Each question is followed by three statements I, II and
III. You have to decide whether the data given in the statements are sufficient to answer the question. In how many days can A alone complete the work?
I. A and B together can complete the work in 12 days.
II. B and C together can complete the work in 20 days.
III. A and C together can complete the work in 15 days.

ATQ, Let daily work rates be: A = a, B = b, C = c (work per day). Total work = 1 unit. From I: a + b = 1/12 …(1) From II: b + c = 1/20 …(2) From III: a + c = 1/15 …(3) We need a (so that days taken by A alone = 1/a). Check sufficiency: Statement I alone: a + b = 1/12 Two variables, one equation ⇒ infinitely many (a, b) ⇒ a not unique. Not sufficient. Statement II alone: b + c = 1/20 ⇒ not sufficient. Statement III alone: a + c = 1/15 ⇒ not sufficient. Now check any two: I + II: (1) a + b = 1/12 (2) b + c = 1/20 Unknowns a, b, c – two equations. We can express a = 1/12 − b, c = 1/20 − b; b is still free ⇒ infinitely many values of a. Not sufficient. I + III: (1) a + b = 1/12 (3) a + c = 1/15 Subtracting: (a + b) − (a + c) = 1/12 − 1/15 ⇒ b − c = 1/60 Again three unknowns (a, b, c) with two independent equations ⇒ a not fixed. Not sufficient. II + III: (2) b + c = 1/20 (3) a + c = 1/15 Subtract: (a + c) − (b + c) = 1/15 − 1/20 ⇒ a − b = 1/60 Again only two independent equations for three variables ⇒ a not unique. Not sufficient. Using all three I, II and III together: From (1): a = 1/12 − b From (2): c = 1/20 − b Substitute into (3): (1/12 − b) + (1/20 − b) = 1/15 ⇒ 1/12 + 1/20 − 2b = 1/15 Compute 1/12 + 1/20 = (5 + 3)/60 = 8/60 = 2/15 So: 2/15 − 2b = 1/15 ⇒ 2b = 2/15 − 1/15 = 1/15 ⇒ b = 1/30 Then from (1): a + 1/30 = 1/12 a = 1/12 − 1/30 = (5 − 2)/60 = 3/60 = 1/20 So A alone takes 1/a = 20 days (unique). Only all three statements together determine a; any two are not enough.