CAT Data Sufficiency Questions & Solutions

A sample of real CAT Data Sufficiency past-year questions, each with a worked solution and the correct answer highlighted. AthenaPrep has 108 Data Sufficiency questions in all — sign up free to practise them timed.

Passage
In a single elimination tournament, any a player is eliminated with a single loss. The tournament is played in multiple rounds subject to the following rules :

(a) If the number of players, say n, in any round is even, then the players are grouped into n/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round.

(b) If the number of players, say n, in any round is odd, then one of them is given a bye, that is he automatically moves on to the next round. The remaining (n–1) players are grouped into (n–1)/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round. No player gets more than one bye in the entire tournament.

Thus, if n is even, then n/2 players move on to the next round while if n is odd, then (n+1)/2 players move on to the next round. The process is continued till the final round, which obviously is played between two players. The winner in the final round is the champion of the tournament.
Q1.CAT 2008
What is the number of Matches played by the champion?

A. The entry list for the tournament consists of 83 players?

B. The champion received one bye.
- Q can be answered from A alone but not from B alone.
- Q can be answered from B alone but not from A alone.
- Q can be answered from A alone as well as from B alone.
- Q can be answered from A and B together but not from any of them alone.
- Q cannot be answered even from A and B together.
Show solution
If we consider statement a then 2 possibilities occur , the champion may or may not receive bye and the answer for both possibilities would be different.
If we consider both statements a & b together we can find the exact answer to the question.
Passage
Directions for the following four questions: Each question is followed by two statements A and B. Indicate your responses based on data sufficiency
Q2.CAT 2007
Rahim plans to draw a square JKLM with a point O on the side JK but is not successful. Why is Rahim unable to draw the square?

A: The length of OM is twice that of OL.

B: The length of OM is 4 cm
- The question can be answered using A alone but not using B alone.
- The question can be answered using B alone but not using A alone.
- The question can be answered using A and B together, but not using either A or B alone.
- The question cannot be answered even using A and B together.
- None of these
Show solution
If the side of the square is x cm, then the maximum length of OM is $\sqrt2x$
The minimum length of OL is x.
So, OM can never be 2 times OL
So, using statement A alone, we can conclude that Rahim is unable to draw the square.
Using statement B alone, we cannot answer the question.
So, option a) is the correct answer.
Q3.CAT 2003
Each packet of SOAP costs Rs. 10. Inside each packet is a gift coupon labelled with one of the letters S, O, A and P. If a customer submits four such coupons that make up the word SOAP, the customer gets a free SOAP packets. Ms. X kept buying packet after packet of SOAP till she could get one set of coupons that formed the word SOAP. How many coupons with label P did she get in the above process?

A. The last label obtained by her was S and the total amount spent was Rs. 210.

B. The total number of vowels obtained was 18.
- The questions can be answered by one of the statements, but not by the other
- The question can be answered using either of the two statements alone.
- The questions can be answered using both the statements together, but cannot be answered using either statement alone.
- The question cannot be answered even by using both the statements together.
Show solution
Considering statements a we can deduce that she got letter S in her 21st purchase in previous 20 puchase she had either A, O or P . Hence statement a is not enough to answer. If we consider statement b we will know that out of 20 purchases 18 are vowel so 2 purchases contained letter P . So both statements are necessary to answer the question.
Q4.CAT 2002
Is $x = y$ ?

A. $(x+y)(1/x + 1/y) = 4$

B. $(x-50)^2 = (y-50)^2$
- The question can be answered by one of the statement alone but not by the other.
- The question can be answered by using either statement alone.
- The question can be answered by using both the statements together, but cannot be answered by using either statement alone.
- The question cannot be answered by either of the statements.
Show solution
Consider statement 1: $(x+y)(1/x + 1/y) = 4$

$(x+y)^2/xy = 4$

$x^2+y^2+2xy = 4xy$

$x^2+y^2+2xy-4xy = 0$

$(x-y)^2 = 0$

x = y.

Consider statement 2: $(x-50)^2 = (y-50)^2$

(x-y)(x+y-100) = 0

Either x = y or x+y = 100.

Statement 1 is sufficient whereas statement 2 is not sufficient to answer the question.
Q5.CAT 2001
On a given day a boat ferried 1,500 passengers across the river in 12 hr. How many round trips did it make?

I. The boat can carry 200 passengers at any time.

II. It takes 40 min each way and 20 min of waiting time at each terminal.
- The question can be answered by one of the statements alone and not by the other.
- The question can be answered by using either statement alone.
- The question can be answered by using both the statements together, but cannot be answered by using either statement alone.
- The question cannot be answered even by using both statements together.
Show solution
The question asked is about the number of round trips made in 12 hours. This can be answered if we know the time of travel and the time of waiting. So, the question can be answered by using statement 2 alone but not by using statement 1 alone.
Q6.CAT 2000
For any two real numbers:

a + b = 1 if both a and b are positive or both a and b are negative.
a + b = -1 if one of the two numbers a and b is positive and the other negative.

What is (2 + 0) + (-5 + -6)?
A. a + b is zero if a is zero
B. a + b = b + a
- the question can be answered by one of. the statements alone, but cannot be answered by using the other statement alone.
- the question can be answered by using either statement alone.
- the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
- the question cannot be answered even by using both statements together.
Show solution
We have (2 + 0) + (-5 + -6)

Now using statement B it can be written as (0 + 2) + (-5 + -6).

Using trhe statement A we can get the answer to the required question.

Hence,the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
Passage
Each question is followed by two statements, A and B. Choose the option based on the given statements and questions
Q7.CAT 1999
There is a circle with centre C at the origin and radius r cm. Two tangents are drawn from an external point D at a distance d cm from the centre. What are the angles between each tangent and the X-axis?
A.The coordinates of D are given
B.The X-axis bisects one of the tangents.
- The question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
- The question can be answered by using either statement alone.
- The question can be answered by using both statements together, but cannot be answered using either statement alone.
- The question cannot be answered even by using both statements together.
Show solution
From statement I, We can draw the following figure
From the figure we can say that in triangle GCD we know 2 sides GC and CD, so we can find the third side
Also we can find the angle GDC(Using trigonometric ratios)

Similarly in triangle CDE, we know CE and DE, so we can find angle CDE.
From this we can get the angle FDE. Now in right angled triangle FDE, we can find the angle DFE.
So we can answer from statement I
Now let us consider statement II
By using trigonometric relations angle CFG can be calculated.
From this we can get angle DFE
Iin triangle CGD we can find angles GDC(Using trigonometric ratios)
Now GDC = CDJ
Thus we can find EDH and from that we can find EHD.
Hence the question can be answered from statement II alone.
Q8.CAT 1998
Radha and Rani appeared in an examination. What was the total number of questions?

I. Radha and Rani together solved 20% of the paper.

II. Radha alone solved 5/3 rd of the paper solved by Rani.
- The question can be answered with the help of any one statement alone but not by the other statement.
- The question can be answered with the help of either of the statements taken individually.
- The question can be answered with the help of both statements together.
- The question cannot be answered even with the help of both statements together.
Show solution
Let the total questions be x and the questions solved by Radha and Rani be y and z respectively.

According to the first statement, y+z = 20% of x

Accoding to the second statement, y = 5/3rd of z

There are 2 equations and three variables. Hence the solution cannot be determined.
Q9.CAT 1997
Is the number completely divisible by 99?
I. The number is divisible by 9 and 11 simultaneously.
II. If the digits of the number are reversed, the number is divisible by 9 and 11.
- The question can be answered with the help of one statement alone.
- The question can be answered with the help of any one statement independently.
- The question can be answered with the help of both statements together.
- The question cannot be answered even with the help of both statements together.
Show solution
99 = 9 * 11 => If the number is divisible by 9 and 11, then the number is divisible by 99.

=> We can answer the question using only statement I.

The divisibility rule of 9 is that the sum of the digits in the number must be divisible by 9.

The divisibility rule of 11 is that the difference of sums of digits in odd places and digits in even places must be a multiple of 11.

According to statement II, if the number is reversed then the number is divisible by 9 and 11 => Divisible by 99.

Now, on reversing the number again, the number is still divible by 99 because the sum of the digits remains the same and also the difference of sums of digits in odd places and digits in even places remains the same.

Hence, the question can be answered using either of the statements alone.
Q10.CAT 1996
What is the number of type-2 widgets produced, if the total number of widgets produced is 20,000? The company produces only two types of widgets.
I. If the production of type-1 widgets increases by 10% and that of type-2 decreases by 6%, the total production remains the same.
II. The ratio in which type-1 and type-2 widgets are produced is 2 : 1.
- if the question cannot be answered even with the help of both the statements taken together.
- if the question can be answered by any one of the two statements.
- if each statement alone is sufficient to answer the question, but not the other one (e.g. statement I alone is required to answer the question, but not statement II and vice versa.
- if both statements I and II together are needed to answer the question.
Show solution
Considering 1st statement alone:
When total number of widgets is 20000
let's say share of type-1 = x
and share of type-2 = y
So x+y = 20000
and 1.1x+0.94y = 20000
Hence we can find value of x and y by statement 1 alone.
Now considering second statement alone;
$x+y = 20000$
$\frac{x}{y} = \frac{2}{1}$
Hence with second statement also, we can find value of x and y
So answer will be B.
Passage
Each question is followed by two statements, I and II. Mark the answer

Q11.CAT 1996
How old is Sachin in 1997?
I. Sachin is 11 years younger than Anil whose age will be a prime number in 1998.
II. Anil's age was a prime number in 1996.
- if the question cannot be answered even with the help of both the statements taken together.
- if the question can be answered by any one of the two statements.
- if each statement alone is sufficient to answer the question, but not the other one (e.g. statement I alone is required to answer the question, but not statement II and vice versa.
- if both statements I and II together are needed to answer the question.
Show solution
Anil's age is a prime number in both 1996 and 1998. Hence, Anil's age in 1996 and 1998 are twin primes ( Prime numbers that differ by 2). Anil's age in 1996 and 1998 can be (11,13),(17,19) ,(29,31),.......and so on. Since a unique set cannot be determined, option A is the right answer.
Q12.CAT 1991
Mr. Murthy takes the morning train to his office from station A to station B, and his colleague Mr.Rahman joins him on the way. There are three stations C, D and E on the way not necessarily in that sequence. What is the sequence of stations?
I. Mr. Rahman boards the train at D.
II. Mr. Thomas, who travels between C & D has two segments of journey in common with Mr. Murthy but none with Mr. Rahman.
- The question can be answered with the help of statement I alone.
- The question can be answered with the help of statement II, alone.
- Both, statement I and statement II are needed to answer the question
- The question cannot be answered even with the help of both the statements.
Show solution
Considering first statement and second statement alone, information will not suffice to find the sequence of stations,
But if we consider both statements together we can tell about the sequence of stations as between A and B, Mr. Thomas travels with Mr. Murthy but not with Mr. Rahman. That means Mr. Thomas must have departed from station C and arrived at station D.
And it is also given that between C and D there are two segments. Hence, E lies between C and D, and Mr. Rahman departed from station D.

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CAT Data Sufficiency Questions & Solutions

Passage

In a single elimination tournament, any a player is eliminated with a single loss. The tournament is played in multiple rounds subject to the following rules :

(a) If the number of players, say n, in any round is even, then the players are grouped into n/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round.

(b) If the number of players, say n, in any round is odd, then one of them is given a bye, that is he automatically moves on to the next round. The remaining (n–1) players are grouped into (n–1)/2 pairs. The players in each pair play a match against each other and the winner moves on to the next round. No player gets more than one bye in the entire tournament.

Thus, if n is even, then n/2 players move on to the next round while if n is odd, then (n+1)/2 players move on to the next round. The process is continued till the final round, which obviously is played between two players. The winner in the final round is the champion of the tournament.

Q1.CAT 2008

What is the number of Matches played by the champion?

A. The entry list for the tournament consists of 83 players?

B. The champion received one bye.

Q can be answered from A alone but not from B alone.
Q can be answered from B alone but not from A alone.
Q can be answered from A alone as well as from B alone.
Q can be answered from A and B together but not from any of them alone.
Q cannot be answered even from A and B together.

Show solution

If we consider statement a then 2 possibilities occur , the champion may or may not receive bye and the answer for both possibilities would be different.
If we consider both statements a & b together we can find the exact answer to the question.

Passage

Directions for the following four questions: Each question is followed by two statements A and B. Indicate your responses based on data sufficiency

Q2.CAT 2007

Rahim plans to draw a square JKLM with a point O on the side JK but is not successful. Why is Rahim unable to draw the square?

A: The length of OM is twice that of OL.

B: The length of OM is 4 cm

The question can be answered using A alone but not using B alone.
The question can be answered using B alone but not using A alone.
The question can be answered using A and B together, but not using either A or B alone.
The question cannot be answered even using A and B together.
None of these

Show solution

If the side of the square is x cm, then the maximum length of OM is $\sqrt2x$
The minimum length of OL is x.
So, OM can never be 2 times OL
So, using statement A alone, we can conclude that Rahim is unable to draw the square.
Using statement B alone, we cannot answer the question.
So, option a) is the correct answer.

Q3.CAT 2003

Each packet of SOAP costs Rs. 10. Inside each packet is a gift coupon labelled with one of the letters S, O, A and P. If a customer submits four such coupons that make up the word SOAP, the customer gets a free SOAP packets. Ms. X kept buying packet after packet of SOAP till she could get one set of coupons that formed the word SOAP. How many coupons with label P did she get in the above process?

A. The last label obtained by her was S and the total amount spent was Rs. 210.

B. The total number of vowels obtained was 18.

The questions can be answered by one of the statements, but not by the other
The question can be answered using either of the two statements alone.
The questions can be answered using both the statements together, but cannot be answered using either statement alone.
The question cannot be answered even by using both the statements together.

Show solution

Considering statements a we can deduce that she got letter S in her 21st purchase in previous 20 puchase she had either A, O or P . Hence statement a is not enough to answer. If we consider statement b we will know that out of 20 purchases 18 are vowel so 2 purchases contained letter P . So both statements are necessary to answer the question.

Q4.CAT 2002

Is $x = y$ ?

A. $(x+y)(1/x + 1/y) = 4$

B. $(x-50)^2 = (y-50)^2$

The question can be answered by one of the statement alone but not by the other.
The question can be answered by using either statement alone.
The question can be answered by using both the statements together, but cannot be answered by using either statement alone.
The question cannot be answered by either of the statements.

Show solution

Consider statement 1: $(x+y)(1/x + 1/y) = 4$

$(x+y)^2/xy = 4$

$x^2+y^2+2xy = 4xy$

$x^2+y^2+2xy-4xy = 0$

$(x-y)^2 = 0$

x = y.

Consider statement 2: $(x-50)^2 = (y-50)^2$

(x-y)(x+y-100) = 0

Either x = y or x+y = 100.

Statement 1 is sufficient whereas statement 2 is not sufficient to answer the question.

Q5.CAT 2001

On a given day a boat ferried 1,500 passengers across the river in 12 hr. How many round trips did it make?

I. The boat can carry 200 passengers at any time.

II. It takes 40 min each way and 20 min of waiting time at each terminal.

The question can be answered by one of the statements alone and not by the other.
The question can be answered by using either statement alone.
The question can be answered by using both the statements together, but cannot be answered by using either statement alone.
The question cannot be answered even by using both statements together.

Show solution

The question asked is about the number of round trips made in 12 hours. This can be answered if we know the time of travel and the time of waiting. So, the question can be answered by using statement 2 alone but not by using statement 1 alone.

Q6.CAT 2000

For any two real numbers:

a + b = 1 if both a and b are positive or both a and b are negative.
a + b = -1 if one of the two numbers a and b is positive and the other negative.

What is (2 + 0) + (-5 + -6)?
A. a + b is zero if a is zero
B. a + b = b + a

the question can be answered by one of. the statements alone, but cannot be answered by using the other statement alone.
the question can be answered by using either statement alone.
the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
the question cannot be answered even by using both statements together.

Show solution

We have (2 + 0) + (-5 + -6)

Now using statement B it can be written as (0 + 2) + (-5 + -6).

Using trhe statement A we can get the answer to the required question.

Hence,the question can be answered by using both the statements together, but cannot be answered by using either statement alone.

Passage

Each question is followed by two statements, A and B. Choose the option based on the given statements and questions

Q7.CAT 1999

There is a circle with centre C at the origin and radius r cm. Two tangents are drawn from an external point D at a distance d cm from the centre. What are the angles between each tangent and the X-axis?

A.The coordinates of D are given

B.The X-axis bisects one of the tangents.

The question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
The question can be answered by using either statement alone.
The question can be answered by using both statements together, but cannot be answered using either statement alone.
The question cannot be answered even by using both statements together.

Show solution

From statement I, We can draw the following figure

From the figure we can say that in triangle GCD we know 2 sides GC and CD, so we can find the third side
Also we can find the angle GDC(Using trigonometric ratios)

Similarly in triangle CDE, we know CE and DE, so we can find angle CDE.
From this we can get the angle FDE. Now in right angled triangle FDE, we can find the angle DFE.
So we can answer from statement I
Now let us consider statement II

By using trigonometric relations angle CFG can be calculated.
From this we can get angle DFE
Iin triangle CGD we can find angles GDC(Using trigonometric ratios)
Now GDC = CDJ
Thus we can find EDH and from that we can find EHD.
Hence the question can be answered from statement II alone.

Q8.CAT 1998

Radha and Rani appeared in an examination. What was the total number of questions?

I. Radha and Rani together solved 20% of the paper.

II. Radha alone solved 5/3 rd of the paper solved by Rani.

The question can be answered with the help of any one statement alone but not by the other statement.
The question can be answered with the help of either of the statements taken individually.
The question can be answered with the help of both statements together.
The question cannot be answered even with the help of both statements together.

Show solution

Let the total questions be x and the questions solved by Radha and Rani be y and z respectively.

According to the first statement, y+z = 20% of x

Accoding to the second statement, y = 5/3rd of z

There are 2 equations and three variables. Hence the solution cannot be determined.

Q9.CAT 1997

Is the number completely divisible by 99?
I. The number is divisible by 9 and 11 simultaneously.
II. If the digits of the number are reversed, the number is divisible by 9 and 11.

The question can be answered with the help of one statement alone.
The question can be answered with the help of any one statement independently.
The question can be answered with the help of both statements together.
The question cannot be answered even with the help of both statements together.

Show solution

99 = 9 * 11 => If the number is divisible by 9 and 11, then the number is divisible by 99.

=> We can answer the question using only statement I.

The divisibility rule of 9 is that the sum of the digits in the number must be divisible by 9.

The divisibility rule of 11 is that the difference of sums of digits in odd places and digits in even places must be a multiple of 11.

According to statement II, if the number is reversed then the number is divisible by 9 and 11 => Divisible by 99.

Now, on reversing the number again, the number is still divible by 99 because the sum of the digits remains the same and also the difference of sums of digits in odd places and digits in even places remains the same.

Hence, the question can be answered using either of the statements alone.

Q10.CAT 1996

What is the number of type-2 widgets produced, if the total number of widgets produced is 20,000? The company produces only two types of widgets.
I. If the production of type-1 widgets increases by 10% and that of type-2 decreases by 6%, the total production remains the same.
II. The ratio in which type-1 and type-2 widgets are produced is 2 : 1.

if the question cannot be answered even with the help of both the statements taken together.
if the question can be answered by any one of the two statements.
if each statement alone is sufficient to answer the question, but not the other one (e.g. statement I alone is required to answer the question, but not statement II and vice versa.
if both statements I and II together are needed to answer the question.

Show solution

Considering 1st statement alone:
When total number of widgets is 20000
let's say share of type-1 = x
and share of type-2 = y
So x+y = 20000
and 1.1x+0.94y = 20000
Hence we can find value of x and y by statement 1 alone.
Now considering second statement alone;
$x+y = 20000$
$\frac{x}{y} = \frac{2}{1}$
Hence with second statement also, we can find value of x and y
So answer will be B.

Passage

Each question is followed by two statements, I and II. Mark the answer

Q11.CAT 1996

How old is Sachin in 1997?
I. Sachin is 11 years younger than Anil whose age will be a prime number in 1998.
II. Anil's age was a prime number in 1996.

if the question cannot be answered even with the help of both the statements taken together.
if the question can be answered by any one of the two statements.
if each statement alone is sufficient to answer the question, but not the other one (e.g. statement I alone is required to answer the question, but not statement II and vice versa.
if both statements I and II together are needed to answer the question.

Show solution

Anil's age is a prime number in both 1996 and 1998. Hence, Anil's age in 1996 and 1998 are twin primes ( Prime numbers that differ by 2). Anil's age in 1996 and 1998 can be (11,13),(17,19) ,(29,31),.......and so on. Since a unique set cannot be determined, option A is the right answer.

Q12.CAT 1991

Mr. Murthy takes the morning train to his office from station A to station B, and his colleague Mr.Rahman joins him on the way. There are three stations C, D and E on the way not necessarily in that sequence. What is the sequence of stations?
I. Mr. Rahman boards the train at D.
II. Mr. Thomas, who travels between C & D has two segments of journey in common with Mr. Murthy but none with Mr. Rahman.

The question can be answered with the help of statement I alone.
The question can be answered with the help of statement II, alone.
Both, statement I and statement II are needed to answer the question
The question cannot be answered even with the help of both the statements.

Show solution

Considering first statement and second statement alone, information will not suffice to find the sequence of stations,
But if we consider both statements together we can tell about the sequence of stations as between A and B, Mr. Thomas travels with Mr. Murthy but not with Mr. Rahman. That means Mr. Thomas must have departed from station C and arrived at station D.
And it is also given that between C and D there are two segments. Hence, E lies between C and D, and Mr. Rahman departed from station D.

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