A and B together can complete a work in 12 days. A alone can complete it in 20 days. If B does the work only for half a day daily, then in how many days A and B together will complete the work ?

### Concept & Partial Day Work Rates To solve for modified work conditions, deduce each individual's full-day work rate (efficiency), adjust it for their new working hours (e.g., half a day), and combine these new rates to find the total time required. ### Step-by-Step Solution * **Find B's Full-Day Rate:** * Combined 1-day work (A + B) = $\frac{1}{12}$ * A's 1-day work = $\frac{1}{20}$ * B's 1-day work = (A + B) - A $$ \frac{1}{12} - \frac{1}{20} = \frac{5 - 3}{60} = \frac{2}{60} = \frac{1}{30} $$ * **Adjust B's Rate:** * If B works for only half a day, B's effective 1-day work is halved: $\frac{1}{2} \times \frac{1}{30} = \frac{1}{60}$ * **Calculate New Combined Rate:** * New combined 1-day work = A's full-day work + B's half-day work $$ \frac{1}{20} + \frac{1}{60} = \frac{3 + 1}{60} = \frac{4}{60} = \frac{1}{15} $$ * **Total Time:** * Since their new combined daily rate is $\frac{1}{15}$, it will take them exactly 15 days to finish the entire work. ### Exam Strategy & Shortcut Use the LCM method for total work. The LCM of 12 and 20 is 60 units. Efficiency of (A + B) = $60 \div 12 = 5$ units/day. Efficiency of A = $60 \div 20 = 3$ units/day. This means B's efficiency is $5 - 3 = 2$ units/day. If B works half-time, B produces $2 \div 2 = 1$ unit/day. New combined efficiency = $3 (\text{from A}) + 1 (\text{from B}) = 4$ units/day. Total time = $\frac{60}{4} = 15$ days. ### Common Pitfall A frequent mistake is finding the time B takes to complete the work alone (30 days), doubling it to 60 days (since B is working half-days), and then averaging 20 and 60 instead of properly adding their reciprocal work rates. ### Final Answer Therefore, the correct answer is **15 days**.