Amar can complete a certain piece of work in 10 days when he works alone. He works on the job for 4 days and then stops, after which Arun finishes the remaining work in 9 days by working alone. If Amar and Arun work together from the beginning, in how many days will they complete the entire work?

Introduction / Context:
This time and work problem involves two workers, Amar and Arun. Amar starts the work and leaves before finishing, and Arun completes the remaining portion. Using this information, we are asked how long both of them together would need if they started and finished the job jointly.

Given Data / Assumptions:

• Amar alone can finish the work in 10 days.
• He works alone on the job for 4 days and then stops.
• Arun then completes the remaining work alone in 9 days.
• We assume both work at constant daily rates.

Concept / Approach:
First, we calculate how much work Amar completes in 4 days, then how much work Arun completes in 9 days. Combined, these two must equal the full job. This allows us to determine Arun's individual rate. With both individual rates known, we can simply add them to obtain the combined rate and then invert to find the time that both together need to complete the job.

Step-by-Step Solution:
Let the total work be 1 job. Amar alone finishes in 10 days, so Amar's rate = 1 / 10 job per day. In 4 days, Amar completes 4 * (1 / 10) = 4 / 10 = 2 / 5 of the job. Remaining work after Amar stops = 1 - 2 / 5 = 3 / 5 of the job. Arun alone completes the remaining 3 / 5 in 9 days. So Arun's rate = (3 / 5) / 9 = 3 / (5 * 9) = 1 / 15 job per day. When Amar and Arun work together, their combined rate = 1 / 10 + 1 / 15. Find a common denominator: 1 / 10 = 3 / 30 and 1 / 15 = 2 / 30, so combined rate = 5 / 30 = 1 / 6 job per day. Time taken together = 1 / (1 / 6) = 6 days.

Verification / Alternative check:
Another way is to take total work as 30 units. Amar's rate becomes 30 / 10 = 3 units per day, and in 4 days he does 12 units. Arun does the remaining 18 units in 9 days, so Arun's rate is 2 units per day, which is equivalent to 1 / 15 of the whole job. Together, they produce 3 + 2 = 5 units per day, so time to finish 30 units is 30 / 5 = 6 days, matching the result.

Why Other Options Are Wrong:
4 days and 3 days are too short because even Amar, the faster worker, alone needs 10 days. 8 days is longer than necessary and does not fit with the combined rate of 1 / 6 job per day. 5 days also does not match the calculated rate and total work.

Common Pitfalls:
Some learners make the mistake of averaging Amar's and Arun's times directly instead of going through their rates. Others forget to subtract Amar's completed portion correctly or miscalculate Arun's rate from the remaining work. Always work with fractions of the total job and derive each worker's rate separately before combining them.

Final Answer:
Amar and Arun together can complete the work in 6 days.