A can do a piece of work in $63\text{ days}$ 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 = $140$.

Ratio of efficiency of A to B = $100:140=5:7$.

Time taken is inversely proportional to efficiency:

$$\frac{\text{Time taken by B}}{\text{Time taken by A}}=\frac{\text{Efficiency of A}}{\text{Efficiency of B}}$$

$$\frac{\text{Time taken by B}}{63}=\frac{5}{7}$$

$$\text{Time taken by B}=\frac{5}{7}\times 63=5\times 9=45\text{ days}$$