

N

, preventing a unique calculation of the relative proportions.

Key Concept

Overlapping Sets and Weighted Averages in Data Sufficiency

Estimated Time:2m 0s

Question 14Question →

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 15Question →

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 16Question →

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 17Question →

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 18Question →

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 19Question →

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 20Question →

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

Overlapping Sets, Statistics, and Data Distributions

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