A man and a boy can do a piece of work in 24 days. If the man works alone for the last 6 days, it is completed in 26 days. How long would the boy take to do it alone? (S.S.C., 2005)

### Concept & Work Breakdown by Phases When a total timeline is given alongside a solo finishing phase, subtract the solo days from the total days to find how long the workers collaborated. Use the fraction of work completed together to deduce the solo worker's speed from the remainder. ### Step-by-Step Solution * **Given:** Man and Boy together take 24 days. The total work takes 26 days if the Man works alone for the last 6 days. * **Calculation:** Since the total job took 26 days and the Man worked alone for the last 6 days, the Man and Boy must have worked together for the initial period. * Time worked together = $26 - 6 = 20$ days. * The Man and Boy's combined 1-day work is $1/24$. * Work completed by them together in 20 days = $20 \times (1/24) = 20/24 = 5/6$. * Remaining work = $1 - 5/6 = 1/6$. * This remaining $1/6$ of the work was completed by the Man alone in 6 days. * So, the Man's 1-day work = $(1/6) / 6 = 1/36$. (This means the man alone takes 36 days). * We need the Boy's 1-day work. * Combined 1-day work = Man's 1-day work + Boy's 1-day work * $1/24 = 1/36 + \text{Boy's 1-day work}$ * Boy's 1-day work = $1/24 - 1/36$ * Find the LCM of 24 and 36, which is 72. * Boy's 1-day work = $(3 - 2) / 72 = 1/72$. * Therefore, the boy alone would take 72 days. ### Exam Strategy & Shortcut They worked together for 20 out of the required 24 days, meaning they finished $20/24 = 5/6$ of the work. The man finished the remaining $1/6$ in 6 days. Thus, the man does $1/6$ work in 6 days, or $1/36$ per day. Total efficiency is $1/24$. Boy = $1/24 - 1/36 = 1/72$. So, 72 days. ### Common Pitfall A frequent error is assuming the man and boy worked together for 26 days minus 6 days of the man alone, but misinterpreting the timeline and calculating the boy's remaining workload backward. Always anchor the fractional work completed. ### Final Answer Therefore, the correct answer is **72 days**.