Velocity Ratio is defined by Classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

 as the distance that the point of Effort moved divided by the distance that the point of Load moved as given from here

Thus VR = Distance(Effort) / Distance(Load) = (S(i) / S(o))

According to the definition of Efficiency

Efficiency

Efficiency in general describes the extent to which time or effort is well used for the intended task or purpose. It is often used with the specific purpose of relaying the capability of a specific application of effort to produce a specific outcome effectively with a minimum amount or quantity of...

:- Efficiency = Energy Output / Energy Input = E(o) / E(i) = (Force Output * Distance of point of application of the Load moved) / (Force Input * Distance of point of application of the Effort moved) = ( F(o) * S(o) ) / ( F(i) * S(i) ). Thus

Efficiency (η) as η = ( F(o) * S(o) ) / ( F(i) * S(i) )
η = (F(o) / F(i)) * (S(o) / S(i))
η = (F(o) / F(i)) / (S(i) / S(o))
η = (F(o) / F(i)) / VR
η = MA / VR , where MA is defined as (F(o) / F(i))
Where MA = Mechanical advantage

Mechanical advantage

Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. Ideally, the device preserves the input power and simply trades off forces against movement to obtain a desired amplification in the output force...

Hence, Efficiency (η) = MA / VR