A and B can do a piece of work in 28 and 35 days respectively. They began to work together but A leaves after some time and B completed the remaining work in 17 days. After how many days did A leave?

### Concept & Work Completion When someone leaves a job early, calculate the work done in the remaining days by the other person, then find how long they worked together on the rest. $$ \text{Total Work} = \text{Work Together} + \text{Remaining Work} $$ ### Step-by-Step Solution * **Given:** A's time = 28 days, B's time = 35 days. * A's 1 day work = $1/28$, B's 1 day work = $1/35$. * B completes remaining work in 17 days. Work done by B in 17 days = $17 \times (1/35) = 17/35$. * Work done by A and B together initially = $1 - 17/35 = 18/35$. * (A + B)'s 1 day work = $1/28 + 1/35 = (5 + 4) / 140 = 9/140$. * Number of days A and B worked together = $\frac{18/35}{9/140}$. * Days = $(18/35) \times (140/9) = 2 \times 4 = 8$ days. ### Exam Strategy & Shortcut Use LCM for total work. LCM of 28 and 35 is 140. Total work = 140 units. A's efficiency = $140/28 = 5$ units/day. B's efficiency = $140/35 = 4$ units/day. B works alone for 17 days: $17 \times 4 = 68$ units. Remaining work = $140 - 68 = 72$ units. This 72 units was done by (A+B) together. (A+B) efficiency = $5+4 = 9$ units/day. Days worked together = $72 / 9 = 8$ days. ### Common Pitfall Setting up the equation incorrectly by assuming they worked together for $(x-17)$ days instead of $x$ days, or confusing who did the remaining work. ### Final Answer Therefore, the correct answer is **8 days**.