What is the difference between two numbers x and y? I. The square of the first number equals 9 times the second: x^2 = 9y. II. The ratio of the first to the second is x : y = 3 : 4. (Assume positive integers to recover solvability.)

Introduction / Context:
We must determine whether the difference x − y can be uniquely found from the statements. By Recovery-First, we clarify that x and y are positive integers (a standard minimal assumption for such DS items).

Given Data / Assumptions:

• I: x^2 = 9y.
• II: x : y = 3 : 4.
• x, y ∈ positive integers.

Concept / Approach:
Each statement alone gives a relation but not a unique pair. Together they may pin down a single solution.

Step-by-Step Solution:
From II: Let x = 3t, y = 4t.Plug into I: (3t)^2 = 9y ⇒ 9t^2 = 9(4t) ⇒ 9t^2 = 36t ⇒ t(9t − 36) = 0 ⇒ t = 4 (discard t = 0 under positive-integer assumption).Thus x = 12, y = 16 ⇒ |x − y| = 4 (difference uniquely 4).

Why Other Options Are Wrong:
I alone leaves infinite solutions (choose any y that makes x = 3√y integral); II alone leaves a family scaled by t; “either” is false; “even both not sufficient” is false because the pair is unique (x = 12, y = 16).

Common Pitfalls:
Including the zero solution (t = 0) leads to non-unique difference; Recovery-First clarifies positivity to avoid that ambiguity.

Final Answer:
Both statements together are necessary.