J = hours it takes Jill by herself
K = hours it takes Jack by himself
so hmm if we look at say 1 hr interval, we can say that in 1hr Jack, who can do the whole thing in 7 hours, has really done only 1/7 th of the whole work, likewise, we can also say that in the same hour, Jill has done only 1/J of the whole work, since working together it takes them 4 hours, then in that one hour they have done together only 1/4 of the whole work.
[tex]\underset{\textit{\LARGE whole work for 1 hr}}{\stackrel{\textit{Jack has done}}{\cfrac{1}{7}}~~ + ~~\stackrel{\textit{Jill has done}}{\cfrac{1}{J}}}~~ = ~~\stackrel{\textit{whole work done}}{\cfrac{1}{4}}[/tex]
[tex]\stackrel{\textit{multiplying both sides by }\stackrel{LCD}{28J}}{28J\left(\cfrac{1}{7}~~ + ~~\cfrac{1}{J} \right)~~ = ~~28J\left( \cfrac{1}{4} \right)}\implies 4J~~ + ~~28~~ = ~~7J \\\\\\ 28=3J\implies \cfrac{28}{3}=J\implies 9\frac{1}{3}=J\impliedby \stackrel{\textit{9 hours and 20 minutes}}{560~minutes}[/tex]
