Data sufficiency (DS) checks students ability to answer a question with the information provided in the question. Here a student is not expected to solve a question but to see whether the given information in the statements I and II are adequate to answer the given question or not. Therefore, these questions require maximum clarity of fundamentals.

Question: - - - - - - -

Statement I: - - - - - -

Statement II: - - - - - -

Direction: Each of the following problems consist of a question followed by two statements, I and II. You must determine whether the information given by the statements is sufficient to answer the question asked. In addition to the information provided in the statements, you should rely on your knowledge of mathematics and ordinary facts (such as the number of seconds in a minute).

(a) if the question can be answered by using only statement I

(b) if the question can be answered by using only statement II.

(c) if the question can be answered by using both statements together, but cannot be answered using either statement alone.

(d) if the question cannot be answered even by using both statements together.

(e) if the question can be answered either statements alone.

Data sufficiency questions are of two types.

Type 1 Questions: These questions start with "with what, who, how,... etc".

Type 2 Questions: These questions start with a helping verb like "is, does, are, etc". So answers for these questions are yes, no, or data insufficient.

Is Raja older than Ramu?

I. Sita is 4 years younger than Raja and 2 years younger than Ramu.

II. The average age Raja and Ramu is 21 years.

Answer: a

Explanation: This is a type 2 question. Let Sita age is 20. Then Raja age is 24, and Ramu age is 22. So clearly Raja is older than Ramu. So Statement I is sufficient to answer this question.

Statement II, however, gives only the average age but nothing about their individual ages.

The correct answer choice is (a).

What is the value of x?

I. ${x^2}$ = 16

II. x > 0

Answer: c

Explanation: From statement I we can see that x = +4 or x = –4. Now you may be tempted to consider this as a valid answer, as we have determined the value of x. But since this is not a unique answer, such an answer is not valid. So statement I alone is not sufficient to answer the question.

Statement II alone does not give a definite answer as there are infinite numbers greater than 0.

By combining both the statements we can conclude that x takes +4. So correct answer is c.

Is Babu older than Raghu?

I. The ratio of Raghu’s age to Babu’s age is 7 : 4.

II. In 6 years Raghu will be half as old as he is now.

Answer: a

If you think that both statements are necessary to answer this question, you fell into the trap. Read the question clearly. It only requires us to determine who is older but not asked us to find their ages. If the ratio of the their ages is 7 : 4, then Raghu, the 7 part, is older than Babu, the 4 part. So we can clearly state that Babu is not older than Raghu. It makes no difference that the answer to the question is No. A ‘no’ is as good as a ‘yes’.

Statement II alone is not sufficient. You can use it to find out how old Raghu is now, but you have no information about his age relative to Babu’s age.

So option a is correct.

What is probability of drawing a red ball, if a bag contains red, blue and green balls.

I. Total balls in the bag are 30

II. The balls are in the ratio 1:2:3

Answer:

Explanation: Here you may think that Statement I is not sufficient to answer as no details about the balls were present. After reading Statement II you may feel that, now I can calculate the number of balls of each color so Statement I and II are required. But to calculate probability we just need the ratio's of the balls. For example, from statement II, let the balls are x, 2x, 3x. Probability to draw a red ball = $\dfrac{x}{{x + 2x + 3x}} = \dfrac{x}{{6x}} = \dfrac{1}{6}$

So only statement II is enough. Choice (b).

What is the volume of a right circular cylinder?

I. The area of the base of the figure is 196.

II. The height of the figure is 16.

Answer: c

Explanation: The volume of a cylinder is the product of the area of the base and its height. i.e., $\pi {r^2}h$. You know the area of the base from statement I and the height from statement II. By multiplying 196 and 16 we get the required answer. But it is not necessary to multiply. So correct option is (c)

How old is Subhash?

I. In 16 more years Subhash will be twice as old as he is today.

II. Four years ago Subhash was

Answer: E

Explanation:

Once again you could go ahead and solve for Subhash’s age, but you do not have to and should not take the time to do so. Statement I alone is sufficient. Set up the equation x + 16 = 2x, where x stands for Subhash’s current age. On solving, we get x = 16. While it does not take long to set up the equation and prove the answer, why do so if you do not have to? Every second counts, and besides, you might set up the wrong equation, get a weird answer, and get frustrated. You waste even more time ‘trying to get it right’.

The same is true for statement II. You can set up the equation x – 4 = $\dfrac{3}{4}x$ . On solving, x = 16. As soon as you know you could solve the problem, you are through. The correct answer to this example is (e) because either statement I alone or statement II alone is sufficient.

If x, y and z are consecutive integers, is y even?

I. x < y < z

II. xz is odd integer.

Answer: (b)

Explanation:

Statement I is not sufficient to answer the question asked. Although statement I describes the order of the integers, it provides no information about which elements of the sequence are even and which are odd.

Statement II, however, is by itself sufficient to determine whether y is even or not. If xz is odd, then both x and z must be odd integers. In the series of 3 consecutive integers, at least one of the integers must be even. Therefore, y must be even. Since statement II alone is sufficient to answer the question asked, but statement I alone is not, the correct answer choice is (b).

Is r even?

I. r + s is odd.

II. s is even.

Answer: (c)

Statement I alone is not sufficient to answer the question. Knowing that r + s is odd will not tell you whether r is even. If s is even, then r is odd, because an even number and an odd number sum up to an odd number. If s is odd, then r is even, for the same reason. Put the two statements together. If s is even, and r + s is odd, then r must be odd. The answer to the question is no, but there is an answer to the problem. The answer to the question ‘Is r even?’ is ‘No, r is not even’. But because you can give an answer, even though that answer is negative, you do have enough data to answer the problem. So correct option is (c)

How many students are enrolled in Sastri's Quant class?

I. If 3 more students sign up for the class and no one drops out, more than 35 students will have enrolled in the class.

II. If 4 students drop out of the class and no more sign-up, fewer than 30 students will have enrolled in the class.

Answer: (c)

Statement I alone is not sufficient to answer the question asked, but I does imply that at least 33 students are enrolled in the class. Statement II alone is not sufficient to answer the question asked, but II does imply that no more than 33 students are enrolled in the class. Although neither statement alone is sufficient to answer the question, the two statements taken together are sufficient to answer the question that the number of students enrolled in the class is 33. Since neither statement alone is sufficient to answer the question but both together are sufficient, the correct answer choice is (c).

Is K a prime number?

I. K > 10

II. K divided by 2 has a remainder 0.

Answer: (c)

Explanation:

Statement I alone is not sufficient. Some numbers greater than 10 are prime, like 11. Some numbers greater than 10 are composite, like 12. Eliminate choice (A).

Statement II alone is not sufficient. When a number divided by 2 has no remainder, that number is even. You may be thinking that there are no even prime numbers and statement II alone is sufficient. But there is one even prime number, i.e. 2. Therefore, statement II alone is not sufficient because the answer could be yes or no. Eliminate choice (B).

At this point most students get lazy and choose (d). Put the two statements together. You know from statement II that the number must be even. You know from statement I that the number must be greater than 10. Because the only even prime number is 2, any even number greater than 2 must be composite, not prime. The answer to the question is ‘no’, c is not prime. Because you can answer the question based on both statements. The answer is (c).

**Question Format:**Question: - - - - - - -

Statement I: - - - - - -

Statement II: - - - - - -

Direction: Each of the following problems consist of a question followed by two statements, I and II. You must determine whether the information given by the statements is sufficient to answer the question asked. In addition to the information provided in the statements, you should rely on your knowledge of mathematics and ordinary facts (such as the number of seconds in a minute).

(a) if the question can be answered by using only statement I

(b) if the question can be answered by using only statement II.

(c) if the question can be answered by using both statements together, but cannot be answered using either statement alone.

(d) if the question cannot be answered even by using both statements together.

(e) if the question can be answered either statements alone.

**Two types of questions:**Data sufficiency questions are of two types.

Type 1 Questions: These questions start with "with what, who, how,... etc".

Type 2 Questions: These questions start with a helping verb like "is, does, are, etc". So answers for these questions are yes, no, or data insufficient.

**Solved Example 1:**Is Raja older than Ramu?

I. Sita is 4 years younger than Raja and 2 years younger than Ramu.

II. The average age Raja and Ramu is 21 years.

Answer: a

Explanation: This is a type 2 question. Let Sita age is 20. Then Raja age is 24, and Ramu age is 22. So clearly Raja is older than Ramu. So Statement I is sufficient to answer this question.

Statement II, however, gives only the average age but nothing about their individual ages.

The correct answer choice is (a).

**Solved Example 2:**What is the value of x?

I. ${x^2}$ = 16

II. x > 0

Answer: c

Explanation: From statement I we can see that x = +4 or x = –4. Now you may be tempted to consider this as a valid answer, as we have determined the value of x. But since this is not a unique answer, such an answer is not valid. So statement I alone is not sufficient to answer the question.

Statement II alone does not give a definite answer as there are infinite numbers greater than 0.

By combining both the statements we can conclude that x takes +4. So correct answer is c.

**Solved Example 3:**Is Babu older than Raghu?

I. The ratio of Raghu’s age to Babu’s age is 7 : 4.

II. In 6 years Raghu will be half as old as he is now.

Answer: a

If you think that both statements are necessary to answer this question, you fell into the trap. Read the question clearly. It only requires us to determine who is older but not asked us to find their ages. If the ratio of the their ages is 7 : 4, then Raghu, the 7 part, is older than Babu, the 4 part. So we can clearly state that Babu is not older than Raghu. It makes no difference that the answer to the question is No. A ‘no’ is as good as a ‘yes’.

Statement II alone is not sufficient. You can use it to find out how old Raghu is now, but you have no information about his age relative to Babu’s age.

So option a is correct.

**Solved Example 4:**What is probability of drawing a red ball, if a bag contains red, blue and green balls.

I. Total balls in the bag are 30

II. The balls are in the ratio 1:2:3

Answer:

Explanation: Here you may think that Statement I is not sufficient to answer as no details about the balls were present. After reading Statement II you may feel that, now I can calculate the number of balls of each color so Statement I and II are required. But to calculate probability we just need the ratio's of the balls. For example, from statement II, let the balls are x, 2x, 3x. Probability to draw a red ball = $\dfrac{x}{{x + 2x + 3x}} = \dfrac{x}{{6x}} = \dfrac{1}{6}$

So only statement II is enough. Choice (b).

**Solved Example 5:**What is the volume of a right circular cylinder?

I. The area of the base of the figure is 196.

II. The height of the figure is 16.

Answer: c

Explanation: The volume of a cylinder is the product of the area of the base and its height. i.e., $\pi {r^2}h$. You know the area of the base from statement I and the height from statement II. By multiplying 196 and 16 we get the required answer. But it is not necessary to multiply. So correct option is (c)

**Solved Example 6:**How old is Subhash?

I. In 16 more years Subhash will be twice as old as he is today.

II. Four years ago Subhash was

^{3}/_{4}times of his present age.Answer: E

Explanation:

Once again you could go ahead and solve for Subhash’s age, but you do not have to and should not take the time to do so. Statement I alone is sufficient. Set up the equation x + 16 = 2x, where x stands for Subhash’s current age. On solving, we get x = 16. While it does not take long to set up the equation and prove the answer, why do so if you do not have to? Every second counts, and besides, you might set up the wrong equation, get a weird answer, and get frustrated. You waste even more time ‘trying to get it right’.

The same is true for statement II. You can set up the equation x – 4 = $\dfrac{3}{4}x$ . On solving, x = 16. As soon as you know you could solve the problem, you are through. The correct answer to this example is (e) because either statement I alone or statement II alone is sufficient.

**Solved Example 7:**If x, y and z are consecutive integers, is y even?

I. x < y < z

II. xz is odd integer.

Answer: (b)

Explanation:

Statement I is not sufficient to answer the question asked. Although statement I describes the order of the integers, it provides no information about which elements of the sequence are even and which are odd.

Statement II, however, is by itself sufficient to determine whether y is even or not. If xz is odd, then both x and z must be odd integers. In the series of 3 consecutive integers, at least one of the integers must be even. Therefore, y must be even. Since statement II alone is sufficient to answer the question asked, but statement I alone is not, the correct answer choice is (b).

**Solved Example 8:**

I. r + s is odd.

II. s is even.

Answer: (c)

Statement I alone is not sufficient to answer the question. Knowing that r + s is odd will not tell you whether r is even. If s is even, then r is odd, because an even number and an odd number sum up to an odd number. If s is odd, then r is even, for the same reason. Put the two statements together. If s is even, and r + s is odd, then r must be odd. The answer to the question is no, but there is an answer to the problem. The answer to the question ‘Is r even?’ is ‘No, r is not even’. But because you can give an answer, even though that answer is negative, you do have enough data to answer the problem. So correct option is (c)

**Solved Example 9:**How many students are enrolled in Sastri's Quant class?

I. If 3 more students sign up for the class and no one drops out, more than 35 students will have enrolled in the class.

II. If 4 students drop out of the class and no more sign-up, fewer than 30 students will have enrolled in the class.

Answer: (c)

Statement I alone is not sufficient to answer the question asked, but I does imply that at least 33 students are enrolled in the class. Statement II alone is not sufficient to answer the question asked, but II does imply that no more than 33 students are enrolled in the class. Although neither statement alone is sufficient to answer the question, the two statements taken together are sufficient to answer the question that the number of students enrolled in the class is 33. Since neither statement alone is sufficient to answer the question but both together are sufficient, the correct answer choice is (c).

**Solved Example 10:**Is K a prime number?

I. K > 10

II. K divided by 2 has a remainder 0.

Answer: (c)

Explanation:

Statement I alone is not sufficient. Some numbers greater than 10 are prime, like 11. Some numbers greater than 10 are composite, like 12. Eliminate choice (A).

Statement II alone is not sufficient. When a number divided by 2 has no remainder, that number is even. You may be thinking that there are no even prime numbers and statement II alone is sufficient. But there is one even prime number, i.e. 2. Therefore, statement II alone is not sufficient because the answer could be yes or no. Eliminate choice (B).

At this point most students get lazy and choose (d). Put the two statements together. You know from statement II that the number must be even. You know from statement I that the number must be greater than 10. Because the only even prime number is 2, any even number greater than 2 must be composite, not prime. The answer to the question is ‘no’, c is not prime. Because you can answer the question based on both statements. The answer is (c).