Question

one mechanic can do a certain job in 4 days, a second mechanic in 6 days, and a third in 8 days. in what time can the three together do this job?

Answer 1

Now the first mechanic takes 4 days to do the job, so after he has worked for 1 day only, he has done 1/4 of the whole thing.

the second mechanic can do it in 6 days, so after 1 day only of work, he has done 1/6 of the whole thing.

the third mechanic can do it in 8 days, so after a day, he has done 1/8 of the total work.

now, if they all three work together, after 1 hour has gone by, the first mechanic has done 1/4 of the whole thing, the second mechanic has done 1/6 of it, and the last one, has done 1/8.

let's say it took all three working together "t" days, that means after 1 hour of work, they all combined have done 1/t of the whole thing, thus

$\bf \stackrel{first~mechanic}{\cfrac{1}{4}}~+~\stackrel{second~mechanic}{\cfrac{1}{6}}~+~\stackrel{third~mechanic}{\cfrac{1}{8}}~=~\stackrel{total~work}{\cfrac{1}{t}}\\\\ -------------------------------\\\\ \textit{let's use our LCD of 24 on the left-hand-side} \\\\\\ \cfrac{6~+~4~+~3}{24}=\cfrac{1}{t}\implies \cfrac{13}{24}=\cfrac{1}{t}\implies t=\cfrac{24}{13}\implies \stackrel{\textit{1day, 20hrs, 18mins}}{t=1\frac{11}{13}}$