Henry Ernest Dudeney/Modern Puzzles/186 - The False Scales/Solution 1
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Modern Puzzles by Henry Ernest Dudeney: $186$
- The False Scales
- A pudding, when placed into one of the pans of a balance, appeared to weigh $4$ ounces more than $\tfrac 9 {11}$ of its true weight,
- What was its true weight?
Solution
Suppose that the false balance has imbalanced pans.
That is, the pans are of different weight.
Then the true weight of the pudding is $154$ ounces.
Proof
Let $W$ ounces be the true weight of the pudding.
We are given that:
Hence we have:
\(\ds W\) | \(=\) | \(\ds \dfrac {\paren {\tfrac {9 W} {11} + 4} + \paren {\tfrac {9 W} {11} + 52} } 2\) | True Weight from False Balance with Imbalanced Pans | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 W\) | \(=\) | \(\ds \dfrac {18 W} {11} + 56\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 22 W\) | \(=\) | \(\ds 18 W + 56 \times 11\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4 W\) | \(=\) | \(\ds 616\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds W\) | \(=\) | \(\ds 154\) | simplifying |
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $186$. -- The False Scales
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $97$. The False Scales