# A balance problem

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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?

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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: