Eighteen men or thirty six boys working 6 hours per day can plough a field in 24 days. In how many days will 24 men and 24 boys working 9 hours per day plough the same field?

Introduction / Context:
This time and work question compares the work capacities of men and boys. We are given that either a group of men or a group of boys can complete a task in the same number of days. From this, we can determine the relative efficiency of a man and a boy. Then we use that relationship to find how long a mixed group of men and boys will take to complete the same job when working different hours per day.

Given Data / Assumptions:

• 18 men working 6 hours per day can plough the field in 24 days.
• 36 boys working 6 hours per day can also plough the field in 24 days.
• We assume all men have the same efficiency and all boys have the same efficiency.
• We need to find the time for 24 men and 24 boys working 9 hours per day to plough the same field.

Concept / Approach:
First, we express the total work in terms of man hours or boy hours. Using the equality of total work from the two given scenarios, we deduce the efficiency ratio of a man to a boy. Then, we compute the total daily work capacity of the mixed group of 24 men and 24 boys working 9 hours per day and divide the total work by this daily capacity to obtain the required number of days.

Step-by-Step Solution:
Step 1: Let the work done per man per hour be m units and per boy per hour be b units.Step 2: From the first case, total work W = 18 men * 6 hours per day * 24 days * m = 18 * 6 * 24 * m.Step 3: From the second case, W = 36 boys * 6 hours per day * 24 days * b = 36 * 6 * 24 * b.Step 4: Equating the two expressions for W: 18 * 6 * 24 * m = 36 * 6 * 24 * b.Step 5: Cancel the common factor 6 * 24 on both sides to get 18m = 36b.Step 6: Therefore m = 2b, so one man is equivalent to two boys in hourly efficiency.Step 7: Now compute total work W explicitly in terms of b. Using the boys expression: W = 36 * 6 * 24 * b = 36 * 144 * b = 5184b units.Step 8: For the new group, we have 24 men and 24 boys working 9 hours per day.Step 9: Convert men to equivalent boys. Since m = 2b, 24 men contribute the same as 24 * 2 = 48 boys per hour.Step 10: Total workers in equivalent boy units = 48 (from men) + 24 (actual boys) = 72 boy units per hour.Step 11: Working 9 hours per day, total daily work capacity = 72 * 9 * b = 648b units per day.Step 12: Number of days required = total work / daily work capacity = 5184b / 648b = 8 days.

Verification / Alternative check:
We can check the proportionality. In the original scenario, if we convert 18 men to equivalent boys, that is 18 * 2 = 36 boys, which matches the second given scenario. The mixed group is larger in equivalent capacity than either original group, and also works more hours per day, so fewer days are expected. Eight days is a reasonable and consistent result after the rate calculations.

Why Other Options Are Wrong:

• 9 days: This would require a slightly smaller daily capacity or a larger total work, which is not consistent with the given data.
• 10 days: This would suggest the mixed group is less efficient than computed and contradict the equality of the original scenarios.
• 6 days: This time is too short for the computed daily capacity and would imply total work smaller than 5184b.

Common Pitfalls:
A common error is to ignore the hours per day and compare only numbers of workers and days. Another mistake is misinterpreting the equality of work done by 18 men and 36 boys, leading to an incorrect ratio of man efficiency to boy efficiency. Always include the hours per day and carefully equate total work for the different groups before moving on to the final calculation.

Final Answer:
The field will be ploughed in 8 days by 24 men and 24 boys working 9 hours per day.