Time and Work

PIPES & CISTERNS:

Inlet Pipe: A pipe which is used to fill the tank is known as Inlet Pipe.

Outlet Pipe: A pipe which can empty the tank is known as an Outlet Pipe.
                                                     Efficiency * Total time taken = Total units of tank filled/emptied
                                                     Assume the tank to be of 1 unit.

• If a pipe can fill the tank in 'x' hours then the part filled per hour(efficiency)= 1/x
• If a pipe can empty a tank in 'y' hours, then the part emptied per hour(efficiency)= 1/y
• If pipe A can fill a tank 'x' hours and pipe B can empty a tank in 'y' hours, if they are both active at the same time, then

                                The part filled per hour  =$$\dfrac{1}{x}-\dfrac{1}{y}$$(if y>x)

                                The part emptied per hour  =$$\dfrac{1}{y}-\dfrac{1}{x}$$(if x>y)

• Efficiency * Total time taken = Total work
  Assume the total work to be 1 unit.
  
  If X can do the work in 'n' days, the fraction of work(efficiency) X does in a day is $$\frac{1}{n}$$ units/day.
• If X can do the work in 'x' days, and Y can do the work in 'y' days, the number of days taken by both of them together to do the work is $$\frac{x*y}{x+y}$$
• If $$A_1$$ men can do $$B_1$$ work in $$C_1$$ days and $$A_2$$ men can do $$B_2$$ work in $$C_2$$ days, then $$\frac{A_1 C_1}{B_1}$$ =$$\frac{A_2 C_2}{B_2}$$
  
  Note: 
  To simplify calculations, we try to get efficiencies as integers. We assume the total work to be some integral multiple of the total time taken. 
  
  Example: If X can do a work in 15 days and Y can do it in 12 days. Find the total days required to complete the work if X and Y both are working together.
  
  Sol. We assume the work to be LCM(15,12)=60 units to get the efficiencies of both X and Y in integers.
         Efficiency of X$$=\dfrac{60}{15}=4$$ units/day. 
  
         Efficiency of Y$$=\dfrac{60}{12}=5$$ units/day.
         So, when they work together the efficiencies would add up$$=4+5=9$$units/day.
         Let the time required to complete the work by X and Y together$$=a$$.
         So, $$9a=60$$.   $$a=\dfrac{60}{9}$$ days.