A can do a piece of work in 63 days and B is $40\%$ more efficient than A. Then the number of days taken by B to do the same work is:

Let the efficiency of A be 100.
Since B is $40\%$ more efficient than A, efficiency of B = $100+40=140$.

The ratio of efficiency of A to B is:

$$\text{Efficiency Ratio (A : B)}=100:140=5:7$$

Since efficiency is inversely proportional to the time taken to complete the work:

$$\text{Time Ratio (A : B)}=7:5$$

Given that A takes 63 days:

$$7\text{ units}=63\text{ days}$$

$$1\text{ unit}=\frac{63}{7}=9\text{ days}$$

Therefore, the number of days taken by B:

$$5\text{ units}=5\times 9=45\text{ days}$$