S, T, and U can complete a piece of work in 40 days, 48 days, and 60 days respectively. They start the work together and receive Rs 10800 in total for completing it. T leaves 2 days before completion and U leaves 5 days before completion, so S alone finishes the remaining work. What is the share of S (in rupees) from the total payment?

Introduction / Context:
This is a work and wages problem that involves three workers S, T, and U with different efficiencies. They all start the job together, but two of them leave before the work is completed, and the remaining person finishes the job alone. The total payment is given and must be divided among them in proportion to the work each person actually completed. This reflects real life situations where payment is linked to contribution.

Given Data / Assumptions:

• S can complete the work alone in 40 days.
• T can complete the work alone in 48 days.
• U can complete the work alone in 60 days.
• All three start the job together.
• T leaves 2 days before the work is completed.
• U leaves 5 days before the work is completed.
• The total payment for the completed work is Rs 10800.
• Payment is proportional to the amount of work done by each person.

Concept / Approach:
First, we treat the total work as 1 job and compute the daily work rates of S, T, and U. Next, we let the total duration of the project be D days and use the fact that each worker contributes their rate multiplied by the number of days they actually work. Setting the sum of these contributions equal to 1 job allows us to solve for D. After that, we calculate how much work S performs and use the ratio of S work to total work to compute S share of the total payment.

Step-by-Step Solution:
Step 1: Let total work be 1 job. Step 2: Rate of S = 1/40 job per day. Step 3: Rate of T = 1/48 job per day. Step 4: Rate of U = 1/60 job per day. Step 5: Let the total duration of the project be D days. Step 6: S works for all D days, T works for (D - 2) days, and U works for (D - 5) days. Step 7: Total work equation: (D/40) + ((D - 2)/48) + ((D - 5)/60) = 1. Step 8: Solving this equation gives D = 18 days. Step 9: Work done by S = D/40 = 18/40 = 9/20 of the job. Step 10: Since total work is 1, S share of payment is (9/20) of Rs 10800. Step 11: S share = (9/20) * 10800 = 9 * 540 = Rs 4860.

Verification / Alternative check:
We can check the work proportions. T works (18 - 2) = 16 days at 1/48 job per day, so T contribution is 16/48 = 1/3 job. U works (18 - 5) = 13 days at 1/60 job per day, so U contribution is 13/60 job. Adding the contributions: S has 9/20 job, T has 1/3 job, and U has 13/60 job. Converting to a common denominator of 60 gives S = 27/60, T = 20/60, and U = 13/60. The sum is (27 + 20 + 13)/60 = 60/60 = 1 job, confirming the calculations and validating S share.

Why Other Options Are Wrong:
Rs 4320, Rs 4500, and Rs 4960 correspond to different and incorrect fractions of the total work; they do not match the actual 9/20 part that S performs. Rs 5400 would mean S does exactly half the job, which is not supported by the computed contributions. Only Rs 4860 correctly reflects S share based on his actual portion of the work done.

Common Pitfalls:
Students may assume that all workers share the money equally, ignoring differences in efficiency and time spent. Another error is miscalculating the total time D or incorrectly solving the fractional equation. Care must be taken when handling multiple time periods and different leaving times for the workers. Always compute each person contribution explicitly and then divide the total payment proportionally.

Final Answer:
The share of S from the total payment is Rs 4860.