Answer
$\dfrac{60}{37} \text{ hours}$
Work Step by Step
Let $x$ be the time it takes to paint the fence when all three boys are working together.
In terms of $1$ unit/part, the rates are related as
\begin{array}{l}\require{cancel}
\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}=\dfrac{1}{x}
.\end{array}
Multiplying both sides by the $LCD=
60x
,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
60x\left(\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}\right)=\left(\dfrac{1}{x}\right)60x
\\\\
15x(1)+12x(1)+10x(1)=60(1)
\\\\
15x+12x+10x=60
\\\\
37x=60
\\\\
x=\dfrac{60}{37}
.\end{array}
Hence, it will take $
\dfrac{60}{37} \text{ hours}
$ to finish painting the fence when all three boys work together.
