A and B can complete a work in 18 days and 15 days respectively. They started doing the work together but after 3 days A had to leave and B alone completed the remaining work. The whole work was completed in

### Concept & Total Work Calculation To solve problems involving workers leaving mid-way, find the total work by taking the Least Common Multiple (LCM) of their individual times. Then, calculate daily efficiencies to determine the work done before someone left and the remaining work. $$ \text{Total Work} = \text{Efficiency} \times \text{Time} $$ ### Step-by-Step Solution * **Given:** A can complete the work in 18 days, B in 15 days. They work together for 3 days. * **Calculation:** Let Total Work = LCM(18, 15) = 90 units. * Efficiency of A = $90 / 18 = 5$ units/day. * Efficiency of B = $90 / 15 = 6$ units/day. * Combined efficiency of A + B = $5 + 6 = 11$ units/day. * Work done by A and B together in 3 days = $11 \times 3 = 33$ units. * Remaining work = $90 - 33 = 57$ units. * After 3 days, A leaves. So, B completes the remaining 57 units alone. * Time taken by B to complete remaining work = $57 / 6 = 19/2 = 9\frac{1}{2}$ days. * The question asks for the total time to complete the *whole* work. * Total time = Time worked together + Time worked by B alone = 3 days + $9\frac{1}{2}$ days = $12\frac{1}{2}$ days. ### Exam Strategy & Shortcut Instead of fractional methods which cause messy additions, always jump straight to LCM values. LCM of 18 and 15 is 90. Efficiencies are 5 and 6. Together they do 11 per day for 3 days (33 units). Remaining 57 is done by B (speed 6) in $57/6 = 9.5$ days. Total time = $3 + 9.5 = 12.5$ days. ### Common Pitfall A very common mistake is finding the time B takes for the *remaining* work ($9\frac{1}{2}$ days) and selecting that as the answer. You must carefully read whether the question asks for the remaining time or the total time. ### Final Answer Therefore, the correct answer is **$12\frac{1}{2}$ days**.