As the name suggest, Data Sufficiency is a typical set of problems asked not only in CAT but in other national and international exams. The conceptual knowledge and skills required to solve Data Sufficiency problems are far easier than that required to solve standard Problem Solving questions. Fundamental knowledge of the principles of arithmetic, algebra and geometry is sufficient. Then why these questions exist??? The data sufficiency questions are designed to test our reasoning and logical ability, more than the mathematical skills.
i.e. If someone asks what is 2x+11. We can’t tell as we do not know the value of x.
Now, someone tells us that x is a positive integer <2.
Can you tell now? Yes
Yes because x is now 1 and 2x +11=13.
General Pattern of Data Sufficiency Problems:
A Data Sufficiency problem consists of a mathematical or logical problem followed by two statements containing information relating to it. The students must decide whether the problem can be solved by using the information from the given statement combined or individually. Choose your answer as 1, 2, 3 or 4 as per the directions given below.
Directions: Each question is followed by two statements, I and II. Answer each question using the following instructions:
Choose 1: if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2: if the question can be answered by using either of the statements alone.
Choose 3: if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 4: if the question cannot be answered even by using both statements together.
In Recent Past, with five options given in most of the exams, options 1 have been broken into two parts.
Now the options look like
Choose 1 if the question can be answered by using statement A alone but not by Statement B alone.
Choose 2 if the question can be answered by using statement B alone but not by Statement A alone.
Choose 3 if the question can be answered by using either of the statements alone.
Choose 4 if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 5 if the question cannot be answered even by using both statements together.
Let us see some examples to understand the above in depth.
[Solved Example]
Example 1:
Directions: The question below is followed by two statements, I and II. Answer each question using the following instructions:
Choose 1 if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 4 if the question cannot be answered even by using both statements together.
Question : Is x-y positive?
Statement 1: x is a natural number
Statement 2: y is negative integer
Solution:
Statement 1 tells us that x is a natural number but doesn’t tell us anything about y so you can’t say about x-y.
Statement 2 tells that y is a negative integer but doesn’t tell you anything about x so you can’t say about x-y.
Now, to check for option 3, we can combine the information from both statements. x is a natural number so x=1,2,3,4-------- whereas y=-1,-2,-3-------
And x-y would always be positive so you can tell x-y is always positive with confidence. Hence Choose 3.
[Points to Remember]
Under time constraints, while analyzing the statements a student normally makes the mistake of carrying over information from one statement to another. Remember data sufficiency is designed to test your patience. You should not jump to a conclusion or make any assumption regarding data. You only know what is written in the question. So while analyzing statement 2 alone, you have to assume that you haven’t read Statement 1 yet.
[/Points to Remember]
Now, let us use the same example with only one variation.
[Solved Example]
Example 2:
Directions: The question below is followed by two statements, I and II. Answer each question using the following instructions:
Choose 1 if the question can be answered by using one of the statements alone,but cannot be answered using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 4 if the question cannot be answered even by using both statements together.
Question: is x-y positive?
Statement 1: x is a negative integer.
Statement 2: y is a natural number.
Solution:
Statement 1 tells us that x is a negative integer but doesn’t tell us anything about y so you can’t say about x-y.
Statement 2 tells that y is a natural number but doesn’t tell you anything about x so you can’t say about x-y.
Now, to check for option 3, we can combine the information from both statements. x is a negative integer so x=-1,-2,-3------- where as y=1,2,3,4------
And x-y would always be negative so you can tell x-y is always negative with confidence.
So which option should you choose? Students make mistake here by choosing 4 because you think x-y should be positive. No, the question is
Is x-y positive and the examiner is looking for a definite answer. In Example 1, it was ‘Yes’ and in Example 2 it is ‘No’. So ‘No’ is also an answer and hence option 3 is right answer even in Example 2.
Now, let us modify the above problem to see a different variation.
[Solved Example]
Example 3:
Directions: The question below is followed by two statements, I and II. Answer each question using the following instructions:
Choose 1 if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 4 if the question cannot be answered even by using both statements together.
Question: Is x-y positive?
Statement 1: x is a integer.
Statement 2: y is a natural number.
Statement 1 tells us that x is an integer but doesn’t tell us anything about y so you can’t say about x-y.
Statement 2 tells that y is a natural number but doesn’t tell you anything about x so you can’t say about x-y.
Now, to check for option 3, we can combine the information from both statements. x is a integer so x=-------, -3, -2, -1, 0, 1, 2, 3------- where as y=1,2,3,4------
Now, depending upon the value that x or y pick in one instance, x-y can be positive or negative so you can’t tell x-y is positive or negative in general.
In this case, the correct option would be 4. So, remember that for a question which is not expecting one definitive answer, even the presence of 2 solutions doesn’t mean that you have got the answer.
[/Solved Example]
Note: In first 3 examples, if there would have been 5 options, nothing to worry. May be the correct options would have been 4, 4 and 5 or something else depending upon the order of option.
The above 3 examples shall be summarised in form of some basic rules while solving DS questions.
Rules to Solve DS Questions
[Points to Remember]
Rule 1. Even ‘no’ can be the answer.
[/Points to Remember]
[Solved Example]
Example 4:
Directions: The question below is followed by two statements, I and II. Answer the question using the following instructions:
Choose 1: if the question can be answered by using statement A alone but not by Statement B alone.
Choose 2: if the question can be answered by using statement B alone but not by Statement A alone.
Choose 3: if the question can be answered by using either of the statements alone.
Choose 4: if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 5: if the question cannot be answered even by using both statements together.
Question: x is a positive integer. Is x=2?
Statement 1: x2-16=0
Statement 2: x is the smallest odd prime number.
Solution:
As per Statement 1, x=4 or -4. Combining with basic data, x is 4. So x is not equal to 2 and you can say with 100% confidence.
Statement 2 tells you that x=3. So x is not equal to 2 and you can say with 100% confidence.
So which option would you choose. Students make mistake to choose 5 as they think that none of the statements gave a x=2. But that was not the question. The examiner asked is x=2 and ‘No’ is also a definite and unique answer.
[/Solved Example]
[Points to Remember]
Rule 2. If a question asks for a numerical value, the question is answerable only if a statement provides data with which one can arrive at a unique value and not a range of values.
[/Points to Remember]
[Solved Example]
Example 5:
Directions: The question below is followed by two statements, I and II. Answer the question using the following instructions:
Choose 1: if the question can be answered by using statement A alone but not by Statement B alone.
Choose 2: if the question can be answered by using statement B alone but not by Statement A alone.
Choose 3: if the question can be answered by using either of the statements alone.
Choose 4: if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 5: if the question cannot be answered even by using both statements together.
Question : x is an integer. What is the value of x
1. x>0
2. x2-25<0
Solution:
As per Statement 1, x is any positive integer so nothing can be found out
From Statement 2, -5< x < 5 again x can be multiple values.
So the correct option is (5). Students think that as they have found a set of values for x, it is fine. No, you were supposed to find only one definite value of x.
[/Solved Example]
[Points to Remember]
Rule 3. Though you are not supposed to assume any data not given, the basic formulas of Algebra, Geometry, and Arithmetic can be used to solve questions.
[/Points to Remember]
[Solved Example]
Example6:
Directions: The question below is followed by two statements, I and II. Answer the question using the following instructions:
Choose 1: if the question can be answered by using statement A alone but not by Statement B alone.
Choose 2: if the question can be answered by using statement B alone but not by Statement A alone.
Choose 3: if the question can be answered by using either of the statements alone.
Choose 4: if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 5: if the question cannot be answered even by using both statements together.
Question: In a triangle ABC, Angle (B)=900 . Fine length ( AC)
Statement 1: length ( AB)= 6
Statement 2: length( BC)=8
Combining the two statements and using Pythagoras Theorem, AC=10
Hence, Option 4 is correct.
[/Solved Example]
[PointstoRemember]
Rule 4. The solution to a question can be different from 2 statements but unique in both cases. In this case, both the options are right.
[/Points to Remember]
[Solved Example]
Example:7
Directions: The question below is followed by two statements, I and II. Answer the question using the following instructions:
Choose 1 if the question can be answered by using statement A alone but not by Statement B alone.
Choose 2 if the question can be answered by using statement B alone but not by Statement A alone.
Choose 3 if the question can be answered by using either of the statements alone.
Choose 4 if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 5 if the question cannot be answered even by using both statements together.
Question: What is the value of x if x2-5x+6=0?
Statement 1.x>2
Statement 2.x<3
Solution:
x2-5x+6=0 is a quadratic equation whose roots are 2 and 3
Now from Statement 1, x>2 hence x=3 so you get a unique value of x=3 thus Statement 1 alone is sufficient
From Statement 2, x<3 hence x=2 so you get a unique value of x=3 thus Statement 2 alone is sufficient
As you can see, both Statements 1 and 2 are individually sufficient to solve the question though you get different values of x from both cases, so the right option is 3.
[/Solved Example]
[Points to Remember]
Rule 5: Generally, you are not supposed to do very long calculations in DS questions. If you find caught yourself into one, you are on wrong track.
[/Points to Remember]
[Solved Example]
Example:8
Directions: The question below is followed by two statements, I and II. Answer the question using the following instructions:
Choose 1 if the question can be answered by using statement A alone but not by Statement B alone.
Choose 2 if the question can be answered by using statement B alone but not by Statement A alone.
Choose 3 if the question can be answered by using either of the statements alone.
Choose 4 if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 5 if the question cannot be answered even by using both statements together.
Question: x is a positive integer. Find the value of x13
Statement 1: x is an even number
Statement 2 : x is a prime number.
Now, from Statement 1, x can be 2,4,6,8 and so on...hence x13 can take different values
From Statement 2, x can be 2,3,5,7.... and hence x13 can take different values
Combining the information of 2 statements with basic data, x=2
So we need to find the value of 213.
Now, many students start finding this value. But do you really need to? As soon as you come to last step, you know that the value of x can be found out with 100% confidence using both statements hence the correct option is 4. Why and where do we need to write the value of 213 ?
[/Solved Example]
Let us see some more examples;
[Solved Example]
Example:9
Directions: The question below is followed by two statements, I and II. Answer the question using the following instructions:
Choose 1 if the question can be answered by using statement A alone but not by Statement B alone.
Choose 2 if the question can be answered by using statement B alone but not by Statement A alone.
Choose 3 if the question can be answered by using either of the statements alone.
Choose 4 if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 5 if the question cannot be answered even by using both statements together.
Q. In an election, does Candidate A get the majority of the votes?
I. Candidate B got 49% of the votes.
II.Candidate A got twice as many votes as B and the difference between their votes was 30%
of the total votes.
Solution:
Statement I: Candidate B got 49% of the votes. But there can be a candidate C with 50.9% votes . Hence, Candidate A may or may not have got the majority of the votes. Hence, statement I alone is not sufficient.
Statement II: A = 2B and A – B = 30% of the total.
2B – B = B 30% of the total.
A = 2B = 60% of the total votes.
Hence, statement II alone is sufficient to answer the question.
Ans= (2)
[/Solved Example]
[Solved Example]
Example 10:
Directions: The question below is followed by two statements, I and II. Answer the question using the following instructions:
Choose 1 if the question can be answered by using statement A alone but not by Statement B alone.
Choose 2 if the question can be answered by using statement B alone but not by Statement A alone.
Choose 3 if the question can be answered by using either of the statements alone.
Choose 4 if the question can be answered by using statements together, but cannot be answered using either of the statements alone.
Choose 5 if the question cannot be answered even by using both statements together.
Q. Tarak is standing 2 steps to the left of a red mark and 3 steps to the right of a blue mark. He tosses a coin. If it comes up head, he moves one step to the right; otherwise he moves one step to the left. He keeps doing this until he reaches one of the two marks, and then he stops. At which mark does he stop?
I. He stops after 21 coin tosses.
II. He obtained three more tails than heads.
Solution: As Tarak is standing 2 steps to the left of a red mark and 3 steps to the right of a
blue mark, he would reach Red mark if he takes even steps and blue if he takes odd steps.(This is the concept that you should know from Number Systems, please see yourself why this would happen)
From Statement 1, He takes 21 steps that are odd. So by taking 12 steps to the left and 9 to the right, he would stop at blue mark. (No other combination is possible if he has to stop after 21 steps). So Statement 1 alone is sufficient.
From Statement 2, as he obtained 3 more tails than heads, there are n different combinations possible . But in any case, he can stop at blue only, so you are sure that he would stop at blue only.
Hence, both the statements are individually sufficient.
Ans=(3)
[/Solved Example]
Now let us solve similar questions which appeared in CAT.
Each question is followed by two statements, A and B. Answer each question using the following instructions: (CAT 2008)
Mark (1) if the question can be answered by using statement A alone but not by using statement B alone.
Mark (2) if the question can be answered by using statement B alone but not by using statement A alone.
Mark (3) if the question can be answered by using either statement alone.
Mark (4) if the question can be answered by using both the statements together but not by either of the statements alone.
Mark (5) if the question cannot be answered on the basis of the two statements
In a single elimination tournament, any 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 in to 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.
Q 11. 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.
a) 1
b) 2
c) 3
d) 4
e) 5
Solution: (d)
If we consider statement A alone, the entry list consists of 83 players. In round 1, one of the 83 players gets a bye and directly moves to the next round. Thus, 42 players move to round 2. Similarly, 21 players move to round 3, 11 to round 4, 6 to round 5, 3 to round 6 and 2 to round 7. The winner of the tournament would have played one match in each of the rounds i.e. 7 matches, provided he does not get a bye. Since we do not know whether the champion got a bye or not, we cannot answer on the basis of statement A alone.
From statement B, the champion received one bye. We cannot determine the number of matches played by him. So, statement B alone is not sufficient.
From A and B together, the player got one bye, so he must have played only 6 matches. So, we can answer the question by using both statements A and B.
Q12. If the number of players, say n, in the first round was between 65 and 128, then what is the exact value of n?
A. Exactly one player received a bye in the entire tournament.
B. One player received a bye while moving on to the fourth round from the third round
a) 1
b) 2
c) 3
d) 4
e) 5
Solution: (d)
From statement A alone, exactly one player received a bye in the tournament. We get many values of n between 65 and 128 that satisfy this condition. E.g. n can have value 124 in round 1 to follow the pattern – 124, 62, 31, 16, 8, 4, 2, 1
N can also have the value 127 following the pattern – 127, 64, 32, 16, 8 , 4, 2, 1
Thus, statement A alone is not sufficient.
From B, one player received a bye while moving from round 3 to 4. Here also we can have multiple values of n. E.g. n can have value 124 in round 1 where 1 player received a bye while moving from round 3 to 4 – 124, 62, 31, 16, 8, 4, 2, 1
Also, n can have value 122 in round 1 – 122, 61, 31, 16, 8, 4, 2, 1
Thus, statement B alone is not sufficient.
Taking A and B together, n can only have the value 124 in round 1, where exactly one player received a bye while moving from round 3 to 4 following the pattern – 124, 62, 31, 16, 8, 4, 2, 1
Thus, we can answer the question using both A and B together.
The following question appeared in (CAT 2000)
Choose 1; if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2; if the question can be answered by using either statement alone.
Choose 3; if the question can be answered by using both statements together, but cannot be answered using either statement alone.
Choose 4; if the question cannot be answered even by using both statements together
Q 13. Consider three real numbers, X, Y and Z. Is Z the smallest of these numbers?
A. X is greater than at least one of Y and Z.
B. Y is greater than at least one of X and Z.
a) 1
b) 2
c) 3
d) 4
Solution: (c)
From A, there are 4 possibilities:
X>Y>Z
X>Z>Y
Z>X>Y
Y>X>Z
Thus, A alone is not sufficient.
From B, there are 4 possibilities:
Y>X>Z
Y>Z>X
X>Y>Z
Z>Y>X
Thus, B alone is not sufficient.
Taking A and B together, we get
Either X>Y>Z or Y>X>Z
In both cases, Z is smallest, hence option (c)
Q 14. Let X be a real number. Is the modulus of X necessarily less than 3?
A. X(X + 3) < 0
B. X(X – 3) > 0
a) 1
b) 2
c) 3
d) 4
Solution: (a)
X(X+3) < 0
=> -3
=> |X| < 3
=> A alone is sufficient
X(X-3) > 0
=> X < 0 or X > 3
=> No unique answer
Hence, option (a)
Q 15. How many people are watching TV programme P?
A. Number of people watching TV programme Q is 1000 and number of people watching both the programmes, P and Q, is 100.
B. Number of people watching either P or Q or both is 1500.
a) 1
b) 2
c) 3
d) 4
Solution: (c)
From A, Q = 1000
P
Q = 100This does not give any information regarding how many people are watching TV.
From B, P U Q = 1500
This statement alone is not sufficient.
Combining A and B, we have
P U Q = P + Q - P
Q1500 = P + 1000 – 100
P = 600, hence option (c)
Q 16. Triangle PQR has angle PRQ equal to 90 degrees. What is the value of PR + RQ?
A. Diameter of the inscribed circle of the triangle PQR is equal to 10 cm.
B. Diameter of the circumscribed circle of the triangle PQR is equal to 18 cm
a) 1
b) 2
c) 3
d) 4
Solution: (c)
From A, PF = PR – FR = PR – OD = PR – 5
QD = QR – DR = QR – OF = QR – 5
=> PE = PF and QE = QD
=> PE = PR – 5 and QE = QR – 5
=> PQ = PE + QE = PR – 5 + QR – 5
=> PR + QR = PQ + 10
=> This statement alone is not sufficient.
From B, diameter of circumscribing circle = hypotenuse
=> PQ – 18
=> PR2 + RQ2 = PQ2
But this does not give PR + RQ
Combining both statements,
PR + QR = PQ + 10 = 18+10 = 28
Hence, option (c)
Q 17. Harshad bought shares of a company on a certain day, and sold them the next day. While buying and selling he had to pay to the broker one percent of the transaction value of the shares as brokerage. What was the profit earned by him per rupee spent on buying the shares?
A. The sales price per share was 1.05 times that of its purchase price.
B. The number of shares purchased was 100.
a) 1
b) 2
c) 3
d) 4
Solution: (a)
From A, cost of buying shares for Harshad is CP + 0.01CP = 1.01CP
Cost of selling shares for Harshad is SP – 0.01 SP = 0.99SP
Profit = cost of selling – cost of buying = 0.99SP – 1.01 CP
= 0.99x1.05CP – 1.01CP
= 0.0295CP
Thus, profit earned per rupee spent on buying shares is 0.0295
From B, we cannot conclude anything, hence option (a).
Q 18. For any two real numbers
a
b = 1 if both a and b are positive or both a and b are negative.= –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 
aa) 1
b) 2
c) 3
d) 4
Solution: (c)
= 0, hence both statements are required.
Q 19. There are two straight lines in the x-y plane with equations ax + by = c , dx + ey = f. Do the two straight lines intersect?
A. a, b, c, d, e and f are distinct real numbers.
B. c and f are non-zero
a) 1
b) 2
c) 3
d) 4
Solution: (d)
Statement A implies that a, b, c, d, e, f are real distinct numbers.
But if a/d = b/e = c/f then lines might be parallel or might not intersect at all.
Thus, A alone is not sufficient.
B implies that equations are not homogenous.
Thus, B is also not sufficient.
Condition for intersecting lines is a/d
b/eEven after combining, the condition is not clear, hence option (d).
Q 20. O is the centre of two concentric circles. ae is a chord of the outer circle and it intersects the inner circle at point; b and d. c is a point on the chord in between b and d. What is the value ofac/ce?
A. bc/cd = 1
B. A third circle intersects the inner circle at b and d and the point c is on the line joining the centres of the third circle and the inner circle
a) 1
b) 2
c) 3
d) 4
Solution: (b)
From A, bc = cd
If c is the mid point of bd, it would also be the mid point of ae as the circles are concentric.
Thus, ac = ae
Thus, A alone is sufficient.
From B, if c is the mid point on the line joining the two centres, it has to bisect chord bd.
Thus, c will also bisect the chord ae as circles are concentric.
=> ac = ce
=> B alone is sufficient, hence option (b).
Q 21. Ghosh Babu has decided to take a non-stop flight from Mumbai to No-man’s-land in South America. He is scheduled to leave Mumbai at 5 am, Indian Standard Time on December 10, 2000. What is the local time at No-man’s-land when he reaches there?
A. The average speed of the plane is 700 kilometres per hour.
B. The flight distance is 10,500 kilometres
a) 1
b) 2
c) 3
d) 4
Solution: (d)
By using both statements together, we can get the duration of the flight, but for the arrival time we require time zone difference between Mumbai and no man’s land.
Hence, option (d).
Q 22. What are the ages of two individuals, X and Y?
A. The age difference between them is 6 years.
B. The product of their ages is divisible by 6.
a) 1
b) 2
c) 3
d) 4
Solution: (d)
From A, X – Y = 6
From B, XY is divisible by 6
Thus many values are possible like (12, 6), (18, 12), (24, 18)
Thus the question cannot be answered even by using the statements together. Hence option (d).
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