Three labourers A, B and C together can finish a certain piece of work in 8 days. A and C together can do the same work in 12 days, while A and B together can do it in 13 1/3 days. If the total payment for the work is Rs. 750, in what ratio should it be divided among A, B and C?

Introduction / Context:
This question asks for the ratio of wages to be paid to three labourers A, B and C based on their individual contributions to a job. They have worked together and in pairs for different durations, and we are given how long various combinations take to complete the full work. Payment must be divided proportional to the amount of work each has done, so we must find their individual work rates. Such problems appear often in aptitude tests involving partnerships and work-sharing.

Given Data / Assumptions:

• A + B + C together can complete the work in 8 days.
• A + C together can complete the work in 12 days.
• A + B together can complete the work in 13 1/3 days (which is 40/3 days).
• Total payment for the job is Rs. 750, to be divided based on individual contributions.
• All work rates are constant over time.

Concept / Approach:
We treat the total work as 1 unit. Let the daily work rates of A, B and C be a, b and c respectively. Then we can write equations for A + B + C, A + C, and A + B based on their respective completion times. Solving these equations gives the individual rates a, b and c. The ratio of these rates equals the ratio of work done by each in any common time interval, which matches the ratio of wages they should receive.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: From A + B + C finishing in 8 days, we have a + b + c = 1/8. Step 3: From A + C finishing in 12 days, we have a + c = 1/12. Step 4: From A + B finishing in 13 1/3 days = 40/3 days, we have a + b = 1 / (40/3) = 3/40. Step 5: To find a, add the last two equations and subtract the first: (a + b) + (a + c) − (a + b + c) = 3/40 + 1/12 − 1/8. Step 6: This simplifies to a = 3/40 + 1/12 − 1/8 = 1/30. Step 7: Then b = (a + b) − a = 3/40 − 1/30 = 1/24, and c = (a + c) − a = 1/12 − 1/30 = 1/20. Step 8: Thus daily work rates of A, B, C are 1/30, 1/24, and 1/20 respectively. Step 9: Multiply all rates by the common denominator 120 to get the ratio A : B : C = 4 : 5 : 6.

Verification / Alternative check:
We can double-check by recomputing each pair and all three. For example, A + C has rate 1/30 + 1/20 = 1/12, matching the given 12 days. A + B has rate 1/30 + 1/24 = 3/40, matching 40/3 days. A + B + C has rate 1/30 + 1/24 + 1/20 = 1/8, matching 8 days. Since all conditions are satisfied, the ratio 4 : 5 : 6 is correct and consistent.

Why Other Options Are Wrong:
Any other ratio, such as 4 : 7 : 5, 5 : 7 : 4, or 5 : 6 : 8, would correspond to different relative rates for A, B and C. Reconstructing completion times using those ratios would not match the given 8 days, 12 days and 40/3 days for the various combinations. Only 4 : 5 : 6 fits all three completion-time conditions simultaneously.

Common Pitfalls:
A frequent mistake is to assume that time is directly proportional to payment without considering individual rates. Another issue is mismanaging the fractional days, particularly 13 1/3 days; it is important to convert this to 40/3 days before taking the reciprocal. Carefully forming and solving the system of equations for the rates a, b and c avoids these issues.

Final Answer:
The wages should be divided among A, B and C in the ratio 4 : 5 : 6.