The Time taken by Anjali to finish a piece of work is four times the time taken by Ron and Three times the time taken by John. If all three of them work together, it takes them 4 days to complete the entire work. How much time Ron alone take to finish the work?

Given data:

Time taken by Anjali to finish work is 4 times the time taken by Ron and 3 times the time taken by John.

Time taken by all three working together = 4 days

Concept:

The reciprocal of the time taken by each person equals the work done by them per day. The sum of their work per day equals the reciprocal of the time taken by them working together.

Calculation:

⇒ Let time taken by Ron be x days.

⇒ Therefore, time taken by Anjali = 4x days, time taken by John = 4/3x days

⇒ Work done by Ron per day = 1/x, by Anjali = 1/4x, by John = 3/4x

⇒ Total work done per day = 1/x + 1/4x + 3/4x = 1/4

⇒ (4 + 1 + 3)/4x = 1/4

⇒ 8/4x = 1/4

⇒ Solving for x gives x = 8

Therefore, Ron takes 8 days to finish the work alone.
Alternate Method 

Time ∝ 1/Efficiecny

days →   Anjali : Ron

                   4    :   1

So

Efficiency →   Anjali : Ron

                          1     :  4

same as,

days →   Anjali : John

                   3    :   1

So

Efficiency →   Anjali : John

                          1     :  3

So

Efficiency →   Anjali : Ron : John

                          1    :   4    :   3

So

Total work = Total days * Total efficiency 

Total work = 4 * (1 + 4 + 3) = 32 units

days required to Ron to complete the work = 32/4 = 8 days