This article is in continuation to our article on “How to solve data sufficiency questions?”. We recommend you to go through it first. In this article, we will take up 5 different questions and apply our standard 3-steps process to help you understand the scenarios in which you get the answer as Option A, B, C, D, or E.

Before scrolling down to the respective solution section, we advise you to solve each question on your own and see whether you are getting the same answer option or not?

## Practice Question 1: Option A

If x is a positive integer, what is the value of x?

(1) x^{3} = 27

(2) 4x + 3y = 15

### Solution –

#### Step 1: Analyse Question Stem

- x is a positive integer.
- x > 0 and

- x can be 1, 2, 3, 4…. and so on.

We need to find the value of x (or, say a **unique **value of x).

#### Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

**Statement 1**: *x ^{3} = 27*

- According to this statement,
*x*^{3}= 27- Taking the cube root on both sides of the above equation, we get,
- Or, x = 3

So, from statement 1 alone we can find the **unique** value of x. Hence, statement 1 is sufficient and we can eliminate answer options B, C and E.

**Statement 2**: *4x + 3y = 15*

- According to this statement,
*4x + 3y = 15*- Or,

- Here, we have 2 variables i.e. x and y and only one equation. No additional information is provided.
- So, y can be a positive integer, negative integer, or fraction.

- Out of the many possibilities that exist, let’s consider the following two simple cases:
- Case 1: if y is a positive integer.
- Let’s say y = 1, then

- Case 2: if y is a fraction.
- Let’s say ,then

- We can see that we are not getting a unique value of x from the above two cases.

- Case 1: if y is a positive integer.

Hence, statement 2 is NOT sufficient.

In this question, __statement (1) alone is sufficient, but statement (2) alone is not sufficient.__

Thus, the correct answer is __Option__**A**.

We further need not analyze the statements by combining them.

So, I hope, it’s clear, how we can get Option A as the answer in a DS question. Now, let’s move to the next question.

## Practice Question 2: Option B

If x is an integer, what is the value of x?

(1) x^{2} = 9

(2) 4x + 3y = 15, where x, and y are positive integers

### Solution

#### Step 1: Analyse Question Stem

- x is an integer.
- It means x can be -3, -2, -1, 0, 1, 2, 3… and so on.

We need to find a **unique** value of x.

#### Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

**Statement 1**: *x ^{2} = 9*

- From this statement,
- Or, x = -3 or 3

Here we are getting two different values of x. So, from statement 1 alone, we can’t get a unique value of x.

Hence, statement 1 is NOT sufficient and we can eliminate answer options A and D.

**Statement 2**: *4x + 3y = 15, where x, and y are positive integers.*

- According to this statement, 4x + 3y = 15 and
- x and y can be 1, 2, 3, 4, 5, …. and so on.

- Now, 4x + 3y = 15
- Or, , let us assume different values of y and see if we can get a unique value of x or not.
- Case 1: If y = 1 then , this value of x is possible.
- Case 2: If y = 2, then , x can’t be a fraction, so we can ignore this value of x.
- Case 3: If y = 3, then , x can’t be a fraction, so we can ignore this value of x.
- Case 4: If y =4, then , x can’t be a fraction, so we can ignore this value of x.
- Case 5: If y = 5, then , x can’t be 0. So we can ignore this value of x.
- If we further increase y, the resulting value of x will be negative which is not possible as x is a positive integer.

- Or, , let us assume different values of y and see if we can get a unique value of x or not.
- Hence, we can see that, for the given conditions, the only possible value of x is 3.

As we can get a unique value of x, hence statement 2 is sufficient.

In this question, __statement (1) alone is not sufficient, but statement (2) alone is sufficient.__

Thus, the correct answer is __Option B.__

We further need not analyse the statements by combining them. As you can see, while analysing statement 2, our focus was solely on that particular statement and we put all our efforts in trying to get the answer from that statement only. This is a very important aspect, some students end up considering Statement 1 while analysing Statement 2 and that’s a completely incorrect approach.

Now, let’s look at a case, where we actually need both the statements, to solve the question.

## Practice Question 3: Option C

If x is an integer, what is the value of x?

(1) x^{2} = 9

(2) 4x + 3y = 15, where x > 0

### Solution

#### Step 1: Analyse Question Stem

- x is an integer.
- It means x can be -3, -2, -1, 0, 1, 2, 3, …and so on.

We need to find a **unique** value of x.

#### Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

**Statement 1**: *x ^{2} = 9*

- From this statement,
- This means, x = -3 or 3

So, from statement 1 alone, we can’t get a unique value of x.

Hence, statement 1 is NOT sufficient and we can eliminate answer options A and D.

**Statement 2**: *4x + 3y = 15, where x > 0*

- According to this statement, 4x + 3y = 15 and x > 0
- Also, x is an integer.
- Now, 4x + 3y = 15 ⇒ ,

- In the above equation, we know that x is a positive integer, however we don’t know the nature of y i.e. whether y is a positive integer, negative integer or fraction? So, let us consider following three cases:
- Case 1: y is a positive integer,
- Let’s say y = 1 then =3,

- Case 2: y is a fraction.
- Let’s say then

- Case 3: y is a negative integer.
- Let’s say y = -3 then

- We can see that more than one values are possible for x.

- Case 1: y is a positive integer,

Hence, statement 2 is also not sufficient and we can eliminate answer Option B.

#### Step 3: Analyse Statements by combining.

- From statement 1: x = 3 or -3
- From statement 2: x > 0 and it can be 2, 3, 6, etc.
- On combining both the statements, we can see that only x = 3 is common in both the statements.
- Thus, x = 3

Hence, __BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.__

Thus, the correct answer is ** Option C**.

So, you can see, only when we don’t get a unique and definite answer from any statement, then ONLY we combine both the statements to get the answer.

## Practice Question 4: Option D

If x is a positive integer, what is the value of x?

(1) x^{3} = 27

(2) 4x + 3y = 15, where y is a positive integer

### Solution

#### Step 1: Analyse Question Stem

- x is a positive integer.
- x > 0 and
- x can be 1, 2, 3, 4,…and so on.

We need to find a **unique** value of x.

Let’s move to the next step.

#### Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

**Statement 1**:

- According to this statement,
- Taking the cube root on both sides of the above equation, we get,
- Or, x = 3

- Taking the cube root on both sides of the above equation, we get,

So, from statement 1 alone we can find a **unique** value of x.

Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E

**Statement 2**: *4x + 3y = 15, where y is a positive integer*

- According to this statement, 4x + 3y = 15, where y > 0 and it is an integer.
- Now, 4x + 3y = 15
- Or, , So, there can be multiple cases:
- Case 1: If y = 1 then = 3, this is possible
- Case 2: If y = 2, then , this is not possible as x can’t be a fraction.
- Case 3: If y = 3, then , this is not possible since x can’t be a fraction.
- Case 4: If y =4, then , we can ignore this value too, as x can’t be a fraction.
- Case 5: If y = 5, then , this is also not possible as x can’t be 0.
- If we further increase the value of y, x will become negative, which is not possible since x is a positive integer.

- Therefore, from the above cases, we can see that the only possible value of x is 3.

- Or, , So, there can be multiple cases:

Hence, statement 2 is also sufficient.

So, __EACH statement ALONE is sufficient.__

Thus, the correct answer is __Option D.__

Now, let’s come to the final question!!

## Practice Question 5: Option E

If x is an integer, what is the value of x?

(1) x^{2} = 9

(2) 4x + 3y = 15

### Solution

#### Step 1: Analyse Question Stem

- x is an integer.
- It means x can be -3, -2, -1, 0, 1, 2, 3, …and so on.

We need to find the **unique** value of x.

#### Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

**Statement 1**:

- According to this statement,
- Or, x = -3 or 3

So, from statement 1 alone, we can’t get a unique value of x.

Hence, statement 1 is not sufficient and we can eliminate answer Options A and D.

**Statement 2**: *4x + 3y = 15*

- According to this statement, 4x + 3y = 15
- Or,

- Here, we have 2 variables i.e. x and y and only one equation.
- No additional information about the nature of y is given i.e. whether y is a positive integer, negative integer or it is a fraction.
- So, let us consider one simple possibility that y is a positive integer. Thus, we will get many cases as y can be 1, 2, 3, 4… and so on. For example:
- Case 1: Let’s say y = 1, then , this is possible
- Case 2: Let’s say y = 4, then , this is not possible as x cannot be a fraction.
- Case 3: Let’s say y = 9, then this is possible.

- So, let us consider one simple possibility that y is a positive integer. Thus, we will get many cases as y can be 1, 2, 3, 4… and so on. For example:

- From the above cases we can see that we are getting different values of x i.e. x can be -3, 3, etc.

Hence, statement 2 is also NOT sufficient and we can eliminate answer B.

#### Step 3: Analyse Statements by combining.

- From statement 1: x = -3 or 3
- From statement 2: x can be 3, -3 etc.
- On combining both the statements, we get, x = -3 or 3

We can observe that even after combining the two statements, we are not getting a unique value of x.

Hence, __Statement (1) and (2) TOGETHER are NOT sufficient.__

Thus, the correct answer is __Option E.__

## The Conclusion

In this article, we focused on how we get an answer as A, B, C, D, or E in data sufficiency type questions. We solved 5 practice questions and learned how to apply the standard 3-steps process to solve DS questions and get the correct answer. You are recommended to solve more DS questions to perfectly understand this 3-step process.

Further, if you have noticed, in all 5 practice questions, you were asked to find a unique value of x. This is one type of DS question which you will get in GMAT. However, there is one more type of DS question. That’s why, to make you more familiar with DS questions and help you to master the DS questions, in our next article, we will discuss two types of DS questions and two common mistakes made by students. So, do read the next article.