Answer
Jerome takes 10.5 hours, Tim takes 9.5 hours
Work Step by Step
Jerome = $x$, Tim = $y$
$1/x + 1/y = 1/5$
$x-1=y$
$1/x + 1/y = 1/5$
$1/x + 1/(x-1) = 1/5$
$1/x + 1/(x-1) = 1/5$
$5x*(1/x + 1/(x-1)) = 1/5*5x$
$5 + 5x/(x-1) = x$
$(x-1)*(5+5x/(x-1)) = x*(x-1)$
$5(x-1) + 5x = x*(x-1)$
$5x-5+5x =x^2-x$
$10x-5 =x^2-x$
$10x-5-10x+5=x^2-x-10x+5$
$0 = x^2-11x+5$
$a=1$, $b=-11$, $c=5$
$x=(-b±\sqrt {b^2-4ac})/2a$
$x=(-(-11)±\sqrt {(-11)^2-4*1*5})/2*1$
$x=(11±\sqrt {121-20})/2*1$
$x=(11±\sqrt {101})/2$
$x=(11±10.0498)/2$
$x=(11±10.0498)/2$
$x=(11+10.0498)/2$
$x=21.0498/2$
$x=10.52$
$x=10.5$
$x=(11±10.0498)/2$
$x=(11-10.0498)/2$
$x=.9502/2$
$x=.4751$
This answer is not valid since this value would mean Tim takes a negative amount of time. We can't have a negative amount of time.
$10.5-1=9.5$
