Algebra- show all work

Question

Pipe A together with Pipe B can fill a tank in 45 minutes. Pipe B and Pipe C can fill the same tank in 60 minutes. Pipe A and Pipe C can fill the tank in 90 minutes. How long will it take for all 3 pipes (together) to fill the tank?

Answer 1

Let the tank capacity be T gallons. Pipe A delivers at a rate of A gallons per minute, pipe B at B gpm, and pipe C at C gpm. In 45 mins pipe A will deliver 45A gallons and pipe B 45B gallons, so 45A+45B=T. Similarly, 60B+60C=T and 90A+90C=T. We need to find t in t(A+B+C)=T. A+B=T/45; B+C=T/60; A+C=T/90. Add these together: 2(A+B+C)=T(1/45+1/60+1/90)=(T/15)(1/3+1/4+1/6)=(T/15)(4+3+2)/12=T/20. Therefore A+B+C=T/40. Now we can calculate t=T/(A+B+C)=40 minutes.