Data Sufficiency Practice Questions — State PSC Exam

Three numbers $x$ , $y$ , and $z$ are positive integers. What is the exact value of $x$ ?

Statement I: The arithmetic mean of $x$ , $y$ , and $z$ is equal to $x + 2$ , where $y \le z$ .
Statement II: The least common multiple (LCM) of $y$ and $z$ is $12$ , and their greatest common divisor (GCD) is $2$ .

Which of the following options correctly describes the sufficiency of the statements?

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II is sufficient to answer the question, even when taken together.

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Show answer & explanation

Answer: Neither Statement I nor Statement II is sufficient to answer the question, even when taken together.

Answer

Neither Statement I nor Statement II is sufficient to answer the question, even when taken together.

The correct answer states that neither statement is sufficient, even when combined. Statement I reduces to

y + z = 2x + 6

, which leaves

x

dependent on the sum

y + z

. Statement II yields two possible pairs for

(y, z)

with

\text{GCD}=2

and

\text{LCM}=12

, namely

(2, 12)

with sum

14

, and

(4, 6)

with sum

10

. Substituting

y + z = 14

gives

x = 4

, while substituting

y + z = 10

gives

x = 2

. Because both

x = 4

and

x = 2

are valid positive integer solutions, a single unique value for

x

cannot be determined.

Step-by-Step Solution

1

Analyze Statement I individually.

The arithmetic mean formula gives

\frac{x + y + z}{3} = x + 2 \implies x + y + z = 3x + 6 \implies y + z = 2x + 6

.

Since

y

and

z

are unknown positive integers,

x

can take multiple values (e.g., if

y+z = 8

,

x=1

; if

y+z = 10

,

x=2

). Thus, Statement I alone is insufficient.

2

Analyze Statement II individually.

Find pairs

(y, z)

such that

y \le z

,

\text{GCD}(y, z) = 2

, and

\text{LCM}(y, z) = 12

.

Since

y \times z = \text{GCD} \times \text{LCM} = 24

, the valid pairs of positive integers with

\text{GCD}=2

and

\text{LCM}=12

are

(2, 12)

and

(4, 6)

. Statement II makes no reference to

x

, so it is insufficient.

3

Evaluate Statements I and II together.

Test both valid pairs from Statement II in the relation

2x = y + z - 6

from Statement I.

Case 1: For

(y, z) = (2, 12)

,

y + z = 14 \implies 2x = 14 - 6 = 8 \implies x = 4

.
Case 2: For

(y, z) = (4, 6)

,

y + z = 10 \implies 2x = 10 - 6 = 4 \implies x = 2

.
Since two distinct values of

x

(

x = 4

and

x = 2

) satisfy all conditions, both statements together are NOT sufficient to determine a unique value for

x

.

Key Concept

Data Sufficiency with Multiple Discrete Solutions in Number Theory

Question 2Question →

What is the value of the positive integer $n$ ?

Statement I: $n + 8 = 15$
Statement II: $3n = 21$

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II alone is sufficient, but both statements together are sufficient to answer the question.

Neither Statement I nor Statement II, even when taken together, is sufficient to answer the question.

Show answer & explanation

Answer: Either Statement I alone or Statement II alone is sufficient to answer the question.

Answer

Either Statement I alone or Statement II alone is sufficient to answer the question.

Evaluating Statement I alone gives

n = 15 - 8 = 7

, which uniquely determines

n

. Evaluating Statement II alone gives

n = \frac{21}{3} = 7

, which also uniquely determines

n

. Since each statement independently provides a unique answer, either statement alone is sufficient.

Step-by-Step Solution

1

Evaluate Statement I independently

From

n + 8 = 15

, subtracting

8

from both sides gives

n = 7

.

This yields a unique value for

n

, so Statement I alone is sufficient.

2

Evaluate Statement II independently

From

3n = 21

, dividing both sides by

3

gives

n = 7

.

This also yields a unique value for

n

, so Statement II alone is sufficient.

3

Determine overall sufficiency choice

Since both Statement I alone and Statement II alone yield the exact unique answer, either statement alone is sufficient.

