Six boys and eight girls can complete a certain job in 10 days, and twenty-six boys and forty-eight women can complete the same job in 2 days. In how many days will fifteen boys and twenty girls together complete the job?

Introduction / Context:
This is a mixed time and work problem involving boys and girls as separate worker types. You are given two scenarios where combinations of boys and girls complete the same job in different times. The goal is to determine how long another combination will take. The idea is to express total work in terms of daily efficiencies of boys and girls and then use both scenarios to find their relationship.

Given Data / Assumptions:

• Six boys and eight girls finish the job in 10 days.
• Twenty-six boys and forty-eight women (treated as equivalent to girls in efficiency) finish the same job in 2 days.
• Each boy has the same efficiency; each girl has the same efficiency.
• Total work is the same in all scenarios.
• We must find the time taken by fifteen boys and twenty girls working together.

Concept / Approach:
Let the daily work done by one boy be b units and by one girl be g units. From each scenario, we can write an equation for the total work. Since the work is the same, we can equate these expressions and solve for the relationship between b and g. Then we calculate the combined daily work for fifteen boys and twenty girls and use Time = Work / Rate to get the required number of days.

Step-by-Step Solution:
Let one boy's daily work = b units and one girl's daily work = g units. From the first group: (6b + 8g) * 10 = W. So W = 60b + 80g. From the second group: (26b + 48g) * 2 = W. So W = 52b + 96g. Equate both expressions for W: 60b + 80g = 52b + 96g. Rearrange: 60b − 52b = 96g − 80g ⇒ 8b = 16g ⇒ b = 2g. Substitute g = b / 2 into W = 60b + 80g ⇒ W = 60b + 80 * (b / 2) = 60b + 40b = 100b. Now consider 15 boys and 20 girls: daily work = 15b + 20g. Replace g: 15b + 20 * (b / 2) = 15b + 10b = 25b. Thus time T = W / daily work = 100b / 25b = 4 days.

Verification / Alternative check:
You can think of each girl as half a boy in efficiency, since b = 2g implies g = 0.5b. Then in the final group, fifteen boys plus twenty girls is equivalent to 15 + 20 * 0.5 = 15 + 10 = 25 boys. Total work W was 100 boy-days, so with 25 equivalent boys, the time is 100 / 25 = 4 days, which matches our result.

Why Other Options Are Wrong:
2 days or 3 days would require a much more powerful team than we have. 5 days would correspond to a lower effective workforce than 20 boys, which is not the case since fifteen boys and twenty girls are equivalent to 25 boys. Hence only 4 days is consistent with the computed efficiencies.

Common Pitfalls:
Students sometimes mix up the second scenario by reading “women” as a new worker type instead of treating them as girls with the same efficiency. Another pitfall is to equate only the brackets (6b + 8g) and (26b + 48g) instead of total work, which ignores the different durations and leads to incorrect relations.

Final Answer:
Fifteen boys and twenty girls together will complete the job in 4 days.