A can lay railway track between two given stations in 16 days and B can do the same job in 12 days. With the help of C, they did the job in 4 days only. Then, C alone can do the job in :

### Concept & Multi-Person Work Rates When two people are joined by a third to complete a task faster, the third person's efficiency is the difference between the total combined efficiency and the sum of the original two people's efficiencies. ### Step-by-Step Solution * **Given:** A's time alone = 16 days. B's time alone = 12 days. Combined time (A + B + C) = 4 days. * **Calculation:** Combined 1-day work = $\frac{1}{4}$ A's 1-day work = $\frac{1}{16}$ B's 1-day work = $\frac{1}{12}$ * C's 1-day work = (Combined 1-day work) - (A's 1-day work + B's 1-day work) C's 1-day work = $\frac{1}{4} - \left(\frac{1}{16} + \frac{1}{12}\right)$ * Find a common denominator to add the fractions inside the parenthesis. The LCM of 16 and 12 is 48. $\frac{1}{16} + \frac{1}{12} = \frac{3}{48} + \frac{4}{48} = \frac{7}{48}$ * Now, subtract this sum from the combined 1-day work ($\frac{1}{4}$ can be written as $\frac{12}{48}$): C's 1-day work = $\frac{12}{48} - \frac{7}{48} = \frac{5}{48}$ * Time taken by C alone = Reciprocal of C's 1-day work = $\frac{48}{5}$ days. * Convert to a mixed fraction: $48 \div 5 = 9$ with a remainder of 3. $\frac{48}{5} = 9\frac{3}{5}$ days. ### Exam Strategy & Shortcut Use the Total Work (LCM) method. Total Work = LCM(16, 12, 4) = 48 units. A's efficiency = $48 / 16 = 3$ units/day. B's efficiency = $48 / 12 = 4$ units/day. (A + B + C)'s combined efficiency = $48 / 4 = 12$ units/day. C's efficiency = Total - (A + B) = $12 - (3 + 4) = 12 - 7 = 5$ units/day. Time for C = Total Work / C's efficiency = $48 / 5 = 9\frac{3}{5}$ days. ### Common Pitfall A common mistake is adding the days $16 + 12$ or calculating the combined time of A and B first, and then making an error when subtracting that combined entity from the 4-day total. Always convert to unit rates/efficiencies first. ### Final Answer Therefore, the correct answer is **$9\frac{3}{5}$ days**.