False Balance Problem

Classic Problem

 * A Cheshire cheese being put into one of the pans of a false balance,
 * was found to weigh $16$ pounds,
 * and when put into the other pan only $9$ pounds.


 * What is the true weight?

Solution
$12$ pounds.

Proof
It is assumed that the reason for the falseness of this balance is that its arms are of unequal lengths.

Let $W$ pounds be the true weight.

Let the arms of the balance be $p$ and $q$.

The vertical force exerted on the pan of the balance is proportional to the length of the arm and the mass of the body being weighed.

Then we have: