Question 1197454
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Mike and gerry can finish building a doghouse in 4 hours. 
It would take mike 6 more hours {{{highlight(cross(that))}}} <U>THAN</U> gerry to finish the job working alone. 
How long would it take each boy if each works alone?
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Let &nbsp;{{{x}}}&nbsp; be the number of hours for gerry to cover the roof, &nbsp;working alone.
If Mike works alone, &nbsp;he can complete this job in &nbsp;{{{x+6}}}&nbsp; hours.
Thus, &nbsp;in one single day Mike covers &nbsp;{{{1/(x+6)}}}&nbsp; part of the roof area, &nbsp;while gerry covers &nbsp;{{{1/x}}}&nbsp; part of the roof area.
Working together, &nbsp;Mike and gerry make &nbsp;{{{1/(x+6)+1/x}}}&nbsp; of the whole work in each single day.
Since they can cover the entire roof in &nbsp;6&nbsp; hours working together, &nbsp;the equation for the unknown value &nbsp;{{{x}}}&nbsp; is as follows:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{4/(x+6) + 4/x = 1}}}.


To simplify this equation, &nbsp;multiply both sides by &nbsp;{{{(x+6)*x}}}, &nbsp;then transfer all terms 
from the right side to the left with the opposite signs, &nbsp;collect the common terms 
and adjust the signs. &nbsp;In this way you get


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{4x + 4(x+6) = x(x+6)}}},
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{4x + 4x +24 = x^2+6x}}},
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x^2+6x-4x-4x-24 = 0}}},
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x^2-2x-24 = 0}}},


You get the quadratic equation. &nbsp;It is factorable


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(x-6)*(x+4) = 0


The equation has two roots: &nbsp;{{{x[1]= 6}}}&nbsp; and &nbsp;{{{x[2]= -4}}}.
The second root &nbsp;{{{x[2]= -2}}}&nbsp; makes no sense and we decline it. 

So, &nbsp;the potentially correct solution is &nbsp;{{{x[1]= 6}}}: &nbsp;gerry covers the roof in &nbsp;6&nbsp; hours working alone.
Hence, Mike can complete the job in 6+6 = 12 hours.


Let us check it. 


Since &nbsp;{{{4/12+4/6}}} = {{{1/3 + 2/3}}} = {{{1}}}, &nbsp;this solution is correct.


<B>Answer</B>. &nbsp;gerry can finish the job in &nbsp;6&nbsp; hours working alone.  &nbsp;Mike can complete the job in &nbsp;12 &nbsp;hours.


Solved.