Answer
$2 \text{ hours}
$
Work Step by Step
Let $x$ be the time it takes to fill the cartons.
In terms of $1$ unit/part, the rates are related as
\begin{array}{l}\require{cancel}
\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7.5}=\dfrac{1}{x}
.\end{array}
Using the $LCD=30x,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
30x\left( \dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7.5} \right)=\left(\dfrac{1}{x}\right)30x
\\\\
6x(1)+5x(1)+4x(1)=30(1)
\\\\
6x+5x+4x=30
\\\\
15x=30
\\\\
x=\dfrac{30}{15}
\\\\
x=2
.\end{array}
Hence, it takes $
2 \text{ hours}
$ to fill the carton when all three machines are working.
