Let rates be $a,b,c$. Given: $a+b=\tfrac{1}{15}$, $b+c=\tfrac{1}{20}$, $a+b+c=\tfrac{1}{10}$.

Using $(a+b)+(a+c)+(b+c)=2(a+b+c)$:

$\tfrac{1}{15}+(a+c)+\tfrac{1}{20}=\tfrac{1}{5}\ \Rightarrow\ a+c=\tfrac{1}{5}-\left(\tfrac{1}{15}+\tfrac{1}{20}\right)=\tfrac{1}{12}$

Time by $A+C = \dfrac{1}{1/12}=12$ days.

Answer: $\boxed{12\text{ days}}$ ✅