Answer
$\frac{F_o}{F_i}=\frac{L_i}{L_o}$.
Work Step by Step
Consider the diagram that accompanies the problem. Assume that the man pushes straight down on the end of the lever for a distance $h_i$, so the input work is $W_i=F_i h_i$. The object moves upward a shorter distance $h_o$, as seen from the diagram, so the output work is $W_o=F_o h_o$. Equate the two amounts of work as suggested.
$$W_i=F_i h_i= W_o=F_o h_o $$
$$\frac{F_o}{F_i}=\frac{h_i}{h_o}$$
However, the diagram shows that we have 2 similar triangles.
$$\frac{h_i}{h_o}=\frac{L_i}{L_o}$$
Therefore, $\frac{F_o}{F_i}=\frac{L_i}{L_o}$ as was to be proved.
