Two workers with a helper: Ramesh and Suresh take a job for ₹ 800. Ramesh alone can finish it in 12 days; Suresh alone in 16 days. With the assistance of a boy, they finish in 6 days. How should the money be divided among Ramesh, Suresh, and the boy, proportional to work done?

Introduction / Context:
Lump-sum wage splitting by contribution requires computing each person's share of the total work. The boy's presence speeds up completion to 6 days; from this we infer his effective rate and then split ₹ 800 accordingly.

Given Data / Assumptions:

• Rate(Ramesh) = 1/12
• Rate(Suresh) = 1/16
• Total rate with boy = 1/6
• Total amount = ₹ 800

Concept / Approach:
Boy's rate = total rate − (Ramesh + Suresh) rate. Then, each share fraction = rate / total rate. Multiply these fractions by ₹ 800 to determine the payments.

Step-by-Step Solution:
Rate(R+S) = 1/12 + 1/16 = 4/48 + 3/48 = 7/48Total rate = 1/6 = 8/48Rate(boy) = 8/48 − 7/48 = 1/48Share fractions (multiply by 6): R = 6*(1/12) = 1/2; S = 6*(1/16) = 3/8; Boy = 6*(1/48) = 1/8Payments: R = 1/2*800 = ₹ 400; S = 3/8*800 = ₹ 300; Boy = 1/8*800 = ₹ 100

Verification / Alternative check:
Fractions sum to 1; payments sum to ₹ 800 — consistent.

Why Other Options Are Wrong:
Options that give the boy ₹ 400 or ₹ 100 but mis-allocate others do not match the exact rate-derived fractions.

Common Pitfalls:
Proportioning by days worked instead of rates; miscomputing the boy's rate by forgetting to subtract both Ramesh and Suresh from the total rate.

Final Answer:
Ramesh = ₹ 400, Suresh = ₹ 300 and boy = ₹ 100