Let a machine overcome a load L by the application of an effort E . In time t , let the displacement of effort be dE and the displacement of load be dL .
‌ Work input ‌‌=‌ Effort ‌×‌ displacement of effort ‌
‌=E×dE
Work output = Load × displacement of load
=L×dL
Efficiency =‌

‌ Work output ‌

‌ Work input ‌

=‌

L×dL

E×dE

=‌

L

E

×‌

dL

dE

=‌

L

E

×‌

1

dE∕dL

But ‌‌‌

‌ Load (L) ‌

‌ Effort (E) ‌

= M.A. and ‌

dE

dL

= V.R.
∴‌‌ Efficiency ‌η‌=‌

‌ M.A. ‌

‌ V.R. ‌

‌ or ‌‌‌ M.A. ‌‌=‌ V.R. ‌×η
Thus, the mechanical advantage of a machine is equal to the product of its efficiency and Velocity Ratio.