Question 196079
Their rates of mowing add to get their rate
working together
In words:
(number of lawns mowed by Shelly)/(time it takes Shelly to mow them) plus
(number of lawns mowed by William)/(time it takes William to mow them) equals
(number of lawns mowed by Shelly and william together)/(time to mow them)
Let Shelly's time to mow 1 lawn = {{{t}}} hrs
Then William's time = {{{t + 5}}} hrs
Time mowing together = {{{5}}} hrs
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{{{1/t + 1/(t+5) = 1/5}}}
Multiply both sides by {{{5t*(t + 5)}}}
{{{5*(t + 5) + 5t = t*(t + 5)}}}
{{{5t + 25 + 5t = t^2 + 5t}}}
{{{t^2 - 5t = 25}}}
Solve by completing the square:
{{{t^2 - 5t + (5/2)^2 = 25 + (5/2)^2}}}
{{{t^2 - 5t + 25/4 = 100/4 + 25/4}}}
{{{(t - 5/2)^2 = 125/4}}}
Take the square root of both sides
{{{t - 5/2 = 5*sqrt(5)/2}}}
{{{t = 5*(1 + sqrt(5))/2}}}
{{{t = 8.090}}} hrs
{{{t + 5 = 13.090}}} hrs 
William's time mowing alone is 13.1 hrs
check answer:
{{{1/t + 1/(t+5) = 1/5}}}
{{{1/8.09 + 1/13.09 = 1/5}}}
{{{.1236 + .0764 = .2}}}
{{{.199994 = .2}}} close enough