Examples of Algebra Problems/Hindu Problem
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Example of Problem in Algebra
- The first man has $16$ azure-blue gems,
- the second man has $10$ emeralds,
- and the third has $8$ diamonds.
- Each among them gives to each of the others $2$ gems of the kind owned by himself;
- and then all $3$ men come to be possessed of equal wealth.
- What are the prices of those azure-blue gems, emeralds and diamonds?
Solution
Let $b$, $e$ and $d$ denote the value of the azure-blue, emerald and diamond respectively.
Then:
- $b : e : d = 2 : 5 : 10$
Without knowing the price of any of the gems in any given monetary units, the best that can be done is to give their ratios.
Proof
We are given:
\(\ds \paren {16 - 4} b + 2 e + 2 d\) | \(=\) | \(\, \ds \paren {10 - 4} e + 2 b + 2 d \, \) | \(\, \ds = \, \) | \(\ds \paren {8 - 4} d + 2 b + 2 e\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 12 b + 2 e + 2 d\) | \(=\) | \(\, \ds 6 e + 2 b + 2 d \, \) | \(\, \ds = \, \) | \(\ds 4 d + 2 b + 2 e\) | |||||||||
\(\ds \leadsto \ \ \) | \(\ds 10 b\) | \(=\) | \(\, \ds 4 e \, \) | \(\, \ds = \, \) | \(\ds 2 d\) | subtracting $2 b + 2 e + 2 d$ from each expression | ||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac b 2\) | \(=\) | \(\, \ds \dfrac e 5 \, \) | \(\, \ds = \, \) | \(\ds \dfrac d {10}\) | dividing by $20$ throughout |
That is:
\(\ds 5 b\) | \(=\) | \(\ds 2 e\) | ||||||||||||
\(\ds 10 e\) | \(=\) | \(\ds 5 d\) |
and the result follows.
$\blacksquare$
Historical Note
This problem is given by David Wells in his Curious and Interesting Puzzles of $1992$ without any indication of its source.
A study of the sources which he quotes may reveal where he got it from, but this is work that is still to be done.
Sources
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Indian Puzzles: $58$