Anuj and Manoj can together paint their house in 30 days. After working for 20 days, Anuj has to go out and Manoj finishes the remaining work in the next 30 days. If Manoj had gone away after 20 days instead of Anuj, then Anuj would have completed the remaining work in

### Concept & Remaining Fractional Work Determine the fraction of work completed during the combined working period to find the rate of the individual who completes the remaining fraction. $$ \text{Total Time} = \frac{\text{Time taken for Remaining}}{\text{Remaining Fraction}} $$ ### Step-by-Step Solution * **Given:** (Anuj + Manoj) can do the work in 30 days. * (Anuj + Manoj)'s 1 day work = $1/30$. * Work done by them together in 20 days = $20 \times (1/30) = 2/3$. * Remaining work = $1 - 2/3 = 1/3$. * Manoj finishes this $1/3$ work in 30 days. * Manoj's total time for the whole work = $30 \times 3 = 90$ days. * Manoj's 1 day work = $1/90$. * Anuj's 1 day work = (Combined 1 day work) - (Manoj's 1 day work) = $1/30 - 1/90 = \frac{3-1}{90} = 2/90 = 1/45$. * Anuj's total time for the whole work = 45 days. * If Manoj had left after 20 days, the remaining work would still be $1/3$. * Time Anuj would take to finish $1/3$ of the work = $(1/3) \times 45 = 15$ days. ### Exam Strategy & Shortcut (A+M) do the job in 30 days. After 20 days, 10 days of their combined work is left. Manoj takes 30 days to do what (A+M) would do in 10 days. So Manoj is $1/3$ as efficient as (A+M) combined. This means Anuj is $2/3$ as efficient as (A+M) combined. Therefore, Anuj is twice as efficient as Manoj. If Manoj takes 30 days to finish the remainder, Anuj (being twice as fast) will take half the time, which is $30 / 2 = 15$ days. ### Common Pitfall Calculating Anuj's total time (45 days) and selecting that as the answer, ignoring that the question asks for the time to complete the *remaining* work. ### Final Answer Therefore, the correct answer is **15 days**.