$8$ men or $12$ women can do a piece of work in $20$ days. The number of days needed to complete the job, if $10$ men and $15$ women are employed to do the same piece of work is:

Given that the work done by $8$ men is equal to the work done by $12$ women:
$$8\text{ men}=12\text{ women}⟹1\text{ man}=\frac{12}{8}\text{ women}=\frac{3}{2}\text{ women}$$

The problem states that $12$ women can complete the work in $20$ days.
Now, we need to find the time taken by $10$ men and $15$ women. Let's convert $10$ men into an equivalent number of women:

$$10\text{ men}=10\times \frac{3}{2}\text{ women}=15\text{ women}$$

Thus, the total team strength in terms of women is:

$$10\text{ men}+15\text{ women}=15\text{ women}+15\text{ women}=30\text{ women}$$

Using the work equivalence formula $M_1{D}_1=M_2{D}_2$:

$$12\times 20=30\times D_2$$

$$240=30\times D_2$$

$$D_2=\frac{240}{30}=8\text{ days}$$