This matches the standard option where either statement independently suffices.

Key Concept

Single-Variable Linear Data Sufficiency

Estimated Time:45s

Question 3Question →

What is the average age of a class of 20 students?

Statement I: The sum of the ages of all 20 students in the class is 300 years.
Statement II: The age of the youngest student in the class is 12 years.

Statement I alone is sufficient to answer the question, while Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, while Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II is sufficient to answer the question.

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Show answer & explanation

Answer: Statement I alone is sufficient to answer the question, while Statement II alone is not sufficient.

Answer

Statement I alone is sufficient to answer the question, while Statement II alone is not sufficient.

The correct choice highlights that Statement I alone provides the total sum of ages for the 20 students. Since average equals total sum divided by the number of items (

\text{Average} = \frac{300}{20} = 15

), Statement I alone is sufficient. Statement II only tells us the age of one specific student, which does not allow us to compute the average for all 20 students.

Step-by-Step Solution

1

Analyze the core formula needed to solve the question

The average age of a group is calculated using the formula:

\text{Average Age} = \frac{\text{Sum of ages of all students}}{\text{Total number of students}}

. The total number of students is given as

20

.

Understanding the required parameters helps evaluate statement sufficiency.

2

Evaluate Statement I individually

Statement I gives the sum of ages

= 300

years.

\text{Average Age} = \frac{300}{20} = 15

years. Thus, Statement I alone yields a unique answer and is sufficient.

Determines if Statement I provides enough data on its own.

3

Evaluate Statement II individually

Statement II gives the age of the youngest student

= 12

years. This tells us nothing about the remaining

19

students, so we cannot determine the sum or the average age. Thus, Statement II alone is not sufficient.

Determines if Statement II provides enough data on its own.

Key Concept

Data Sufficiency - Evaluation of Individual Statements in Average Calculations

Question 4Question →

A rectangular garden is surrounded on all four sides by a concrete walking path of uniform width $w$ meters. What is the area of the concrete walking path in square meters?

Statement I: The perimeter of the outer boundary of the walking path is $80\text{ meters}$ .
Statement II: The area of the rectangular garden is $300\text{ square meters}$ , and its length is $5\text{ meters}$ greater than its width.

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient, but Statement II alone is not sufficient.

Statement II alone is sufficient, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient.

Neither Statement I nor Statement II is sufficient, even when combined.

Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

Show answer & explanation

Answer: Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

Answer

Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

The correct answer states that both statements together are sufficient, but neither alone is sufficient. Statement I leaves the path width and garden dimensions undetermined, while Statement II gives the garden dimensions without the path width. Combining them allows the path width and consequently the exact path area to be calculated uniquely.

Step-by-Step Solution

1

Analyze Statement I independently

Let the length and width of the garden be

L

and

W

meters, and the uniform path width be

w

meters. The outer dimensions are

(L + 2w)

and

(W + 2w)

. The outer perimeter is

2[(L + 2w) + (W + 2w)] = 2(L + W + 4w) = 80

, which simplifies to

L + W + 4w = 40

. The area of the path is

(L + 2w)(W + 2w) - LW = 2w(L + W + 2w)

. Since

L + W

and

w

can take multiple values satisfying

L + W + 4w = 40

, the area of the path cannot be uniquely determined. Hence, Statement I alone is NOT sufficient.

Evaluating whether Statement I alone fixes a single numerical value for the path area.

2

Analyze Statement II independently

Statement II states that

L \cdot W = 300

and

L = W + 5

. Substituting gives

(W + 5)W = 300

, so

W^2 + 5W - 300 = 0

. Factoring gives

(W + 20)(W - 15) = 0

, yielding

W = 15\text{ m}

and

L = 20\text{ m}

. However, no information is given about the path width

w

. Thus, the path area

2w(35 + 2w)

cannot be computed without

w

. Hence, Statement II alone is NOT sufficient.

Evaluating whether Statement II alone fixes a single numerical value for the path area.

3

Analyze Statements I and II combined

From Statement II,

L = 20\text{ m}

and

W = 15\text{ m}

, so

L + W = 35\text{ m}

. Substitute this into the relation from Statement I:

35 + 4w = 40 \Rightarrow 4w = 5 \Rightarrow w = 1.25\text{ meters}

. The area of the path is

2(1.25)(35 + 2.5) = 2.5 \times 37.5 = 93.75\text{ square meters}

, which is a unique value. Therefore, both statements together are sufficient.

Determining if combining both statements resolves all unknown variables uniquely.

Key Concept

Evaluating individual statement sufficiency for multi-variable geometric systems prior to combination.

Question 5Question →

Is the positive integer $n$ divisible by $36$ ?

Statement I: $n^2$ is divisible by $108$ .
Statement II: $n^3$ is divisible by $576$ .

Which of the following options correctly describes the sufficiency of the statements?

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II is sufficient to answer the question, and additional data is needed.

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Show answer & explanation

Answer: Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Answer

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating both statements together shows that Statement I requires

n

to be a multiple of

18

(

2^1 \times 3^2

) and Statement II requires

n

to be a multiple of

12

(

2^2 \times 3^1

). The least common multiple of

18

and

12

is

36

, which proves that

n

is guaranteed to be divisible by

36

. Neither statement alone provides both prime factor requirements.

Step-by-Step Solution

1

Analyze Statement I individually

Statement I states that

n^2

is divisible by

108 = 2^2 \times 3^3

. In the prime factorization of a square

n^2

, all exponents must be even numbers. Thus,

n^2

must contain at least

2^2

and

3^4

. Taking square roots,

n

must be a multiple of

2^1 \times 3^2 = 18

. If

n = 18

,

n

is NOT divisible by

36

. If

n = 36

,

n

IS divisible by

36

. Because we get both 'No' and 'Yes' answers, Statement I alone is NOT sufficient.

We must test if Statement I uniquely determines whether

n

is divisible by

36

.

2

Analyze Statement II individually

Statement II states that

n^3

is divisible by

576 = 2^6 \times 3^2

. In the prime factorization of a cube

n^3

, all exponents must be multiples of

3

. Thus,

n^3

must contain at least

2^6

and

3^3

. Taking cube roots,

n

must be a multiple of

2^2 \times 3^1 = 12

. If

n = 12

,

n

is NOT divisible by

36

. If

n = 36

,

n

IS divisible by

36

. Because we get both 'No' and 'Yes' answers, Statement II alone is NOT sufficient.

We must test if Statement II uniquely determines whether

n

is divisible by

36

.

3

Combine Statement I and Statement II

From Statement I,

n

contains at least

3^2

in its prime factorization. From Statement II,

n

contains at least

2^2

in its prime factorization. Combining these requirements,

n

must contain at least

2^2 \times 3^2 = 36

. Therefore,

n

is guaranteed to be divisible by

36

. Both statements together yield a definitive 'Yes'.

Evaluating both statements together combines the minimal necessary powers of each prime factor.

Key Concept

Data Sufficiency evaluation of prime factor exponents and divisibility rules

Estimated Time:2m 0s

Question 6Question →

A train running at a constant speed crosses a stationary platform of length $200\text{ meters}$ . What is the speed of the train in $\text{km/h}$ ?

Statement I: The train takes $20\text{ seconds}$ to completely cross the platform.
Statement II: The train takes $8\text{ seconds}$ to cross a telegraph pole standing beside the track.

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I alone nor Statement II alone is sufficient, and both statements together are not sufficient to answer the question.

Both Statement I and Statement II together are necessary and sufficient to answer the question.

Show answer & explanation

Answer: Both Statement I and Statement II together are necessary and sufficient to answer the question.

Answer

Both Statement I and Statement II together are necessary and sufficient to answer the question.

Evaluating each statement individually shows that neither Statement I nor Statement II alone provides enough information to determine the speed of the train because the length of the train remains unknown. However, combining both statements yields a system of two independent linear equations with two variables (length and speed of the train), allowing us to solve uniquely for the speed of the train (

60\text{ km/h}

). Therefore, both statements together are necessary and sufficient.

Step-by-Step Solution

1

Analyze Statement I alone.

