True Weight from False Balance/Imbalanced Pans

Theorem
Let $B$ be a body whose weight is $W$.

Let $B$ be weighed in a false balance with imbalanced pans.

Let the readings of the weight of $B$ be $a$ and $b$ when placed in opposite pans.

Then:
 * $W = \dfrac {a + b} 2$

Proof
We have that the false balance has pans such that one weighs more than the other.

Let the lengths of the arms of the false balance be $x$.

Let one of the pans weigh $m$ more than the other.

Placing $B$ in the lighter pan gives:
 * $W x = \paren {a + m} x$

and placing $B$ in the heavier pan gives:
 * $\paren {W + m} x = b x$

Then: