Data Sufficiency • Single and two-statement questions
Data Sufficiency: Full-Length Notes
Topics Covered:
Single and Two-Statement Questions
1. What is Data Sufficiency?
Data Sufficiency is a reasoning and quantitative aptitude topic that tests your ability to determine whether the information provided is enough to answer a given question—not to actually solve it, but to judge if it can be solved with the given data1245. This skill is vital for exams because it checks both your conceptual understanding and your analytical ability.
2. Structure of Data Sufficiency Questions
A question is asked.
One or more statements (usually two) are provided.
You must decide if the statements, alone or together, are sufficient to answer the question.
You do NOT need to actually solve the problem—just say if it can be solved with the information given1245.
3. Standard Answer Choices
Most exams use these five options (though the wording may vary):
A. The data in statement I alone is sufficient, but statement II alone is not sufficient.
B. The data in statement II alone is sufficient, but statement I alone is not sufficient.
C. The data in either statement I alone or statement II alone is sufficient.
D. The data in both statements together is sufficient, but neither statement alone is sufficient.
E. The data in both statements together is not sufficient.
4. Types of Data Sufficiency Questions
A. Single-Statement Questions
Only one statement is provided.
You must determine if that statement alone is enough to answer the question.
Example:
Q: Is x an even number?
Statement: x is divisible by 2.
Analysis: Yes, this is sufficient.
B. Two-Statement Questions
Two statements (I and II) are provided.
You must check each statement individually and then together.
Example:
Q: What is the value of x?
I. x + 3 = 7
II. 2x = 8
Analysis:
Statement I: x = 4 (sufficient)
Statement II: x = 4 (sufficient)
Either alone is sufficient.
5. Step-by-Step Approach
Step 1: Understand the Question
What is being asked? (A value, a relationship, a property, etc.)
Step 2: Analyze Each Statement Separately
Check if statement I alone gives a unique answer.
Check if statement II alone gives a unique answer.
Step 3: Analyze Both Statements Together
If neither alone is sufficient, check if using both together is sufficient.
Step 4: Choose the Correct Option
Use the standard answer choices.
6. Key Rules & Tips
Do not make assumptions beyond the data given.
Do not solve the question—just check if it can be solved.
If a statement gives multiple possible answers, it is NOT sufficient.
If a statement gives a unique answer, it IS sufficient.
If neither alone is sufficient, but both together are, choose the appropriate option.
If even both together are not sufficient, mark as “not sufficient.”
7. Examples
Example 1: (Quantitative)
Q: What is the value of x?
I. x + 3 = 8
II. 2x = 10
I alone: x = 5 (sufficient)
II alone: x = 5 (sufficient)
Answer: Either alone is sufficient (Option C).
Example 2: (Logical Reasoning)
Q: Who is the tallest among A, B, and C?
I. A is taller than B.
II. C is taller than A.
I alone: Only A > B; C’s height unknown.
II alone: Only C > A; B’s height unknown.
Both together: C > A > B, so C is tallest.
Answer: Both statements together are sufficient (Option D).
Example 3: (Ranking)
Q: What is Ramesh’s rank from the top in a class of 30?
I. Ramesh is 10th from the bottom.
II. Only 19 students are above Ramesh.
I alone: 30 - 10 + 1 = 21 (rank from top = 21). Sufficient.
II alone: 19 above means Ramesh is 20th from top. Sufficient.
Answer: Each alone is sufficient (Option C).
Example 4: (Blood Relation)
Q: How is A related to B?
I. A is the son of C, who is the mother of B.
II. B is the daughter of C.
I alone: A is C’s son, C is B’s mother, so A is B’s brother. Sufficient.
II alone: Only B is C’s daughter; nothing about A. Not sufficient.
Answer: Statement I alone is sufficient (Option A).
Example 5: (Age)
Q: What is the age of Rahul?
I. Rahul is 5 years younger than his brother, who is 20.
II. Rahul’s mother is 25 years older than Rahul.
I alone: Rahul = 15 (sufficient).
II alone: Only gives relation, not value (not sufficient).
Answer: Statement I alone is sufficient (Option A).
8. Common Mistakes to Avoid
Guessing without checking each statement.
Assuming data not given.
Not checking if both statements together are needed.
Overlooking the possibility that either statement alone could be sufficient.
9. Practice Tips
Practice questions from all reasoning and quantitative topics, as data sufficiency can be based on any concept1245.
Always use the step-by-step approach: check I, check II, check both together.
Read options carefully; sometimes wording varies between exams.
10. Summary Table: Standard Answer Choices
| Option | Meaning |
|---|---|
| A | Statement I alone is sufficient, II alone is not |
| B | Statement II alone is sufficient, I alone is not |
| C | Either statement alone is sufficient |
| D | Both statements together are sufficient, neither alone is sufficient |
| E | Both statements together are not sufficient |
References:
[Testbook: Data Sufficiency Reasoning]1
[BYJU’S: Data Sufficiency]2
[PrepInsta: Data Sufficiency]4
[Hitbullseye: Data Sufficiency]5
Mastering data sufficiency is about knowing when you have enough information—not about solving the problem!
1.
Q: What is the value of x?
I. x + 2y = 6
II. 3x + 6y = 18
Explanation:
Each statement alone gives one equation in two variables, not sufficient.
Both together: Second is just 3 times the first, so still only one equation.
Answer: 5 (Not sufficient even together)4.
2.
Q: Is x an even number?
I. x is divisible by 2.
II. x is a multiple of 4.
Explanation:
I alone: Yes, x is even.
II alone: All multiples of 4 are even.
Answer: 3 (Either alone is sufficient).
3.
Q: What is the value of y?
I. y² = 25
II. y > 0
Explanation:
I alone: y = 5 or -5.
II alone: No info about y.
Together: y² = 25 and y > 0 ⇒ y = 5.
Answer: 4 (Both together sufficient).
4.
Q: Is number n a prime?
I. n is odd.
II. n has only two positive divisors.
Explanation:
I alone: Not sufficient; odd numbers can be composite.
II alone: This is the definition of a prime number.
Answer: 2 (II alone sufficient).
5.
Q: What is the sum of two numbers a and b?
I. a - b = 3
II. a = 7
Explanation:
I alone: Not sufficient.
II alone: Not sufficient.
Together: a = 7, so 7 - b = 3 ⇒ b = 4, so sum = 11.
Answer: 4 (Both together sufficient).
6.
Q: Who is the tallest among A, B, C?
I. A is taller than B.
II. C is taller than A.
Explanation:
I alone: Only A > B.
II alone: Only C > A.
Together: C > A > B, so C is tallest.
Answer: 4 (Both together sufficient).
7.
Q: What is the average of X, Y, and Z?
I. Y’s salary is half of (X + Z).
II. X and Y together earn Rs. 40 more than Z, and Z earns Rs. 500.
Explanation:
I alone: Not sufficient.
II alone: X + Y = 540, Z = 500, so total = 1040, average = 1040/3.
Answer: 2 (II alone sufficient)4.
8.
Q: Is (X – 5) even? X is a real number.
I. X – 15 is an integer.
II. X – 10 is an odd integer.
Explanation:
I alone: X is integer, but (X – 5) could be odd/even.
II alone: X = odd + 10 = odd + even = odd, so (X – 5) = odd – 5 = even.
Answer: 2 (II alone sufficient)4.
9.
Q: What is the value of X + Y?
I. X – 2Y = 5
II. X² – 25 = 4XY – 4Y²
Explanation:
I alone: Not sufficient.
II alone: Not sufficient.
Together: II can be rewritten as (X – 2Y)² = 25, so X – 2Y = ±5, but only one value from I, so X – 2Y = 5. Still, two variables, one equation.
Answer: 5 (Not sufficient even together)4.
10.
Q: Who is the wife of Z?
I. H is the only daughter of X. K is the paternal uncle of X.
II. K is the brother-in-law of X.
III. K and Z are brothers.
Explanation:
Only with all three statements together can you establish the relation.
Answer: All three together are sufficient6.
11.
Q: What is Monica’s position with respect to Rahul in a row of 25 students?
I. Monica is 12th from right end, Rahul is 20th from left.
II. Monica is 4th from right, Rahul is 8th from left.
Explanation:
I alone: With total, can find relative position.
II alone: Without total, cannot.
Answer: 1 (I alone sufficient)5.
12.
Q: Who has secured the least marks among P, Q, R, S, T?
I. S has less marks than only R and T.
II. Q has more marks than P.
Explanation:
I alone: S is 3rd from top, so P or Q is at the bottom.
II alone: Only Q > P.
Together: P has least marks.
Answer: 1 (I alone sufficient)5.
13.
Q: What is the length of the shortest side of a triangle with sides 25 cm and 24 cm?
I. Angles are in the ratio 1:2:3.
II. The length of the perpendicular from the opposite vertex to the longest side is 6.72 cm.
Explanation:
Neither alone is sufficient; both together still not enough without more info.
Answer: 5 (Not sufficient even together)4.
14.
Q: What is the value of x?
I. x + 2y = 6
II. 3x + 6y = 18
Explanation:
Both equations are multiples; only one equation in two variables.
Answer: 5 (Not sufficient even together)4.
15.
Q: What is the present age of Rahul?
I. Rahul is 25 years younger than his mother.
II. Rahul’s mother was born in 1970.
Explanation:
I alone: Not sufficient.
II alone: Not sufficient.
Together: If current year is known, can find mother’s age, then Rahul’s.
Answer: 5 (Not sufficient even together; need current year)1.
16.
Q: What is the total weight of 10 identical poles?
I. Each pole weighs 15 kg.
II. The combined weight of 5 poles is 75 kg.
Explanation:
I alone: 10 × 15 = 150 kg.
II alone: 5 poles = 75 kg ⇒ 1 pole = 15 kg, so 10 × 15 = 150 kg.
Answer: 3 (Either alone sufficient)1.
17.
Q: What is the value of x?
I. x² = 16
II. x is positive.
Explanation:
I alone: x = 4 or -4.
II alone: Not sufficient.
Together: x = 4.
Answer: 4 (Both together sufficient).
18.
Q: Is y an integer?
I. y² is an integer.
II. y³ is an integer.
Explanation:
I alone: y could be irrational (e.g., √2) and y² = 2.
II alone: y could be irrational.
Together: Still possible for y to be irrational.
Answer: 5 (Not sufficient).
19.
Q: Is n a multiple of 5?
I. n ends with 0.
II. n is divisible by 10.
Explanation:
I alone: Yes, ends with 0 ⇒ multiple of 5.
II alone: Yes, divisible by 10 ⇒ divisible by 5.
Answer: 3 (Either alone sufficient).
20.
Q: What is the area of a rectangle?
I. Length is 10 cm.
II. Breadth is 5 cm.
Explanation:
I alone: Not sufficient.
II alone: Not sufficient.
Together: Area = 10 × 5 = 50 cm².
Answer: 4 (Both together sufficient).
21.
Q: Is x > y?
I. x = y + 1
II. x² > y²
Explanation:
I alone: x > y.
II alone: Not always (x = -2, y = 1 ⇒ 4 > 1, but x < y).
Answer: 1 (I alone sufficient).
22.
Q: Is the triangle equilateral?
I. All angles are equal.
II. All sides are equal.
Explanation:
Either alone is sufficient.
Answer: 3 (Either alone sufficient).
23.
Q: What is the value of a three-digit number N?
I. The sum of its digits is 12.
II. The number is divisible by 6.
Explanation:
Neither alone nor together is sufficient; many numbers fit.
Answer: 5 (Not sufficient).
24.
Q: Is A the mother of B?
I. A is the wife of C, who is the father of B.
II. B is the son of C.
Explanation:
I alone: Yes, A is B’s mother.
II alone: No info about A.
Answer: 1 (I alone sufficient).
25.
Q: What is the value of x?
I. 2x + 3 = 11
II. x² = 16
Explanation:
I alone: x = 4
II alone: x = 4 or -4
Answer: 1 (I alone sufficient).
26.
Q: Is p an odd number?
I. p² is odd.
II. p is not divisible by 2.
Explanation:
I alone: Yes, only odd numbers have odd squares.
II alone: Yes, not divisible by 2 ⇒ odd.
Answer: 3 (Either alone sufficient).
27.
Q: Is the number 153 divisible by 9?
I. The sum of its digits is 9.
II. 153 divided by 9 gives an integer.
Explanation:
I alone: 1 + 5 + 3 = 9 ⇒ divisible by 9.
II alone: Yes.
Answer: 3 (Either alone sufficient).
28.
Q: What is the perimeter of a square?
I. The length of one side is 6 cm.
II. The area is 36 cm².
Explanation:
I alone: 4 × 6 = 24 cm.
II alone: √36 = 6, so perimeter = 24 cm.
Answer: 3 (Either alone sufficient).
29.
Q: Is the number x a perfect square?
I. x is an integer.
II. √x is an integer.
Explanation:
I alone: Not sufficient.
II alone: Yes, definition of perfect square.
Answer: 2 (II alone sufficient).
30.
Q: Is y a negative number?
I. y² > 0
II. y < 0
Explanation:
I alone: Not sufficient (could be positive or negative).
II alone: Yes.
Answer: 2 (II alone sufficient).
31.
Q: What is the value of z?
I. z³ = 27
II. z is positive.
Explanation:
I alone: z = 3 or -3
II alone: Not sufficient.
Together: z = 3
Answer: 4 (Both together sufficient).
32.
Q: Is n even?
I. n = 2k, where k is an integer.
II. n is divisible by 4.
Explanation:
I alone: Yes, definition of even.
II alone: Yes, all multiples of 4 are even.
Answer: 3 (Either alone sufficient).
33.
Q: What is the sum of two consecutive integers?
I. The first integer is x.
II. The second integer is x + 1.
Explanation:
Either alone is not sufficient; together: sum = x + x + 1 = 2x + 1
But x is not given, so not sufficient.
Answer: 5 (Not sufficient).
34.
Q: Is the triangle right-angled?
I. One angle is 90°.
II. The sides are 3, 4, 5.
Explanation:
I alone: Yes.
II alone: Yes (3² + 4² = 5²).
Answer: 3 (Either alone sufficient).
35.
Q: What is the median of the set {2, 4, 6, 8, x}?
I. x = 10
II. The mean is 6
Explanation:
I alone: Set is {2, 4, 6, 8, 10} ⇒ median is 6.
II alone: 2 + 4 + 6 + 8 + x = 30 ⇒ x = 10, so median is 6.
Answer: 3 (Either alone sufficient).
36.
Q: Who is older, R or S?
I. R is older than T.
II. T is older than S.
Explanation:
Together: R > T > S ⇒ R is older than S.
Answer: 4 (Both together sufficient).
37.
Q: What is the value of x?
I. x is a prime number between 10 and 20.
II. x is even.
Explanation:
I alone: x could be 11, 13, 17, 19.
II alone: x could be any even.
Together: Only 2 primes between 10 and 20 are even: only 2, but not in range. So, no such x.
Answer: 5 (Not sufficient).
38.
Q: What is the area of a circle?
I. The radius is 7 cm.
II. The diameter is 14 cm.
Explanation:
Either alone is sufficient.
Answer: 3 (Either alone sufficient).
39.
Q: Is x > 0?
I. x² = 25
II. x is positive.
Explanation:
I alone: x = 5 or -5.
II alone: Yes.
Together: x = 5.
Answer: 2 (II alone sufficient).
40.
Q: What is the value of y?
I. y + 3 = 10
II. y² = 49
Explanation:
I alone: y = 7
II alone: y = 7 or -7
Answer: 1 (I alone sufficient).
41.
Q: Is n a multiple of 3?
I. n is divisible by 6.
II. n is even.
Explanation:
I alone: Yes, divisible by 6 ⇒ divisible by 3.
II alone: Not sufficient.
Answer: 1 (I alone sufficient).
42.
Q: What is the value of p + q?
I. p = 2q
II. q = 5
Explanation:
Both together: p = 10, q = 5, so sum = 15.
Answer: 4 (Both together sufficient).
43.
Q: Is the number x divisible by 2?
I. x is divisible by 4.
II. x is odd.
Explanation:
I alone: Yes.
II alone: No.
Answer: 1 (I alone sufficient).
44.
Q: What is the greatest among a, b, c?
I. a > b
II. b > c
Explanation:
Together: a > b > c ⇒ a is greatest.
Answer: 4 (Both together sufficient).
45.
Q: Is the sum of two numbers even?
I. Both numbers are odd.
II. Both numbers are even.
Explanation:
Either alone: Odd + odd = even; even + even = even.
Answer: 3 (Either alone sufficient).
46.
Q: What is the value of x?
I. 2x = 8
II. x² = 16
Explanation:
I alone: x = 4
II alone: x = 4 or -4
Answer: 1 (I alone sufficient).
47.
Q: Is y negative?
I. y² = 9
II. y < 0
Explanation:
I alone: y = 3 or -3
II alone: Yes
Together: y = -3
Answer: 2 (II alone sufficient).
48.
Q: What is the value of a three-digit number N?
I. The sum of its digits is 6.
II. The number is less than 200.
Explanation:
Neither alone nor together is sufficient.
Answer: 5 (Not sufficient).
49.
Q: What is the value of z?
I. z is an integer between 1 and 10.
II. z is a prime number.
Explanation:
Together: z could be 2, 3, 5, 7.
Answer: 5 (Not sufficient).
50.
Q: Is x a positive integer?
I. x > 0
II. x is an integer.
Explanation:
Together: Yes.
Answer: 4 (Both together sufficient).
For more practice and detailed solutions, see resources at IndiaBIX1, Hitbullseye25, Smartkeeda3, Testbook48, and BYJU’S6. If you need full step-by-step solutions for any specific question, just ask!
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