​Ramesh takes twice as long as Naresh and Mahesh together to complete a job. Naresh takes three times as long as Ramesh and Mahesh together to complete a job. If all the three working together can complete the job in 30days, then find the number of days Ramesh alone will take to finish the job.

Correct option is A

Given:

Time taken by Ramesh to complete a job = Twice time taken by Naresh and Mahesh together

Time taken by Naresh to complete a job = Thrice time taken by Ramesh and Mahesh together

Time taken by all working together =30 days

Formula Used:

Let total work be 1.

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

Solution:

Let time taken by Ramesh, Naresh and Mahesh be r, n, m respectively.

Then,

Work done  by Ramesh in one day will be $$\frac{Total\ Work}{Total\ time\ } =\frac{1}{r}$$​​

Work done  by Naresh in one day will be $$\frac{Total\ Work}{Total\ time} =\frac{1}{n}$$​​

Work done  by Mahesh will be $$\frac{Total\ Work}{Total\ time} =\frac{1}{m}$$​​

Let time taken by Naresh and Mahesh together to complete the work = $$x$$​​

Naresh and Mahesh one day’s work $$\frac{1}{n} + \frac{1}{m}=\frac{1}{x}$$​​

Then time taken Ramesh to complete the work = $$2x$$​​

Ramesh one day’s work = $$\frac{1}{2x}$$​​

According to question:

​$$\frac{1}{r} + \frac{1}{n} + \frac{1}{m} =\frac{1}{30}$$

​$$\frac{1}{2x} + \frac{1}{x} = \frac{1}{30}$$​

​$$\frac{1+2}{2x} =\frac{1}{30}$$​​

​$$x= 30 \times \frac{3}{2} = 45 \ days$$​​

Time taken by Ramesh to complete the job = $$\bf 2x = 90 \ days$$  

Alternate Method: 

According to the question; 

Efficiency ratio;

r : n + m = 1 : 2 

and , n : r + m = 1 : 3  

making ratio equal as total efficiency should be same in both case; 

r : n + m = 4 : 8 

and , n : r + m = 3 : 9    

So, individual efficiencies ; r = 4 , n = 3, m = 5 

total work to be done = $$30 \times( 4+3+5) = 30 \times 12$$ 

time taken by Ramesh alone = $$\frac{30 \times 12}{4} =$$ 90 days​