P and Q undertake a job for a total payment of Rs. 1200. P alone can complete the work in 15 days, while Q alone can complete it in 12 days. With the help of R, all three finish the work in 6 days. What is the share of R in the total payment?

Introduction / Context:
This work and wages question involves three people contributing to a job for which a fixed amount is paid. Since wages are typically distributed in proportion to the work each person performs, we must determine how much work R contributes relative to P and Q and then allocate the total payment accordingly.

Given Data / Assumptions:
• P alone can complete the job in 15 days.
• Q alone can complete the job in 12 days.
• Together with R, they finish the job in 6 days.
• Total payment for the job is Rs. 1200.
• Wages are proportional to the amount of work done.

Concept / Approach:
We convert each worker time into a daily work rate expressed as a fraction of the total job. Adding the rates of P, Q and R gives the total combined rate when all work together. From this, we isolate R rate. Next, we compute the work done by each worker during the 6 days and use the ratio of these amounts to split the Rs. 1200 among them.

Step-by-Step Solution:
Let the total work be 1 job. Daily rate of P = 1 / 15 of the job per day. Daily rate of Q = 1 / 12 of the job per day. When P, Q and R all work together, they complete the job in 6 days, so combined daily rate = 1 / 6 job per day. Let R daily rate be r. Then 1 / 15 + 1 / 12 + r = 1 / 6. Compute 1 / 15 + 1 / 12 with common denominator 60: 1 / 15 = 4 / 60, 1 / 12 = 5 / 60, so sum = 9 / 60 = 3 / 20. Thus 3 / 20 + r = 1 / 6. So r = 1 / 6 − 3 / 20 = (10 − 9) / 60 = 1 / 60 job per day. Work done in 6 days: P does 6 × 1 / 15 = 2 / 5 of the job, Q does 6 × 1 / 12 = 1 / 2 of the job, R does 6 × 1 / 60 = 1 / 10 of the job. Thus their work ratio P : Q : R = 2 / 5 : 1 / 2 : 1 / 10. Converting to a common denominator 10 gives 4 : 5 : 1. Total parts = 4 + 5 + 1 = 10 parts, so each part = 1200 ÷ 10 = Rs. 120. Therefore R share = 1 part = Rs. 120.

Verification / Alternative check:
We can check P and Q shares: P gets 4 parts = 4 × 120 = Rs. 480, Q gets 5 parts = 5 × 120 = Rs. 600, and R gets Rs. 120. The total is 480 + 600 + 120 = Rs. 1200, which matches the total payment, confirming the apportionment.

Why Other Options Are Wrong:
Larger values such as Rs. 240 or Rs. 360 would make R contribution look disproportionately large compared to his actual small work fraction of 1 / 10. These values would destroy the correct 4 : 5 : 1 ratio of P : Q : R work contributions.

Common Pitfalls:
A frequent mistake is to divide the total amount equally among P, Q and R, ignoring their different efficiencies and contributions. Another error is miscalculating the combined rate or forgetting to multiply individual rates by 6 days when finding total work done by each person.

Final Answer:
R share of the payment is Rs. 120.