If A can do a job in 8 days and b can do the same job in 12 days, how long would it take them both to complete the job working together?

Question

anonymous · Answer

8 + 12 = 20 
would take 20 days.  
$$\color{red}{\text{ok joemath, take it away}}$$

anonymous · Answer

$$\frac{1}{\frac{1}{8}+\frac{1}{12}}$$

anonymous · Answer

just kidding. solve 
$$(\frac{1}{8}+\frac{1}{12})t=1$$  for t

anonymous · Answer

one day i am going to figure out a snap way to explain this.

anonymous · Answer

ends up being 
$$3x+4x=24$$ but why this is obvious eludes me. joemath?

anonymous · Answer

yeah >.< i just think of it like:
d = rt
the rate is inversely proportional to time, so i need to get the reciprocals of their times to get the "rates".   Then i add the rates together, but i want time as an answer, so i take the reciprocal of my answer......

its gross.

anonymous · Answer

you should have seem me with the jellybean problem.  took maybe half an hour!

anonymous · Answer

sounds intense o.O

anonymous · Answer

no joemath no one thinks like that. must be some obvious way to do it.  i mean look at the answer.  right hand side of equal sign in lcm of 8 and 12.  left hand side is 

$$\frac{24}{8}+\frac{24}{12}$$ which for one thing means my solution is wrong.  but still there must be some obvious reason for this yes?

anonymous · Answer

i guess it has to do with the fact that you would use the LCM to get a common denominator for the fractions?  and by the way, I think like that :(
lol