The ratio of efficiencies of $A$ and $B$ for doing a certain work is $5:8$. Working together they can complete the work in $20$ days. $A$ alone will complete $25\%$ of that work in:

Given the efficiency ratio of $A$ and $B$ is $5:8$.Let the efficiency of $A=5\text{ units/day}$ and $B=8\text{ units/day}$.Total efficiency of $(A+B)=5+8=13\text{ units/day}$.Since they complete the work in $20$ days, the total work is:
$$\text{Total Work}=13\times 20=260\text{ units}$$
We need to find the time taken by $A$ to complete $25\%$ of the total work:
$$\text{Work to be completed}=25\%\text{ of }260=\frac{1}{4}\times 260=65\text{ units}$$
Time taken by $A$:
$$\text{Time}=\frac{\text{Work}}{\text{Efficiency of }A}=\frac{65}{5}=13\text{ days}$$