Rational exp/funct word problem?

Question

In an ice cream cone factory, machine A can produce a case of ice cream cones in 4 minutes, machine B in 3 minutes, and machine C in 6 minutes. If all three machines are running, how long will it take to produce a case? please show work, thanks

Answer 1

1 case/4 min + 1 case/3 min + 1 case/6 min = 1 case/x min
3/12 + 4/12 + 2/12 = 1/x
9/12 = 1/x
12 = 9x
4 = 3x
4/3 = x = 1 1/3 minutes or 1 minute and 20 seconds.

Answer 2

work backwards and measure each machine by how many cases it produces in a minute:

A is 1/4 case per minute, B is 1/3 and C is 1/6

If all three work together, the number of cases produced is:
1/4 + 1/3 + 1/6= (3+4+2)/12 = 3/4 case

So to produce 1 case, it would take 1/(3/4) = 1 and 1/3 minutes

Answer 3

Machine A produces 1/4 of a case per minute
Machine B produces 1/3 of a case per minute
Machine C produces 1/6 of a case per minute
Working together, they produce 1/4+1/3+1/6 case per minute.
1/4+1/3+1/6 = 3/12 + 4/12 + 2/12 = 9/12 = 3/4
They produce 3/4 case per minute, so it will take the 4/3 of a minute to produce one case.

Answer 4

x/4+x/3+x/6 = 1
Multiply by 12
3x + 4x + 2x = 12
9x = 12
x = 12/9 = 4/3 minutes