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?

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 $q$.

$W = \sqrt {p q}$

Thus we have:

$\ds W$

$=$

$\ds \sqrt {16 \times 9}$

$\ds $

$=$

$\ds 4 \times 3$

$\ds $

$=$

$\ds 12$

$\blacksquare$