Data Sufficiency Analysis: Is (x+y) an Integer?

This question asks whether the expression \( (x+y) \) is an integer. We are given two statements involving \( x \) and \( y \), and we need to determine if either statement alone or both statements together are sufficient to answer the question.

Understanding the Question and Statements

• Question: Is \( (x+y) \) an integer? We need a definite YES or NO answer.
• Statement I: \( (2x+y) \) is an integer. Let's represent this as \( 2x+y = k \), where \( k \) is some integer.
• Statement II: \( (x+2y) \) is an integer. Let's represent this as \( x+2y = m \), where \( m \) is some integer.

Analyzing Statement I Alone

Statement I tells us that \( 2x+y \) is an integer. Can this alone tell us if \( x+y \) is an integer?

Let's test some possible values for \( x \) and \( y \) such that \( 2x+y \) is an integer:

• If \( x=0.5 \) and \( y=1 \), then \( 2x+y = 2(0.5) + 1 = 1 + 1 = 2 \). Since 2 is an integer, this satisfies Statement I. In this case, \( x+y = 0.5 + 1 = 1.5 \), which is NOT an integer.
• If \( x=1 \) and \( y=0 \), then \( 2x+y = 2(1) + 0 = 2 \). Since 2 is an integer, this satisfies Statement I. In this case, \( x+y = 1 + 0 = 1 \), which IS an integer.

Since we can find cases where \( 2x+y \) is an integer, but \( x+y \) is sometimes an integer and sometimes not, Statement I alone is not sufficient to answer the question.

Analyzing Statement II Alone

Statement II tells us that \( x+2y \) is an integer. Can this alone tell us if \( x+y \) is an integer?

Let's test some possible values for \( x \) and \( y \) such that \( x+2y \) is an integer:

• If \( x=1 \) and \( y=0.5 \), then \( x+2y = 1 + 2(0.5) = 1 + 1 = 2 \). Since 2 is an integer, this satisfies Statement II. In this case, \( x+y = 1 + 0.5 = 1.5 \), which is NOT an integer.
• If \( x=0 \) and \( y=1 \), then \( x+2y = 0 + 2(1) = 2 \). Since 2 is an integer, this satisfies Statement II. In this case, \( x+y = 0 + 1 = 1 \), which IS an integer.

Since we can find cases where \( x+2y \) is an integer, but \( x+y \) is sometimes an integer and sometimes not, Statement II alone is not sufficient to answer the question.

Analyzing Both Statements Together

Now, let's assume both Statement I and Statement II are true. We have the following system of equations:

\( 2x+y = k \) (where \( k \) is an integer)

\( x+2y = m \) (where \( m \) is an integer)

We want to know if \( x+y \) is an integer. Let's try to combine the two equations to isolate or find information about \( x+y \).

If we add the two equations, we get:

\( (2x+y) + (x+2y) = k + m \)

\( 3x + 3y = k + m \)

\( 3(x+y) = k + m \)

\( x+y = \frac{k+m}{3} \)

Since \( k \) and \( m \) are integers, \( k+m \) is also an integer. However, for \( x+y \) to be an integer, \( k+m \) must be a multiple of 3.

Let's test if \( k+m \) is always a multiple of 3 based on the statements:

• Consider the case where \( x = \frac{1}{3} \) and \( y = \frac{1}{3} \).
  • Statement I: \( 2x+y = 2(\frac{1}{3}) + \frac{1}{3} = \frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1 \). This is an integer (\( k=1 \)).
  • Statement II: \( x+2y = \frac{1}{3} + 2(\frac{1}{3}) = \frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1 \). This is an integer (\( m=1 \)).

  Both statements are satisfied. In this case, \( x+y = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \), which is NOT an integer. Note that \( k+m = 1+1=2 \), which is not a multiple of 3.

• Consider the case where \( x = \frac{2}{3} \) and \( y = \frac{1}{3} \).
  • Statement I: \( 2x+y = 2(\frac{2}{3}) + \frac{1}{3} = \frac{4}{3} + \frac{1}{3} = \frac{5}{3} \). This is NOT an integer. This combination of \( x,y \) does not satisfy Statement I.

  (Let's try a different example where \( x+y \) is an integer)

• Consider the case where \( x = \frac{1}{3} \) and \( y = \frac{2}{3} \).
  • Statement I: \( 2x+y = 2(\frac{1}{3}) + \frac{2}{3} = \frac{2}{3} + \frac{2}{3} = \frac{4}{3} \). Not an integer.
• Consider the case where \( x = 1 \) and \( y = 0 \).
  • Statement I: \( 2x+y = 2(1)+0 = 2 \). Integer (\( k=2 \)).
  • Statement II: \( x+2y = 1+2(0) = 1 \). Integer (\( m=1 \)).

  Both statements are satisfied. In this case, \( x+y = 1+0 = 1 \), which IS an integer. Note that \( k+m = 2+1=3 \), which is a multiple of 3.

We have found scenarios where both statements are true, but \( x+y \) is not an integer (e.g., \( x=1/3, y=1/3 \Rightarrow x+y=2/3 \)) and scenarios where both statements are true and \( x+y \) is an integer (e.g., \( x=1, y=0 \Rightarrow x+y=1 \)).

Since we cannot get a definite YES or NO answer for the question "Is \( (x+y) \) an integer?" even by using both statements together, the information is not sufficient.

Conclusion

Based on the analysis:

• Statement I alone is not sufficient.
• Statement II alone is not sufficient.
• Both Statement I and Statement II together are not sufficient.

Therefore, the question cannot be answered even by using both statements together.

This aligns with option (d).

Revision Table: Key Concepts for Data Sufficiency

Additional Information: Solving Data Sufficiency Questions

When approaching data sufficiency problems, especially those involving algebraic expressions and properties like being an integer, consider the following steps:

1. Understand the Question: Precisely what are you trying to determine? Is it a specific value, a range of values, or a property (like being an integer, positive, negative, etc.)?
2. Analyze Statement I Alone: Assume only Statement I is true. Can you find values for the variables that satisfy Statement I and lead to different answers to the question? If yes, Statement I alone is NOT sufficient. If no (i.e., every set of values satisfying Statement I leads to the same answer), then Statement I alone IS sufficient.
3. Analyze Statement II Alone: Repeat the process for Statement II, assuming only Statement II is true.
4. Analyze Both Statements Together: Assume both Statement I and Statement II are true simultaneously. Can you combine the information from both statements? Can you find values for the variables that satisfy *both* statements and lead to different answers to the question? If yes, both statements together are NOT sufficient. If no (i.e., every set of values satisfying both statements leads to the same answer), then both statements together ARE sufficient.
5. Select the Correct Option: Based on your analysis of steps 2, 3, and 4, choose the option that describes the sufficiency of the statements.

Remember to test edge cases and different types of numbers (integers, fractions, decimals) that could potentially satisfy the statements.