Step 1: Understanding the Question:
This is a standard Time and Work problem involving joint and individual efficiencies.
We know the total time taken when both work together, and we are given a scenario where they work together for some time before one works alone to finish the task. We need to deduce the boy's individual work rate.
Step 2: Key Formula or Approach:
Work done = Rate \( \times \) Time.
Total work is considered as 1 complete unit.
Rate of (Man + Boy) = \( \frac{1}{\text{Time taken together}} \).
Step 3: Detailed Explanation:
A man and a boy together complete the whole work in 24 days.
This means their combined one-day work (efficiency) is \( \frac{1}{24} \) of the total work.
In the modified scenario, the work is completed in a total of 26 days.
The problem states that for the last 6 days, the man works entirely alone.
This implies that both the man and the boy worked together for the initial period of \( 26 - 6 = 20 \) days.
We can calculate the amount of work completed by both of them in those 20 days: \( 20 \times \frac{1}{24} = \frac{20}{24} \).
Simplifying the fraction \( \frac{20}{24} \), we find that they completed \( \frac{5}{6} \) of the total work together.
The remaining work to be done is \( 1 - \frac{5}{6} = \frac{1}{6} \) of the total work.
This remaining \( \frac{1}{6} \) of the work was completed by the man alone in the last 6 days.
Therefore, the man's one-day work (his individual efficiency) is \( \frac{\frac{1}{6}}{6} = \frac{1}{36} \).
Now, we have the combined efficiency of the man and the boy, and the individual efficiency of the man.
Boy's one-day work = Combined one-day work - Man's one-day work.
Boy's one-day work = \( \frac{1}{24} - \frac{1}{36} \).
To subtract these fractions, we find the lowest common multiple (LCM) of 24 and 36, which is 72.
Boy's one-day work = \( \frac{3}{72} - \frac{2}{72} = \frac{1}{72} \).
Since the boy completes \( \frac{1}{72} \) of the work in one day, he will take exactly 72 days to complete the entire work alone.
Step 4: Final Answer:
The boy will take 72 days to complete the work alone.