Ram and Shyam undertake a piece of work for a total payment of ₹5600. Ram alone can finish the work in 5 days, while Shyam alone can finish the same work in 9 days. If they work together and share the payment in proportion to the work done by each, what is the difference (in rupees) between the amounts received by Ram and Shyam?

Introduction / Context:
This is a Time and Work problem combined with profit sharing. Ram and Shyam are paid a fixed total amount for a job, and they share this amount in proportion to the work each has actually done. Since we are given the times taken by each to complete the job alone, we can derive their work rates and use these to determine the ratio in which they should share the payment. Finally, we compute the difference between their shares.

Given Data / Assumptions:

• Ram alone completes the work in 5 days.
• Shyam alone completes the work in 9 days.
• Total payment for the work = ₹5600.
• They work together for the entire duration and share payment proportional to their work done (i.e., their efficiencies).
• Total work is taken as 1 unit.

Concept / Approach:
The key is that when two people work together, their contributions to the work are proportional to their individual rates of work. If a person can do the work in T days, their rate is 1 / T of the work per day. The ratio of Ram's rate to Shyam's rate is therefore (1 / 5) : (1 / 9), which simplifies to 9 : 5. This is the ratio in which they should share the total money. We then find each share and compute the difference between them.

Step-by-Step Solution:
Let total work W = 1 unit. Ram's daily work rate = 1 / 5 units. Shyam's daily work rate = 1 / 9 units. Ratio of their rates (Ram : Shyam) = (1 / 5) : (1 / 9) = 9 : 5. Total payment = ₹5600. Sum of ratio parts = 9 + 5 = 14 parts. Value of one part = 5600 / 14 = ₹400. Ram's share = 9 * 400 = ₹3600. Shyam's share = 5 * 400 = ₹2000. Difference between their shares = 3600 - 2000 = ₹1600.

Verification / Alternative check:
You can also check using actual work done per unit time. Suppose they work together for T days and finish the work: (1 / 5 + 1 / 9) * T = 1. Their individual contributions are (1 / 5) * T and (1 / 9) * T. The ratio of these contributions is (T / 5) : (T / 9) = 9 : 5, confirming that our ratio is correct and independent of T. Therefore, the payment must be shared in the ratio 9 : 5, and the resulting difference in money must be ₹1600.

Why Other Options Are Wrong:
Option A (₹1800), Option B (₹2400) and Option D (₹2200) correspond to incorrect share ratios. If any of these were correct, the total payment would not be consistently shared in the 9 : 5 ratio. By explicitly calculating the shares based on the correct ratio and comparing them with the total amount, we see that only a difference of ₹1600 matches the given total payment of ₹5600.

Common Pitfalls:
A common mistake is to assume that the payment should be divided according to the times they would take to complete the work alone (5 and 9) instead of their rates (1 / 5 and 1 / 9). Remember that contribution depends on how much work each does per unit time, not on how long they would individually take. Always convert times to rates and then compare rates to determine the correct sharing ratio.

Final Answer:
The difference between the amounts received by Ram and Shyam is ₹1600.