Madhu takes twice as much time as Uma to complete a work and Rahul does it in the same time as Madhu and Uma together. If all three working together can finish the work in 6 days, then the time taken by Madhu to finish the work is = ?
Correct Answer: 36 days
Suppose Uma takes x days to complete a work
Then, Madhu takes 2x days to complete the work
Uma's 1 day's work = $$\frac{1}{x}$$
Madhu's 1 day's work = $$\frac{1}{{2x}}$$
Rahul's 1 day's work = (Madhu + Uma)'s 1 day's work
$$\eqalign{ & = \frac{1}{{2x}} + \frac{1}{x} \cr & = \frac{3}{{3x}}{\text{ }} \cr} $$
(Madhu + Uma + Rahul)'s 1 day's work
$$\eqalign{ & = \frac{3}{{3x}} + \frac{1}{{2x}} + \frac{1}{x} = \frac{6}{{3x}} = \frac{3}{x} \cr & \therefore \frac{3}{x} = \frac{1}{6} \cr & \Rightarrow x = 18{\text{ }} \cr} $$
Hence, Madhu takes (2 × 18) = 36 days to complete the work
Then, Madhu takes 2x days to complete the work
Uma's 1 day's work = $$\frac{1}{x}$$
Madhu's 1 day's work = $$\frac{1}{{2x}}$$
Rahul's 1 day's work = (Madhu + Uma)'s 1 day's work
$$\eqalign{ & = \frac{1}{{2x}} + \frac{1}{x} \cr & = \frac{3}{{3x}}{\text{ }} \cr} $$
(Madhu + Uma + Rahul)'s 1 day's work
$$\eqalign{ & = \frac{3}{{3x}} + \frac{1}{{2x}} + \frac{1}{x} = \frac{6}{{3x}} = \frac{3}{x} \cr & \therefore \frac{3}{x} = \frac{1}{6} \cr & \Rightarrow x = 18{\text{ }} \cr} $$
Hence, Madhu takes (2 × 18) = 36 days to complete the work