A can do a piece of work in $25$ days. After working for $5$ days, he took help of B and completed the work in $8$ days. If both A and B had worked together from the beginning, in how many days would they have completed the work?

Let the total work be $25$ units.Efficiency of A = $\frac{25}{25}=1$ unit/day.In $5$ days, work done by A = $5\times 1=5$ units.Remaining work = $25-5=20$ units.This remaining $20$ units of work is completed by A and B together in $8$ days.Combined efficiency of A and B:
$$\text{Efficiency of (A + B)}=\frac{\text{Remaining Work}}{\text{Days}}=\frac{20}{8}=2.5\text{ units/day}$$
If both A and B had worked together from the beginning, time taken to complete the total work:
$$\text{Time}=\frac{\text{Total Work}}{\text{Combined Efficiency}}=\frac{25}{2.5}=10\text{ days}$$