
Work/ratio
GRE practice
A group of 4 pumps are filling a tank. Each of the 3 smaller pumps works at 2/3rds the rate of the largest pump. If all 4 pumps work at the same time, they should fill the tank in what fraction of the time that it would have taken the largest pump if it operated alone?

EDIT: NVM, I don't think I did the work correctly.

Solution
Lets say that the rate of the large pump is $\displaystyle R$ and the rate of the small pump is $\displaystyle r$. If $\displaystyle r$ is$\displaystyle 2/3 $the rate of the large pump ($\displaystyle R$) than $\displaystyle r$ is equal to $\displaystyle (2/3)R$. If three of the pumps are small pumps and one is a large pump than $\displaystyle R+3(2/3)R = $ Total Rate Compared to Large Pump Rate. That equation simplifies to $\displaystyle 3R$ which means that all four pumps works at three times the rate of the large pump. So all four pumps work at the same rate as three large pumps. If you only have one pump than it does onethird of the job that all four pumps do in the same amount of time. Therefore the large pump works at $\displaystyle 1/3$ the speed of all four pumps together.

Make up simple example: tank size = 90 ; pumps = 30:20:20:20
large pump alone: 90/30 = 3
all pumps : 90/90 = 1
so 1/3