A and B are engaged to do a piece of work for Rs. 1540. A alone can do the work in 12 days and B alone can do it in 8 days. Working together with the help of C, they complete the work in 2 days. Assuming payment is divided in proportion to the amount of work done by each, what is the difference between the wages of C and B?

Introduction / Context:
This is a wages and work sharing question. Three workers, A, B, and C, complete a job together and are paid a fixed total amount. The problem provides the times taken by A and B individually and the time taken when all three work together. From this information we infer each person's contribution to the work and then distribute wages proportionally. Finally we compare the wages of C and B to find the difference. Such problems test understanding of proportional division based on actual work done.

Given Data / Assumptions:

• Total payment for the job is Rs. 1540.
• A alone completes the work in 12 days.
• B alone completes the work in 8 days.
• With the help of C, A and B together complete the job in 2 days.
• Payment is shared in proportion to the amount of work done by each person.

Concept / Approach:
First we compute the daily work rates of A and B individually. Then we compute the combined rate of A, B, and C using the fact that they finish one complete job in 2 days. Subtracting A's and B's rates from the combined rate gives C's rate. Next we calculate the fraction of the job done by each person in the 2 days that the job lasts. These fractions are then used to divide the total payment proportionally. Finally we subtract B's share from C's share to find the required difference in wages.

Step-by-Step Solution:
Step 1: Let the total work be 1 job. A's rate is 1 / 12 per day and B's rate is 1 / 8 per day. Step 2: A, B, and C together finish the job in 2 days, so their combined rate is 1 / 2 per day. Step 3: C's rate is the combined rate minus the rates of A and B. So C's rate = 1/2 minus 1/12 minus 1/8. Step 4: Compute 1/12 + 1/8 = 1/12 + 3/24 = 2/24 + 3/24 = 5/24. Thus C's rate = 1/2 minus 5/24 = 12/24 minus 5/24 = 7/24 per day. Step 5: In 2 days, A does 2 times 1/12 = 1/6 of the job, B does 2 times 1/8 = 1/4 of the job, and C does 2 times 7/24 = 7/12 of the job. Step 6: Total payment is Rs. 1540. B's share is (1/4) of 1540 which equals Rs. 385. C's share is (7/12) of 1540 which equals Rs. 1540 times 7 divided by 12 = Rs. 898.33 approximately. Step 7: Difference between C's wages and B's wages is approximately 898.33 minus 385 = Rs. 513.33.

Verification / Alternative check:
We can verify the fractions add to 1. A's share of work is 1/6, B's is 1/4, and C's is 7/12. Converting to a common denominator of 12, we get 2/12, 3/12, and 7/12, which sum to 12/12 = 1. This confirms that our work fractions are consistent. The proportional division of money is then straightforward. Calculating each share and checking that they sum to 1540 also confirms the correctness of the wage distribution.

Why Other Options Are Wrong:

• Rs. 980 and Rs. 810 are much larger than the computed difference between C and B and do not match the proportional division.
• Rs. 420 is smaller than the true difference and would imply incorrect work fractions or miscalculated wages.
• Only Rs. 513.33 matches the precise difference based on the exact fractions of work done.

Common Pitfalls:
Some learners mistakenly use the individual times 12 and 8 directly as weights for payment, ignoring C's contribution. Another pitfall is to forget that the job lasts only 2 days when all three are working, so total work done by each is daily rate multiplied by 2. It is crucial to compute exact fractions of work first and then apply proportional division to avoid such errors.

Final Answer:
The difference between the wages of C and B is approximately Rs. 513.33 in favour of C.