Calculate work done in joint days

Since A and B together can complete the work in $5$ days, their combined 1-day work is $\frac{1}{5}$. They work together for $4$ days, so the amount of work completed is $$4 \times \frac{1}{5} = \frac{4}{5}$.

$\text{Work done} = \frac{4}{5}$

Find remaining work

The remaining fraction of the work after $4$ days is $$1 - \frac{4}{5} = \frac{1}{5}$.

$\text{Remaining work} = \frac{1}{5}$

Determine A's 1-day work (efficiency)

A completes this remaining $\frac{1}{5}$ of the work in $2$ days. Therefore, A's 1-day work is $$\frac{1/5}{2} = \frac{1}{10}$.

$\text{A's efficiency} = \frac{1}{10}$

Calculate total days for A alone

Since A's daily rate of work is $\frac{1}{10}$ of the total work per day, A can complete the entire work alone in $\frac{1}{1/10} = 10$ days.

$\text{Total days} = 10$