A can build up a wall in 8 days while B can break it in 3 days. A has worked for 4 days and then B joined to work with A for another 2 days only. In how many days will A alone build up the remaining part of the wall?

### Concept & Negative Work When an entity destroys or reverses the work being done, their work rate is considered negative. $$ \text{Net Work} = \text{Positive Work Rate} - \text{Negative Work Rate} $$ ### Step-by-Step Solution * **Given:** A can build in 8 days, B can break in 3 days. * A's 1 day work = $1/8$ * B's 1 day work = $-1/3$ (negative because B is destroying the wall) * A works alone for 4 days: Work done = $4 \times (1/8) = 1/2$. * A and B work together for 2 days: Work done = $2 \times (1/8 - 1/3) = 2 \times (\frac{3 - 8}{24}) = 2 \times (-5/24) = -5/12$. * Net work completed so far = $1/2 - 5/12 = 6/12 - 5/12 = 1/12$. * Remaining work to be done = $1 - 1/12 = 11/12$. * Now, A alone must complete this $11/12$ of the wall. * Time taken by A alone = $\frac{\text{Remaining Work}}{\text{A's Rate}} = \frac{11/12}{1/8} = \frac{11}{12} \times 8 = 22/3 = 7\frac{1}{3}$ days. ### Exam Strategy & Shortcut Use the LCM method. LCM of 8 and 3 is 24. Let total work = 24 units. A's efficiency = $+3$ units/day. B's efficiency = $-8$ units/day. A works for 4 days: $4 \times 3 = 12$ units. A and B work for 2 days: $2 \times (3 - 8) = 2 \times (-5) = -10$ units. Total work done = $12 - 10 = 2$ units. Remaining work = $24 - 2 = 22$ units. A finishes the remaining work alone: $22 / 3 = 7\frac{1}{3}$ days. ### Common Pitfall Forgetting to assign a negative value to B's work rate, or miscalculating the net remaining work by adding B's destructive effort instead of subtracting it. ### Final Answer Therefore, the correct answer is **$7\frac{1}{3}$ days**.