If 'x' and 'y' are positive integers, what is the value of x - y? (Data Sufficiency) (1) x + y = 10 (2) x^2 - y^2 = 20
- AStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- BStatement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- CEACH statement ALONE is sufficient.
- DBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Show answer & explanationAnswer & explanation
Correct answer: D. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: x + y = 10. Many pairs of positive integers satisfy this (e.g., x=1, y=9; x=2, y=8), leading to different values for x-y (e.g., -8, -6). Not sufficient. Statement (2) alone: x^2 - y^2 = 20. This can be factored as (x - y)(x + y) = 20. Since x and y are positive integers, x-y and x+y must also be integers. We don't know x+y, so we can't find x-y uniquely (e.g., if x+y=10, x-y=2; if x+y=5, x-y=4). Not sufficient. Combining (1) and (2): We have x + y = 10 and (x - y)(x + y) = 20. Substitute x + y = 10 into the second equation: (x - y)(10) = 20. This gives x - y = 2. Since x and y are positive integers, this solution is valid (x=6, y=4). Thus, both statements together are sufficient.
Why the other options are wrong
- A. Incorrect. Statement (1) alone does not lead to a unique value for x-y.
- B. Incorrect. Statement (2) alone does not lead to a unique value for x-y without knowing x+y.
- C. Incorrect. Neither statement alone is sufficient.
Difference of Squares (Data Sufficiency)
The algebraic identity a^2 - b^2 = (a - b)(a + b) is often useful in Data Sufficiency problems, especially when one of the factors (a+b or a-b) is provided or can be derived.
- Factoring a difference of squares simplifies expressions.
- Can connect seemingly unrelated equations.
- Useful in problems involving sums and differences of variables.
Memory trick: Look for squares; they often hide a useful sum and difference.