Let the length of the train be

L\text{ meters}

and its speed be

v\text{ m/s}

. The equation is

L + 200 = 20v

. Since there are two unknowns (

L

and

v

), Statement I alone is not sufficient.

The length of the train is an unknown variable.

2

Analyze Statement II alone.

Crossing a telegraph pole means distance traveled equals the length of the train. The equation is

L = 8v

. Since there are two unknowns (

L

and

v

), Statement II alone is not sufficient.

The speed cannot be determined without knowing the length of the train.

3

Evaluate Statement I and Statement II together.

Substitute

L = 8v

into the first equation:

8v + 200 = 20v \implies 12v = 200 \implies v = \frac{50}{3}\text{ m/s}

. Converting to

\text{km/h}

:

v = \frac{50}{3} \times \frac{18}{5} = 60\text{ km/h}

. Thus, both statements together give a unique answer.

Two independent equations are sufficient to solve for two unknown variables.

Key Concept

Data Sufficiency evaluation for relative motion and distance-speed problems using linear equations.

Estimated Time:1m 30s

Question 7Question →

Two business partners, $A$ and $B$ , invested in a joint venture. What is the total profit earned by the business at the end of one year?

Statement I: Partner $A$ invested $\$6,000$ for the entire year, while Partner $B$ invested $\$9,000$ for $8$ months.
Statement II: Partner $A$ 's share of the annual profit is $\$2,400$ .

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II is sufficient to answer the question.

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Show answer & explanation

Answer: Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Answer

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating Statement I alone gives the profit-sharing ratio between Partner A and Partner B as

(6000 \times 12) : (9000 \times 8) = 1 : 1

, which is insufficient by itself to find the monetary profit. Statement II alone gives Partner A's profit share as

\$2,400

, which is also insufficient without knowing the proportion. Combining both statements shows that both partners receive equal shares, so total profit is

\$2,400 \times 2 = \$4,800

. Therefore, both statements together are necessary and sufficient.

Step-by-Step Solution

1

Evaluate Statement I alone

Profit sharing ratio of A to B =

(6000 \times 12) : (9000 \times 8) = 72,000 : 72,000 = 1 : 1

.

Profit is distributed in proportion to the product of capital invested and time period. Since no dollar amounts of profit are given, total profit cannot be determined from Statement I alone.

2

Evaluate Statement II alone

Partner A's profit share =

\$2,400

.

Without knowing Partner B's share or the ratio between their shares, the total profit cannot be calculated from Statement II alone.

3

Evaluate Statement I and Statement II together

Since ratio of shares is

1:1

and A's share is

\$2,400

, B's share is also

\$2,400

. Total profit =

\$2,400 + \$2,400 = \$4,800

.

Combining the ratio from Statement I and the monetary value from Statement II gives a unique answer to the question.

Key Concept

Partnership profit distribution ratio and data sufficiency evaluation

Question 8Question →

A vessel contains a mixture of milk and water. What is the initial total volume of the mixture in the vessel?

Statement I: The initial ratio of milk to water in the vessel is $3:2$ . When $10\text{ liters}$ of water is added to the mixture, the ratio of milk to water becomes $1:1$ .
Statement II: If $15\text{ liters}$ of the original mixture is removed and replaced with $15\text{ liters}$ of pure water, the quantity of milk remaining in the vessel is $27\text{ liters}$ .

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient, but Statement II alone is not sufficient.

Statement II alone is sufficient, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient.

Neither Statement I nor Statement II is sufficient.

Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

Show answer & explanation

Answer: Statement I alone is sufficient, but Statement II alone is not sufficient.

Answer

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement I alone gives a complete quantitative relationship: with initial quantities

3x

and

2x

, adding

10\text{ liters}

of water leads to

3x = 2x + 10

, yielding

x = 10

and a unique total initial volume of

50\text{ liters}

. Statement II alone leaves two unknown variables (initial milk fraction and total volume), so it is not sufficient.

Step-by-Step Solution

1

Evaluate Statement I alone.

Initial volume is calculated as

50\text{ liters}

.

