A and B can separately do a piece of work in 20 and 15 days respectively. They worked together for 6 days, after which B was replaced by C. If the work was finished in next 4 days, then the number of days in which C alone could do the work will be

### Concept & Residual Work and Replacement When a worker is replaced mid-project, calculate the total work completed by the initial team, find the remaining work, and determine the new worker's efficiency based on the new team's completion time. ### Step-by-Step Solution * **Determine Total Work and Initial Efficiencies:** * Let Total Work = LCM of 20 and 15 = 60 units. * A's efficiency = $60 \div 20 = 3 \text{ units/day}$. * B's efficiency = $60 \div 15 = 4 \text{ units/day}$. * **Calculate Work Done in First 6 Days:** * A and B combined efficiency = $3 + 4 = 7 \text{ units/day}$. * Work done in 6 days = $6 \times 7 = 42 \text{ units}$. * Remaining work = $60 - 42 = 18 \text{ units}$. * **Analyze the Final Phase:** * B is replaced by C, so the new team is A and C. * They complete the remaining 18 units in 4 days. * Combined efficiency of A + C = $18 \div 4 = 4.5 \text{ units/day}$. * **Find C's Efficiency and Time:** * We know A's efficiency is 3. * $3 + \text{C's efficiency} = 4.5 \implies \text{C's efficiency} = 1.5 \text{ units/day}$. * Time for C alone to complete 60 units = $60 \div 1.5 = 40 \text{ days}$. ### Exam Strategy & Shortcut Use fractions if you prefer avoiding LCM. In 6 days, A does $\frac{6}{20} = \frac{3}{10}$ of the work, B does $\frac{6}{15} = \frac{2}{5} = \frac{4}{10}$. Total done = $\frac{7}{10}$. Remaining = $\frac{3}{10}$. In the next 4 days, A does $\frac{4}{20} = \frac{2}{10}$. This means C must do the final $\frac{1}{10}$ of the total work in those 4 days. If C does $\frac{1}{10}$ of the work in 4 days, C takes $4 \times 10 = 40$ days for the whole work. ### Common Pitfall A frequent mistake is forgetting that A continues to work alongside C for the last 4 days. Some calculate C's efficiency as if C alone finishes the remaining work in 4 days, leading to a wrong efficiency of $4.5$ and an incorrect final time. ### Final Answer Therefore, the correct answer is **40**.