Henry Ernest Dudeney/Modern Puzzles/187 - Weighing the Goods/Solution
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Modern Puzzles by Henry Ernest Dudeney: $187$
- Weighing the Goods
- A tradesman whose morals had become corrupted during the war by a course of profiteering went to the length of introducing a pair of false scales.
- It will be seen from the diagram that one arm is longer than the other,
- though they are purposely drawn so as to give no clue as to the answer.
- As a consequence, it happened that in one of the cases exhibited eight of the little packets
- (it does not matter what they contain)
- exactly balanced three of the canisters,
- while in the other case one packet appeared to be of the same weight as $6$ canisters.
- Now, as the true weight of one canister was known to be exactly one ounce, what was the true weight of the eight packets?
Solution
- $8$ packets weigh $12$ ounces.
Proof
Let the arms of the balance be $x$ and $y$.
Let $P$ and $C$ ounces be the weights of package and canister respectively.
From the laws of physics and the rubric of the question:
\(\ds 3 C x\) | \(=\) | \(\ds 8 P y\) | eight of the little packets ... exactly balanced three of the canisters, | |||||||||||
\(\ds P x\) | \(=\) | \(\ds 6 C y\) | one packet appeared to be of the same weight as $6$ canisters. | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds P\) | \(=\) | \(\ds 6 C \dfrac y x\) | |||||||||||
\(\ds 3 C\) | \(=\) | \(\ds 8 P \dfrac y x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 C\) | \(=\) | \(\ds 8 \paren {6 C \dfrac y x} \dfrac y x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 1\) | \(=\) | \(\ds 16 \dfrac {y^2} {x^2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac y x\) | \(=\) | \(\ds \dfrac 1 4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds P\) | \(=\) | \(\ds \dfrac {6 C} 4\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 C} 2\) |
We are given that $C = 1$.
So $P = \dfrac 3 2$ and so $8 P = 12$.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $187$. -- Weighing the Goods
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $98$. Weighing the Goods