Sumit and Amit undertake a piece of work for Rs 7200. Sumit alone can do the work in 5 days and Amit alone can do it in 4 days. If they work together and share the money in proportion to the work done, what is the difference in the amounts they receive in rupees?

Introduction / Context:
This work and wages problem tests your understanding of how payments are shared when two people work together at different individual rates. The key idea is that money should be divided in the ratio of the work done by each person, which is directly proportional to their rate of doing work when they work for the same amount of time.

Given Data / Assumptions:

• Total payment for completing the work = Rs 7200.
• Sumit alone can complete the work in 5 days.
• Amit alone can complete the work in 4 days.
• They work together on the whole job and share the money according to the work done.
• Required: difference in their shares in rupees.

Concept / Approach:
Work rate is the amount of work done per unit time. If Sumit completes the entire work in 5 days, his rate is 1/5 of the work per day. If Amit completes the work in 4 days, his rate is 1/4 of the work per day. When they work together for the same duration, the ratio of work done by each is equal to the ratio of their rates. We can use this ratio to split the payment and then find the difference between their shares.

Step-by-Step Solution:
Step 1: Compute Sumit's rate of work as 1/5 of the job per day. Step 2: Compute Amit's rate of work as 1/4 of the job per day. Step 3: The ratio of their work rates is Sumit : Amit = (1/5) : (1/4). Step 4: Simplify the ratio by multiplying both parts by 20 to clear denominators: (1/5) × 20 = 4 and (1/4) × 20 = 5. Step 5: Therefore, the ratio of their work and hence their earnings is 4 : 5. Step 6: Let Sumit's share be 4k and Amit's share be 5k. The total payment is 4k + 5k = 9k = 7200. Step 7: Solve for k: k = 7200/9 = 800. Step 8: Sumit's share is 4k = 4 × 800 = Rs 3200. Step 9: Amit's share is 5k = 5 × 800 = Rs 4000. Step 10: The difference in their shares is 4000 - 3200 = Rs 800.

Verification / Alternative check:
You can check that 3200 + 4000 = 7200, so the total is correct. The ratio 3200 : 4000 simplifies to 4 : 5, which matches the ratio of their work rates. This confirms that the distribution is correct and that the difference of Rs 800 is accurate.

Why Other Options Are Wrong:
1200, 1400 and 1600 correspond to incorrect assumptions about the ratio of their earnings, such as 3 : 4 or 2 : 3, which do not match the given completion times. The value 600 might occur from a miscalculation when dividing 7200 by 9 or subtracting the shares. Only 800 fits the correct ratio and total payment.

Common Pitfalls:
A typical mistake is to divide the money in the ratio of the days taken (5 : 4) instead of the rate of work, which is the inverse. Another mistake is to calculate the time for which they work together and then incorrectly use those times, even though the total work is one complete job. Remember that for equal working time, shares are proportional to rates, not to individual completion times.

Final Answer:
The difference in the amounts they receive is Rs 800.