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.
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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
- 1821: John Jackson: Rational Amusement for Winter Evenings
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Rational Amusements for Winter Evenings: $154$