Aman can complete a piece of work in 18 days if he works 10 hours every day. If he wants to finish the same work in 15 days instead, how many hours per day should he work?

Introduction / Context:
This question combines time and work with daily working hours. Aman initially follows a certain schedule and completes the work in a given number of days. He then wants to adjust his daily working hours so that the total project duration becomes shorter. This question helps practice the idea that total work in such cases can be measured as total man hours, which is the product of hours per day and number of days.

Given Data / Assumptions:

• Aman completes the work in 18 days when he works 10 hours per day.
• He wants to complete the same work in 15 days.
• The amount of work and Amans efficiency remain constant.
• We must find the new number of hours per day he should work.
• Total work is measured in man hours, that is, hours per day multiplied by days.

Concept / Approach:
When the same person does the same job with the same efficiency, the total amount of work in man hours remains constant. If we know the current combination of days and daily hours, we can compute total man hours. Any new combination of days and hours that yields the same total man hours will complete the job. So, we calculate total man hours using the original schedule and then divide by the new number of days to get required hours per day.

Step-by-Step Solution:
Step 1: Compute the total work in man hours using the initial schedule. Step 2: Aman works 10 hours per day for 18 days, so total man hours = 10 * 18. Step 3: 10 * 18 = 180 man hours of total work. Step 4: Now Aman wants to complete the same work in 15 days. Step 5: Let the required daily working hours be H hours per day. Step 6: Total work remains 180 man hours, so 15 * H = 180. Step 7: Solve for H: H = 180 / 15 = 12 hours per day.

Verification / Alternative check:
We can verify by substituting the found value back. If Aman works 12 hours per day for 15 days, total work = 12 * 15 = 180 man hours. This equals the original total of 180 man hours from 10 hours per day for 18 days. Since the total work is unchanged and the calculations match, the answer 12 hours per day is confirmed.

Why Other Options Are Wrong:
10 hours per day is the original schedule and would not reduce the total days to 15. Working 14 or 16 hours per day would result in more than 180 man hours, meaning he would be doing extra work. Working 18 hours per day is even more extreme and unnecessary. Only 12 hours per day exactly fits the requirement of completing the same work in 15 days.

Common Pitfalls:
One common mistake is to assume that reducing the days by a certain fraction means increasing the hours by the same fraction, which is not always correct without checking total man hours. Another error is forgetting that the persons efficiency is constant, which is why total man hours must stay the same. Always express total work as a product of days and hours, then apply basic algebra carefully.

Final Answer:
Aman needs to work 12 hours per day to finish the work in 15 days.