- A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- B. The Question can be answered by using either Statement alone
- C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- D. The Question cannot be answered even by using both the Statements together
Answer: D
Explanation
We need to determine if (x+y) is an integer.
**Statement I: (2x+y) is an integer.**
– Let x = 0.5, y = 1. Then 2x+y = 2(0.5)+1 = 1+1 = 2 (integer). But x+y = 0.5+1 = 1.5 (not an integer).
– Let x = 1, y = 1. Then 2x+y = 2(1)+1 = 3 (integer). And x+y = 1+1 = 2 (integer).
Since (x+y) can be an integer or not, Statement I alone is insufficient.
**Statement II: (x+2y) is an integer.**
– Let x = 1, y = 0.5. Then x+2y = 1+2(0.5) = 1+1 = 2 (integer). But x+y = 1+0.5 = 1.5 (not an integer).
– Let x = 1, y = 1. Then x+2y = 1+2(1) = 3 (integer). And x+y = 1+1 = 2 (integer).
Since (x+y) can be an integer or not, Statement II alone is insufficient.
**Combining Statement I and Statement II:**
We have: (2x+y) = K (an integer) and (x+2y) = M (an integer).
Adding these two equations: (2x+y) + (x+2y) = K + M
3x + 3y = K + M
3(x+y) = K + M.
Since K and M are integers, K+M is also an integer. So, 3(x+y) is an integer.
However, if 3(x+y) is an integer, (x+y) itself is not necessarily an integer. For example, if x=1/3 and y=1/3:
– 2x+y = 2/3 + 1/3 = 1 (integer).
– x+2y = 1/3 + 2/3 = 1 (integer).
– But x+y = 1/3 + 1/3 = 2/3 (not an integer).
Therefore, even with both statements combined, we cannot definitively say if (x+y) is an integer. This question tests data sufficiency and properties of integers and rational numbers.