A started a ,work and left after working for 2 days. Then B was called and he finished the work in 9 days. had A left the work after working for 3 days, B would have finished the remaining work in 6 days. In how many days can each of them, working alone, finish the whole work ?
Correct Answer: 5 days, 15 days
Suppose A takes x days to finish the work alone and B take y days to finish the work alone.
$$\eqalign{ & {\text{Then,}}\frac{2}{x} + \frac{9}{y} = 1.....(i) \cr & {\text{And,}}\frac{3}{x} + \frac{6}{y} = 1 \cr & \Leftrightarrow \frac{1}{x} + \frac{2}{y} = \frac{1}{3} \cr & \Leftrightarrow \frac{2}{x} + \frac{4}{y} = \frac{2}{3}.....({\text{ii)}} \cr & {\text{Subtracting (ii) from (i), }} \cr & {\text{We get}}:\frac{5}{y} = \frac{1}{3}{\text{or }}y = 15 \cr & {\text{Putting }}y = 15{\text{ in (i), }} \cr & {\text{We get}}:\frac{2}{x} = \frac{2}{5}{\text{or }}x = 5 \cr} $$
Hence, A alone takes 5 days while B alone takes 15 days to finish the work.
$$\eqalign{ & {\text{Then,}}\frac{2}{x} + \frac{9}{y} = 1.....(i) \cr & {\text{And,}}\frac{3}{x} + \frac{6}{y} = 1 \cr & \Leftrightarrow \frac{1}{x} + \frac{2}{y} = \frac{1}{3} \cr & \Leftrightarrow \frac{2}{x} + \frac{4}{y} = \frac{2}{3}.....({\text{ii)}} \cr & {\text{Subtracting (ii) from (i), }} \cr & {\text{We get}}:\frac{5}{y} = \frac{1}{3}{\text{or }}y = 15 \cr & {\text{Putting }}y = 15{\text{ in (i), }} \cr & {\text{We get}}:\frac{2}{x} = \frac{2}{5}{\text{or }}x = 5 \cr} $$
Hence, A alone takes 5 days while B alone takes 15 days to finish the work.