In how many days will Y alone complete the work? I. X and Y together finish in 8 days. II. Y and Z together finish in 10 days. III. X and Z together finish in 12 days.

Introduction / Context:
Identical structure to the prior rates problem; now solve for Y’s solo rate.

Solution Sketch:
Let x, y, z be daily rates. We have x + y = 1/8, y + z = 1/10, x + z = 1/12. Then 2y = (1/8 + 1/10 − 1/12) = 17/120 ⇒ y = 17/240 ⇒ Y alone takes 240/17 days.

Sufficiency:
All three equations are required to isolate y uniquely.

Final Answer:
I, II and III.