A can do a piece of work in 14 days which B can do in 21 days. They begin together but 3 days before the completion of the work, A leaves off. The total number of days to complete the work is

### Concept & Leaving Work Early When someone leaves a few days before completion, express their working time as $(Total Time - Days left early)$ and equate the sum of their total work contributions to 1. $$ \text{Work by A} + \text{Work by B} = 1 $$ ### Step-by-Step Solution * **Given:** A takes 14 days, B takes 21 days. * A's 1 day work = $1/14$, B's 1 day work = $1/21$. * Let the total number of days to complete the work be $T$. * B worked for all $T$ days. * A left 3 days before completion, so A worked for $(T - 3)$ days. * Work done by A + Work done by B = 1 Total Work $(T - 3) \times (1/14) + T \times (1/21) = 1$ * Find the LCM of denominators (14, 21), which is 42. * Multiply the entire equation by 42: $3(T - 3) + 2T = 42$ $3T - 9 + 2T = 42$ $5T = 42 + 9$ $5T = 51$ * $T = 51/5 = 10\frac{1}{5}$ days. ### Exam Strategy & Shortcut Use the assumed completion method. Assume A did NOT leave and worked for those last 3 days. Total standard work = LCM of 14 and 21 = 42 units. A's efficiency = $42 / 14 = 3$ units/day. B's efficiency = $42 / 21 = 2$ units/day. If A stayed for those 3 days, extra work done = $3 \text{ days} \times 3 \text{ units/day} = 9$ units. New total assumed work = $42 + 9 = 51$ units. This new total is completed by both A and B working together the entire time. Combined efficiency = $3 + 2 = 5$ units/day. Total time = $51 / 5 = 10\frac{1}{5}$ days. ### Common Pitfall Setting up the equation incorrectly by assuming A worked for $T$ days and B worked for $T+3$ days, which misrepresents the timeline of the project completion. ### Final Answer Therefore, the correct answer is **$10\frac{1}{5}$**.