Raman can complete a certain work in 5 days, Jatin can complete it in 7 days and Sachin can complete it in 9 days. If all three work together and earn a total of ₹2860 for completing the work, what is Raman's share (in rupees) of the total payment?

Introduction / Context:
This question is another example of work and wages. Three persons, Raman, Jatin and Sachin, can complete the same work in different times. They do the work together and are paid a lump sum. The payment is to be divided in proportion to the work done by each, which is directly proportional to their individual efficiencies (rates of work). We are asked to find Raman's share of the total amount ₹2860.

Given Data / Assumptions:

• Raman alone finishes the work in 5 days.
• Jatin alone finishes the work in 7 days.
• Sachin alone finishes the work in 9 days.
• Total payment = ₹2860.
• All three work together and share the payment in proportion to their work done (i.e., their rates).
• Total work is treated as 1 unit.

Concept / Approach:
We first compute the work rates for each person: 1 / 5, 1 / 7 and 1 / 9 of the job per day for Raman, Jatin and Sachin respectively. Their shares in the money must be in the ratio of these rates. We then find the sum of these rates and multiply the total amount by Raman's fraction of the total rate to determine his exact monetary share.

Step-by-Step Solution:
Let total work W = 1 unit. Raman's daily rate = 1 / 5 units. Jatin's daily rate = 1 / 7 units. Sachin's daily rate = 1 / 9 units. Total combined rate = 1 / 5 + 1 / 7 + 1 / 9. Compute using fractions: LCM of 5, 7 and 9 is 315. 1 / 5 = 63 / 315, 1 / 7 = 45 / 315, 1 / 9 = 35 / 315. Combined rate = (63 + 45 + 35) / 315 = 143 / 315. Raman's share of total rate = (1 / 5) / (1 / 5 + 1 / 7 + 1 / 9) = (1 / 5) / (143 / 315). (1 / 5) = 63 / 315, so Raman's fraction = 63 / 315 ÷ 143 / 315 = 63 / 143. Raman's share of money = (63 / 143) * 2860. Compute: 2860 * 63 / 143 = 1260. Therefore, Raman receives ₹1260.

Verification / Alternative check:
As a check, we can also compute Jatin's and Sachin's shares and see if they sum correctly to ₹2860. Jatin's fraction = (1 / 7) / (1 / 5 + 1 / 7 + 1 / 9) = 45 / 143; Sachin's fraction = 35 / 143. Using these, Jatin's share = 2860 * 45 / 143 = 900, Sachin's share = 2860 * 35 / 143 = 700. Summing 1260 + 900 + 700 = 2860, which matches the total payment, confirming the calculation.

Why Other Options Are Wrong:
Options B (₹700), C (₹900) and D (₹870) correspond to the shares of Sachin, Jatin or some incorrect fraction, not to Raman's share. Only ₹1260 agrees with the ratio of work rates and the total payment.

Common Pitfalls:
Students sometimes divide the payment inversely proportional to the number of days (5, 7, 9) without converting to rates, but luckily in this case using 1 / 5 : 1 / 7 : 1 / 9 or equivalently 63 : 45 : 35 leads to the same ratio. However, arithmetic with large LCMs like 315 can cause mistakes. Carefully handling each fraction and simplifying ratios before multiplying with the total money helps avoid errors.

Final Answer:
Raman's share of the total payment is ₹1260.