Six men and eight boys can complete a piece of work in ten days, while twenty six men and forty eight boys can complete the same work in two days. In how many days will fifteen men and twenty boys together complete the entire work?

Introduction / Context:
This is a classic time and work question involving two different types of workers, men and boys, each with different efficiencies. The problem provides two different team combinations that complete the same job in different times. From these combinations we can infer the individual work rates of a man and a boy. Once those rates are known, we can combine them again for a new group size and compute the total time needed to finish the work. Such questions commonly appear in aptitude tests and competitive examinations.

Given Data / Assumptions:

• Six men and eight boys complete the work in 10 days.
• Twenty six men and forty eight boys complete the same work in 2 days.
• All men work at the same constant rate per day.
• All boys work at the same constant rate per day.
• We need the time taken by fifteen men and twenty boys together to finish the job.

Concept / Approach:
The key concept is total work equals rate multiplied by time. Let one man do m units of work per day and one boy do b units per day. We write two equations from the two groups given, both equal to one complete job. Solving the system of linear equations gives values of m and b. Then we compute the daily work rate of fifteen men and twenty boys, add the contributions of men and boys, and finally take the reciprocal to find the number of days needed to complete one job.

Step-by-Step Solution:
Step 1: Let daily work of one man be m and that of one boy be b. Step 2: From the first group, (6m + 8b) times 10 days equals 1 job, so 6m + 8b = 1 / 10. Step 3: From the second group, (26m + 48b) times 2 days equals 1 job, so 26m + 48b = 1 / 2. Step 4: Solving these equations gives m = 0.01 and b = 0.005. Step 5: Daily work of fifteen men and twenty boys is 15m + 20b = 15(0.01) + 20(0.005) = 0.15 + 0.10 = 0.25 of the job per day. Step 6: Time taken is 1 divided by 0.25, which equals 4 days.

Verification / Alternative check:
We can quickly verify by checking total work units. Total work can be considered as 1 unit. Fifteen men and twenty boys complete 0.25 units per day. Over 4 days, they would complete 4 times 0.25 which equals 1 unit of work. This means the whole job is done exactly in 4 days with no remainder. The consistency between the derived rates and the original equations confirms that the system of equations has been solved correctly.

Why Other Options Are Wrong:

• 6 days would correspond to a daily rate of only 1 / 6, which is smaller than 0.25, thus contradicting the rate obtained from the original data.
• 7 days makes the daily rate even smaller and therefore does not satisfy the equations derived from the initial groups.
• 5 days would require a daily rate of 1 / 5, which does not match the computed combined rate of 0.25 per day.

Common Pitfalls:
One common mistake is to try to compare the number of men and boys as if they had equal efficiency. Another error is incorrectly forming or solving the simultaneous equations. Some learners also forget to divide by the number of days when converting to rates. Being systematic about defining variables, writing equations, and solving them step by step prevents these misunderstandings and leads to the correct answer efficiently.

Final Answer:
Fifteen men and twenty boys working together will complete the entire work in 4 days.