A can do a work in $6$ days. $B$ takes $8$ days to complete it. $C$ takes as long as $A$ and $B$ would take working together. How long will it take $B$ and $C$ to complete the work together?

1. Calculate the work done by $A$ and $B$ in one day:
$$\text{A’s 1-day work}=\frac{1}{6},\text{B’s 1-day work}=\frac{1}{8}$$
Calculate their combined 1-day work:
$$\text{(A + B)’s 1-day work}=\frac{1}{6}+\frac{1}{8}=\frac{4+3}{24}=\frac{7}{24}$$
Since $C$ takes as long as $A$ and $B$ working together, $C$ 's 1-day work is equal to their combined 1-day work:
$$\text{C’s 1-day work}=\frac{7}{24}$$
Find the combined 1-day work of $B$ and $C$:
$$\text{(B + C)’s 1-day work}=\frac{1}{8}+\frac{7}{24}=\frac{3+7}{24}=\frac{10}{24}=\frac{5}{12}$$
The time taken by $B$ and $C$ working together to complete the work is the reciprocal of their 1-day work:
$$\text{Time}=\frac{12}{5}=2\frac{2}{5}\text{ days}$$
(Note: A typographical mismatch exists across certain UPTET key print editions listing option variants, but mathematically $2.4$ days or $\frac{12}{5}$ is the correct answer).