Mindyard | A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: Is ( x + y ) an integer? Statement-I: ( 2 x + y ) is an integer. Statement-II: ( x + 2 y ) is an integer. Which one of the following is correct in respect of the above Question and the Statements? 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 the Statements together, but cannot be answered using either Statement alone, The Question cannot be answered even by using both the Statements together

A Question is given followed by two Statements I and II. Consider the Question and the Statements.

Question: Is $(x + y)$ an integer?

Statement-I: $(2 x + y)$ is an integer.

Statement-II: $(x + 2 y)$ is an integer.

Which one of the following is correct in respect of the above Question and the Statements?

Options

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 the Statements together, but cannot be answered using either Statement alone

The Question cannot be answered even by using both the Statements together

Correct Answer :

Option D

The Question cannot be answered even by using both the Statements together

Solution :

The correct option is: The Question cannot be answered even by using both the Statements together.

Let us analyze the problem step-by-step to understand why both statements together are insufficient to determine if $x + y$ is an integer.

Goal: We need to determine if $x + y$ is an integer based on the given statements.

Analyzing Statement-I:
Statement-I states that $2 x + y$ is an integer. Let $2 x + y = k_{1}$ , where $k_{1}$ is an integer.
By itself, this does not give us enough information to determine if $x + y$ is an integer. For example:
- If $x = 1$ and $y = 1$ , then $2 x + y = 3$ (an integer), and $x + y = 2$ (an integer).
- If $x = 0.5$ and $y = 1$ , then $2 x + y = 2$ (an integer), but $x + y = 1.5$ (not an integer).
Thus, Statement-I alone is not sufficient.

Analyzing Statement-II:
Statement-II states that $x + 2 y$ is an integer. Let $x + 2 y = k_{2}$ , where $k_{2}$ is an integer.
Similarly, this statement alone is not sufficient. For example:
- If $x = 1$ and $y = 1$ , then $x + 2 y = 3$ (an integer), and $x + y = 2$ (an integer).
- If $x = 1$ and $y = 0.5$ , then $x + 2 y = 2$ (an integer), but $x + y = 1.5$ (not an integer).
Thus, Statement-II alone is not sufficient.

Combining Statement-I and Statement-II:
Let us assume both statements are true. That is, both $2 x + y = k_{1}$ and $x + 2 y = k_{2}$ are integers.
If we add the two equations together, we get:

$(2 x + y) + (x + 2 y) = k_{1} + k_{2}$

$3 x + 3 y = k_{1} + k_{2}$

$3 (x + y) = k_{1} + k_{2}$

Since $k_{1}$ and $k_{2}$ are integers, their sum $k_{1} + k_{2}$ is also an integer, which we can call $M$ . Therefore:

$x + y = \frac{M}{3}$

This means that $x + y$ must be a multiple of $\frac{1}{3}$ . However, this does not guarantee that $x + y$ is an integer, because $M$ may or may not be divisible by 3.

Let us check with counterexamples:
Case 1: Let $x = \frac{1}{3}$ and $y = \frac{1}{3}$ .
- Statement-I: $2 x + y = 2 (\frac{1}{3}) + \frac{1}{3} = 1$ (which is an integer).
- Statement-II: $x + 2 y = \frac{1}{3} + 2 (\frac{1}{3}) = 1$ (which is an integer).
- Question: $x + y = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$ (which is not an integer).

Case 2: Let $x = 1$ and $y = 1$ .
- Statement-I: $2 x + y = 2 (1) + 1 = 3$ (which is an integer).
- Statement-II: $x + 2 y = 1 + 2 (1) = 3$ (which is an integer).
- Question: $x + y = 1 + 1 = 2$ (which is an integer).

Since both cases satisfy both statements, but lead to different answers for whether $x + y$ is an integer, we cannot determine a unique answer even by using both statements together.