A can do a piece of work in 10 days while B can do it in 15 days.  They begin together but 2 days before the completion of the work.  A leaves off.  The total number of days for the work to completed by B in ____ days.

A can do work in 10 days $\Rightarrow$ Work done by A in 1 day $=\frac{1}{10}$

B can do work in 15 days $\Rightarrow$ Work done by B in 1 day $=\frac{1}{15}$

Work done by A and B in 1 day $=\frac{1}{10}+\frac{1}{15}=\frac{3+2}{30}=\frac{5}{30}=\frac{1}{6}$

Work left $=\frac{2}{6}=\frac{1}{3}$

Number of days to complete the work for $B=\frac{\frac{1}{3}}{\frac{1}{15}}=5$ days