A man and a boy, working together, can finish a task in 24 days. If, for the last 6 days, the man works alone, then the task can be finished in 26 days. In how many days can the boy alone finish the task?

Correct option is D

Given:

Time taken to finish by a man and a boy working together = 24 days

If man works alone in last 6 days then time taken is 26 days

Formula Used:

Let total work be 1.

Then work done 1 day $$=\frac{1}{Total\ time\ to\ work}$$​​

Solution:

Time taken by a man and a boy = 24 days

Let the time taken by man to work alone be x

And time taken by boy to work alone be y

Then

​$$\frac{1}{x} + \frac{1}{y} = \frac{1}{24}$$​  ….(1)

If man does the work alone for last 6 days then work is completed in 26 days

Then work done by man =$$\frac{1}{y} \times 26 = \frac{26}{x}$$​​

Work done by boy $$= \frac{1}{y} \times 20 = \frac{20}{y}$$​​

Then

​$$\frac{26}{x} + \frac{20}{y} =1$$​​

​$$\frac{6}{x} + \frac{20}{x} + \frac{20}{y} = 1$$​​

​$$\frac{6}{x} + 20 \left(\frac {1}{x} + \frac{1}{y}\right) = 1$$​​

$$\frac{6}{x} + 20 \times \frac{1}{24} = 1$$​       [Using eq(1)]

$$\frac{6}{x} = \frac{5}{6} = 1$$​

​$$\frac{6}{x} = \frac{1}{6}$$​​

​$$x= 36 days$$​​

Time taken by boy will be

​$$\frac{1}{36} + \frac{1}{y} = \frac{1}{24}$$​​

​$$\frac{1}{y} = \frac{1}{24} - \frac{1}{36}$$​​

​$$\frac{1}{y} = \frac{3-2}{72}$$​​

​$$y= 72 days$$​​