Tushar can complete a piece of work in 6 hours and Amar can complete the same work in 10 hours. If both of them work together on the entire job, how much time will they take to finish it?

Introduction / Context:
This question tests the concept of combined work and time, where two workers with different individual times to complete a job work together. The core idea is to add their rates of working rather than their times, and from the combined rate derive the required time for completing the job together.

Given Data / Assumptions:

• Tushar can complete the work alone in 6 hours.
• Amar can complete the same work alone in 10 hours.
• Both Tushar and Amar start the work together and continue without any break until the work is finished.
• The work is uniform and their rates of working remain constant throughout.

Concept / Approach:
When dealing with time and work problems, it is convenient to consider the work done per hour by each worker, which is called the rate. If one worker completes a job in T hours, then the rate of that worker is 1 / T of the job per hour. For workers working together, the total rate is obtained by adding individual rates. Once the combined rate is known, the total time taken is the reciprocal of that combined rate.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit.Step 2: Rate of Tushar = 1 / 6 of the work per hour.Step 3: Rate of Amar = 1 / 10 of the work per hour.Step 4: Combined rate when both work together = 1 / 6 + 1 / 10.Step 5: Compute 1 / 6 + 1 / 10 = (5 + 3) / 30 = 8 / 30 = 4 / 15 of the work per hour.Step 6: Time taken when working together = 1 / (4 / 15) = 15 / 4 hours.Step 7: Convert 15 / 4 hours into hours and minutes. 15 / 4 = 3.75 hours, which is 3 hours and 0.75 * 60 minutes = 45 minutes.

Verification / Alternative check:
We can verify by checking how much of the work they complete in 3 hours 45 minutes. In 3.75 hours, Tushar completes (1 / 6) * 3.75 = 0.625 of the job and Amar completes (1 / 10) * 3.75 = 0.375 of the job. Together they complete 0.625 + 0.375 = 1.0 of the job, which confirms that they finish the entire work in 3 hours 45 minutes.

Why Other Options Are Wrong:

• 3 hours: At 3 hours, their combined work would be (1 / 6 + 1 / 10) * 3 = (4 / 15) * 3 = 4 / 5 of the job, which is not complete.
• 3 hours 15 minutes: This is 3.25 hours, which leads to total work (4 / 15) * 3.25, less than 1, so the job is still incomplete.
• 3 hours 30 minutes: This is 3.5 hours, which gives total work (4 / 15) * 3.5, also less than 1, so the work is not fully done.

Common Pitfalls:
One common mistake is to take the average of the two times, such as (6 + 10) / 2 = 8 hours, which is completely incorrect because time values are not additive in this way. Another error is forgetting to convert fractional hours into minutes properly. Always work with rates first and then convert the final time into the desired units.

Final Answer:
The work will be completed in 3 hours 45 minutes when Tushar and Amar work together.