Let the initial volume of milk be

3x

liters and water be

2x

liters, making the total initial volume

5x

liters. Adding

10\text{ liters}

of water gives the equation

\frac{3x}{2x + 10} = \frac{1}{1}

, which simplifies to

3x = 2x + 10 \implies x = 10

. Thus, total initial volume

5x = 50\text{ liters}

. Hence, Statement I alone is sufficient.

2

Evaluate Statement II alone.

Insufficient to find total initial volume.

Let the total initial volume be

V

liters and the fraction of milk in the initial mixture be

f

. Removing

15\text{ liters}

of mixture removes

15f

liters of milk. Replacing it with water adds no milk. The remaining milk equation is

f(V - 15) = 27

. Since both

f

and

V

are unknown,

V

cannot be uniquely determined. Hence, Statement II alone is not sufficient.

Key Concept

Data Sufficiency evaluation of mixture ratios and algebraic equations

Estimated Time:1m 30s

Question 9Question →

What is the exact value of the two-digit positive integer $N$ ?

Statement (I): The sum of the tens digit and the units digit of $N$ is equal to $12$ .
Statement (II): Reversing the digits of $N$ yields a new two-digit integer that is $36$ greater than $N$ .

Which of the following statements correctly describes the sufficiency of the given data to answer the question?

Statement (I) alone is sufficient, but Statement (II) alone is not sufficient.

Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Either Statement (I) alone or Statement (II) alone is sufficient.

Neither Statement (I) nor Statement (II) is sufficient.

Both Statement (I) and Statement (II) together are sufficient, but neither statement alone is sufficient.

Show answer & explanation

Answer: Both Statement (I) and Statement (II) together are sufficient, but neither statement alone is sufficient.

Answer

Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating each statement independently reveals that neither statement alone narrows

N

down to a single value. Combining both equations (

x + y = 12

and

y - x = 4

) yields a single unique pair

x = 4

and

y = 8

, giving

N = 48

. Therefore, both statements together are required and sufficient.

Step-by-Step Solution

1

Represent the two-digit integer algebraically.

Let

N = 10x + y

, where

x

is the tens digit (

1 \leq x \leq 9

) and

y

is the units digit (

0 \leq y \leq 9

).

Setting up standard place-value variables allows formal analysis of both statements.

2

Evaluate Statement (I) alone.

From Statement (I),

x + y = 12

. Possible pairs

(x,y)

are

(3,9), (4,8), (5,7), (6,6), (7,5), (8,4), (9,3)

. Thus,

N

could be

39, 48, 57, 66, 75, 84,

or

93

.

Since multiple valid values of

N

exist, Statement (I) alone is NOT sufficient.

3

Evaluate Statement (II) alone.

The reversed number is

10y + x

. Given

(10y + x) - (10x + y) = 36 \implies 9(y - x) = 36 \implies y - x = 4

. Possible pairs

(x,y)

are

(1,5), (2,6), (3,7), (4,8), (5,9)

, giving

N = 15, 26, 37, 48,

or

59

.

Since multiple valid values of

N

exist, Statement (II) alone is NOT sufficient.

4

Evaluate Statement (I) and Statement (II) together.

System of linear equations:

x + y = 12

and

y - x = 4

. Adding the two equations gives

2y = 16 \implies y = 8

. Substituting

y = 8

gives

x = 4

. Thus,

N = 48

uniquely.

Both statements combined provide a unique value for

N

, making them together sufficient.

Key Concept

Data Sufficiency in Digit-Based Integer Problems

Estimated Time:1m 30s

Question 10Question →

Is the integer $x$ odd?

Statement (I): $x^2 + 3x$ is an even integer.
Statement (II): $x + 5$ is an even integer.

Which of the following options is correct?

Statement (I) alone is sufficient, but Statement (II) alone is not sufficient.

Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Either Statement (I) alone or Statement (II) alone is sufficient.

Neither Statement (I) nor Statement (II) is sufficient.

Both Statement (I) and Statement (II) together are sufficient, but neither statement alone is sufficient.

