Question 626926
Suppose that A can mow the lawn in {{{t}}} hours.
Then, B can mow the lawn in {{{t+1}}} hours.


In 1 hour, A can mow {{{1/t}}} fraction of the lawn.
In 1 hour, B can mow {{{1/(t+1)}}} fraction of the lawn.


Therefore, in 1 hour, A and B can together mow {{{1/t + 1/(t+1)=(2*t+1)/(t*(t+1))}}} fraction of the lawn.


Thus, A and B can together mow the entire lawn in {{{t*(t+1)/(2*t+1)}}} hours.


Given, {{{t*(t+1)/(2*t+1)=20/9}}}

{{{9*t*(t+1)=20*(2*t+1)}}}

{{{9*t^2+9t=40*t+20}}}

{{{9*t^2-31*t-20=0}}}

{{{9*t^2-36*t+5*t-20=0}}}

{{{9*t*(t-4)+5*(t-4)=0}}}

{{{(t-4)*(9*t+5)=0}}}


Product of two quantities is zero if at least one of them is zero.

Therefore, either {{{t-4=0}}} or {{{9*t+5=0}}}.


Since t is time and can't be negative so {{{t-4=0}}} or {{{t=4}}} is the solution.

Ans: Gardener A can mow the lawn alone in 4 hours.