False Balance Problem

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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:

\(\ds W p\) \(=\) \(\ds 16 q\)
\(\ds W q\) \(=\) \(\ds 9 p\)
\(\ds \leadsto \ \ \) \(\ds W^2 p q\) \(=\) \(\ds 16 \times 9 p q\)
\(\ds \leadsto \ \ \) \(\ds W\) \(=\) \(\ds \sqrt {16 \times 9}\)
\(\ds \) \(=\) \(\ds 4 \times 3\)
\(\ds \) \(=\) \(\ds 12\)

$\blacksquare$


Sources