C can complete a work in 2/3 of the time A takes to complete it, and B can…

Concept

When two or more people work together for the same duration, the money they earn is shared in proportion to the work each does — that is, in proportion to their work efficiency. Since efficiency and the time taken to complete a job alone are inversely proportional, converting each person's time into an efficiency ratio gives the ratio in which any payment for the joint work should be split.

Application

1. Let A's time to do the work alone = A units. Then C's time = (2/3)A, since C takes 2/3 of A's time.

2. B's time = (3/4) of C's time = (3/4) x (2/3)A = (1/2)A.

3. So the ratio of times taken by A, B and C = A : (1/2)A : (2/3)A. Multiplying through by 6 to clear fractions gives A : B : C (time) = 6 : 3 : 4.

4. Efficiency is inversely proportional to time, so the efficiency (work-rate) ratio of A : B : C = 1/6 : 1/3 : 1/4. Multiplying through by 12 (the LCM of 6, 3, 4) gives A : B : C (efficiency) = 2 : 4 : 3.

5. Since they worked together for the same duration, the total payment of Rs. 9180 is shared in this same 2 : 4 : 3 ratio, over 2 + 4 + 3 = 9 parts.

6. B's share = (4/9) x 9180 = Rs. 4080.

Cross-check

A's share = (2/9) x 9180 = Rs. 2040 and C's share = (3/9) x 9180 = Rs. 3060. Adding all three shares: 2040 + 4080 + 3060 = 9180, which matches the total amount paid — confirming B's share of Rs. 4080 is correct.