Examples of Algebra Problems/Hindu Problem

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:

That is:

and the result follows.