Evaluating data sufficiency to find the average of a specific subset
Question
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: (2x + y) is an integer. Statement-II: (x + 2y) is an integer.
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
Explanation
Evaluate the algebraic statements to see if they can isolate (x+y) as an integer:
Statement I states 2x + y = I_1 (where I_1 is an integer). This can be written as x + (x + y) = I_1. If x is a fraction (e.g., x = 0.5, y = 0), then 2(0.5) + 0 = 1 (Integer), but x + y = 0.5 (Not an integer). If x and y are both integers, x+y is an integer. Because both outcomes are possible, Statement I alone is insufficient.
Statement II states x + 2y = I_2. Similarly, using y = 0.5, x = 0 yields a non-integer sum for x+y. Insufficient.
Combine both statements: Add the two integer equations together: (2x + y) + (x + 2y) = I_1 + I_2 \implies 3x + 3y = I_3 \implies 3(x + y) = I_3. This proves that 3(x + y) is an integer. However, knowing that ⟨MATH⟩3(x+y)⟨/MATH⟩ is an integer does not guarantee that ⟨MATH⟩(x+y)⟨/MATH⟩ itself is an integer (e.g., if x+y = 1/3, then 3(1/3) = 1, which is an integer, but 1/3 is not). Therefore, the statements remain insufficient even when combined.
Answer: (d).
Question details
Year
2024
Paper
CSAT
Question
Q68
Section
Data Interpretation & Sufficiency
Sub-topic
Averages
Type
Statement-based
Difficulty
Medium
Source hint
Data sufficiency evaluation
Same sub-topic — other years
Averages has appeared in multiple CSAT papers:
See all questions on Averages
Browse every tagged question across all years