What is the sum of the age of Ram and Mohan? Statement I : The age of Ram is 6…

A data-sufficiency statement (or combination of statements) is sufficient only when it pins down ONE unique value for what is asked — here, the sum of the two ages. A single equation relating two unknowns, or a bare ratio between two unknowns, can never do this alone; two independent equations in two unknowns are needed, and if two given relations turn out to be algebraically identical, combining them adds no new information.

Let R = Ram's age and M = Mohan's age; we need R + M.

1. Statement I gives R = M + 6 — one equation in two unknowns, so it cannot fix R + M on its own.

2. Statement II gives 40% of M = 30% of R, i.e. 0.4M = 0.3R, so R = (4/3)M — again just one ratio-type equation in two unknowns, insufficient alone.

3. Statement III gives (R/2) : (M/3) = 2 : 1, i.e. 3R = 4M, so R = (4/3)M — the SAME relation as Statement II, so it too cannot fix R + M alone, and it adds nothing new once Statement II is already known.

4. Combining Statement I with Statement II: substitute R = M + 6 into R = (4/3)M to get M + 6 = (4/3)M, giving M = 18 and R = 24, so R + M = 42 — a single unique value, so this combination is sufficient.

Checking M = 18, R = 24: R − M = 24 − 18 = 6 (matches Statement I); 40% of 18 = 7.2 and 30% of 24 = 7.2 (matches Statement II); and (24/2) : (18/3) = 12 : 6 = 2 : 1 (matches Statement III too, confirming II and III really are the same constraint).

So among the given combinations, Statement I together with Statement II determines the unique sum R + M = 42; because Statement III conveys the identical relation as Statement II, Statement I together with Statement III would determine the same unique sum too — but that particular combination is not one of the answer choices offered here, so among the choices actually given, only 'Statement I and Statement II together' correctly describes a sufficient combination.