Question 891095
Aly and A.J., time for A.J. to do the job alone is t.


The description gives rates:
Aly, {{{1/(t-2)}}}
A.J., {{{1/t}}}
Combined Aly & A.J., {{{1/12}}} working together.


The combined rate is simply the sum of their rates.
{{{1/(t-2)+1/t=1/12}}}
LCD is {{{12t(t-2)}}}, so multiply both sides by this expression.
{{{12t+12(t-2)=t(t-2)}}}
{{{12t+12t-24=t^2-2t}}}
{{{26t-24=t^2}}}
{{{t^2-26t+24=0}}}
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{{{t=(26+- sqrt(26^2-4*24))/2}}}
{{{t=(26+- sqrt(580))/2}}}
Using decimal approximation
{{{highlight(t=(26+- 24.033)/2)}}}
{{{highlight(t=25)}}} or {{{cross(t=0.9835)willNotWork}}}
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A.J. could do the job alone in 25 hours.
(More accurately,25.02 hours.)