A is thrice as good a work man as B and takes 10 days less to do a piece of…

Concept: When two people do the same piece of work, the ratio of the TIME they take to finish it is the INVERSE of the ratio of their efficiencies (work rates). If one person is k times as efficient as another, that person takes 1/k times as long.

Application:

1. 1. A is thrice as efficient as B, so the efficiency ratio A : B = 3 : 1, which makes the time ratio A : B = 1 : 3.

2. 2. Let A's time = t days and B's time = 3t days (keeping the 1 : 3 ratio).

3. 3. A takes 10 days less than B, so 3t - t = 10, i.e., 2t = 10, giving t = 5.

4. 4. B's time = 3t = 3 × 5 = 15 days.

Cross-check (algebraic method):

1. 1. Let B take d days, so A takes (d - 10) days.

2. 2. Since A is thrice as efficient, A's rate is 3 times B's rate: 1/(d - 10) = 3 × (1/d).

3. 3. d = 3(d - 10) = 3d - 30, so 2d = 30, giving d = 15.

4. 4. This matches the ratio method, confirming B alone can finish the work in 15 days.