A balance has unequal arms and pans of unequal weight. It is used to weigh three objects. The first object balances against a weight $A$, when placed in the left pan and against a weight $a$, when placed in the right pan. The corresponding weights for the second object are $B$ and $b$. The third object balances against a weight $C$, when placed in the left pan. What is its true weight?

A balance scale will balance when the torques exerted on both sides cancel out. On each of the two sides, the total torque will be

In fact, we don’t exactly care what

We can just give the names

Our task is to compute

So the answer is: