'A' can build up a structure in 8 days and 'B' can break it in 3 days. 'A' worked building for 4 days and then 'B' joined and start breaking while 'A' kept building for another 2 days. In how many days will 'A' alone build up the remaining part of the structure?

The structure is considered as 1 complete job.

A completes the structure in 8 days, so A's work rate per day $= \dfrac{1}{8}$

B breaks the entire structure in 3 days, so B's work rate per day $= \dfrac{1}{3}$

---

A works alone for 4 days.

Work done by A in 4 days:

$= 4 \times \dfrac{1}{8}$

$= \dfrac{4}{8}$

$= \dfrac{1}{2}$

After 4 days, half the structure is built.

---

A and B work together for 2 days.

When working together, A builds while B breaks.

Net work per day $= \dfrac{1}{8} - \dfrac{1}{3}$

Converting to common denominator:

$= \dfrac{3}{24} - \dfrac{8}{24}$

$= -\dfrac{5}{24}$

The negative value indicates the structure is getting destroyed overall.

Net work in 2 days:

$= 2 \times \left(-\dfrac{5}{24}\right)$

$= -\dfrac{5}{12}$

---

Total work completed after 6 days:

$= \dfrac{1}{2} - \dfrac{5}{12}$

$= \dfrac{6}{12} - \dfrac{5}{12}$

$= \dfrac{1}{12}$

Remaining work to complete the structure:

$= 1 - \dfrac{1}{12}$

$= \dfrac{11}{12}$

---

Time required for A to complete the remaining work:

$\text{Time} = \dfrac{\text{Work}}{\text{Rate}}$

$= \dfrac{11}{12} \div \dfrac{1}{8}$

$= \dfrac{11}{12} \times 8$

$= \dfrac{88}{12}$

$= \dfrac{22}{3}$ days

Therefore, A will need $\dfrac{22}{3}$ days to build the remaining part of the structure.