I kept on getting this wrong until I started to make Work, Rate Time problems as a sentence
What we know.
> 6 Machines (m) work
x (unknown) rate for 12 days to complete (=) 1 job
>> hence 6 machines will do 1/12 of the job in 1 day
a rate of X
>> hence 1 machine will do 1/(12*6) of the job
a rate of X
>>> hence x = 1(12*6)
>>> hence X = 1/(72)
> (6 original machines + new machine[s]) work
x (same rate) for 8 days to complete (=) 1 job
>> hence (6 original machines + new machine[s]) will do 1/8 of the job in 1 days
>> hence 1 machine will do 1/8* (6 original machines + new machine[s]) of the in 1 day
>>> hence X = 1/(8* (6+NM)
>>> hence X = 1/(48 + 8NM)
Now we solve
Finally, since X is the same in both we can do the simple equation
Part 1 X = Part 2 X
> 1/(72) = 1/(48 + 8NM)
> 1*(48 + 8NM) =1*(72)
> 8NM = 72-48
> 8NM = 24
> NM = 24/8
> NM = 3