Show answer & explanation

Answer: Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Answer

Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Evaluating Statement (I):

x^2 + 3x = x(x+3)

. The product of any integer

x

and

(x+3)

is always even because one of the two numbers is always even. Thus, Statement (I) provides no specific information about whether

x

is odd or even, making it insufficient. Evaluating Statement (II):

x + 5 = \text{even}

. Subtracting the odd integer

5

from an even integer yields an odd integer, so

x

must be odd. Hence, Statement (II) alone is sufficient to answer the question definitively.

Step-by-Step Solution

1

Evaluate Statement (I) individually.

Rewrite the expression as

x^2 + 3x = x(x + 3)

. If

x

is even, then

x(x + 3) = \text{even} \times \text{odd} = \text{even}

. If

x

is odd, then

x(x + 3) = \text{odd} \times \text{even} = \text{even}

. Thus,

x^2 + 3x

is always even regardless of whether

x

is odd or even.

Since the statement holds true for all integers

x

, it cannot determine whether

x

is odd. Therefore, Statement (I) alone is NOT sufficient.

2

Evaluate Statement (II) individually.

The statement gives that

x + 5

is an even integer. Since

5

is an odd integer,

\text{Odd} + \text{Odd} = \text{Even}

, which implies

x

must be an odd integer.

This yields a definitive 'Yes' answer to the question 'Is

x

odd?'. Therefore, Statement (II) alone IS sufficient.

3

Conclude the data sufficiency evaluation.

Statement (II) alone is sufficient to answer the question, but Statement (I) alone is not sufficient.

Each statement was evaluated independently first, rendering statement combination unnecessary.

Key Concept

Data Sufficiency Parity Analysis

Estimated Time:1m 30s

Question 11Question →

A large water reservoir is equipped with two inlet pipes, Pipe $P$ and Pipe $Q$ , each filling the reservoir at its own constant rate. How many hours will it take to fill the empty reservoir if both pipes operate simultaneously from the start?

Statement (I): Pipe $P$ alone can fill the empty reservoir in $15\text{ hours}$ .
Statement (II): Pipe $Q$ fills the reservoir at a rate that is $50\%$ higher than the filling rate of Pipe $P$ .

Which of the following options correctly evaluates the sufficiency of the statements?

Statement (I) alone is sufficient to answer the question, while Statement (II) alone is not sufficient.

Statement (II) alone is sufficient to answer the question, while Statement (I) alone is not sufficient.

Either Statement (I) alone or Statement (II) alone is sufficient to answer the question.

Neither Statement (I) nor Statement (II) is sufficient, and even together they are not sufficient to answer the question.

Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Show answer & explanation

Answer: Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Answer

Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating Statement (I) alone gives only the individual performance of Pipe P, which is insufficient to determine the combined time. Statement (II) alone provides only a relative ratio between the rates of Pipe P and Pipe Q without any concrete time metric, making it insufficient on its own. When both statements are combined, Statement (I) provides the base rate for Pipe P and Statement (II) allows calculation of Pipe Q's rate, leading to a unique answer of 6 hours for the combined operation.

Step-by-Step Solution

1

Evaluate Statement (I) alone

Pipe

P

's rate is

\frac{1}{15}

of the reservoir per hour. However, no information is given regarding Pipe

Q

's rate.

Statement (I) alone is insufficient to calculate the combined time.

2

Evaluate Statement (II) alone

Pipe

Q

's rate is

1.5

times Pipe

P

's rate, meaning Rate(

Q

)

= 1.5 \times

Rate(

P

).

Statement (II) alone gives only a relative ratio of work rates, with no numerical time value given, so it is insufficient.

3

Evaluate Statements (I) and (II) together

From Statement (I), Rate(

P

)

= \frac{1}{15}

reservoir/hour. From Statement (II), Rate(

Q

)

= 1.5 \times \frac{1}{15} = \frac{1}{10}

reservoir/hour. Combined rate

= \frac{1}{15} + \frac{1}{10} = \frac{1}{6}

reservoir/hour. Thus, total combined time

= 6\text{ hours}

.

Combining both statements yields a unique and definitive answer.

Key Concept

Data Sufficiency in Work and Time / Rate Problems

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