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 . Thus, the information we have tells us that, for some constants :
In fact, we don’t exactly care what are. By subtracting from all equations and dividing by , we get:
We can just give the names and to the quantities and .
Our task is to compute in terms of , , , , and . This can be done by solving for and in terms of ,,, and eliminating them from the implicit expression for in the last equation. Perhaps there is a shortcut, but this will work:
So the answer is: