What is the expected answer for the mass ratio in a lever with lengths a and A?

In a lever, the mass ratio is determined by the lengths of the arms on either side of the fulcrum, which translates to the principle of moments or levers. According to the principle of moments, an object will be in equilibrium when the clockwise moments equal the counterclockwise moments. This can be expressed mathematically as:

(Mass of the load) x (Distance from the fulcrum to the load) = (Mass of the effort) x (Distance from the fulcrum to the effort).

If we denote the mass of the load as M, the mass of the effort as m, the distance from the fulcrum to the load as A, and the distance from the fulcrum to the effort as a, we can rearrange the equation to establish the mass ratio:

M/A = m/a.

This can be rewritten to express the mass ratio as:

M/m = a/A.

Thus, the correct relationship for the mass ratio in terms of the lengths of the lever arms is M/m = a/A. The lengths of the lever arms are inversely proportional to the masses if the moments of the lever are balanced. This is why this answer is the expected answer in the context of lever mechanics.