Rate, time and work

Question

A group of four pumps is designed to fill a tank such that three pumps work at 2/3rd the rate of the fourth pump. If all 4 pumps work at the same time, they should fill the tank in what fraction of the time taken by the largest pump working alone?

Answer 1

Let R be the rate of the largest pump so 2R/3 is the rate for each smaller pump.
Assume an arbitrary volume, V, for the tank.
Let T1 be the time to fill the tank with the large pump.
     T1 x R =V --> T1 = V/R
The effective rate for all 4 pumps is

      R4 R + (2/3)R + (2/3)R + (2/3)R = 3R
Let T4 be the time to fill the tank using all 4 pumps.
      T4 x 3R = V --> T4 = V/3R
So T4 / T1 is the fraction of the time taken by all 4 pumps of the time for the large pump alone.
Specifically, (V/3R) / (V/R) =  1/3