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:

\(\ds \paren {16 - 4} b + 2 e + 2 d\)

\(=\)

\(\, \ds \paren {10 - 4} e + 2 b + 2 d \, \)

\(\, \ds = \, \)

\(\ds \paren {8 - 4} d + 2 b + 2 e\)

\(\ds \leadsto \ \ \)

\(\ds 12 b + 2 e + 2 d\)

\(=\)

\(\, \ds 6 e + 2 b + 2 d \, \)

\(\, \ds = \, \)

\(\ds 4 d + 2 b + 2 e\)

\(\ds \leadsto \ \ \)

\(\ds 10 b\)

\(=\)

\(\, \ds 4 e \, \)

\(\, \ds = \, \)

\(\ds 2 d\)

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: $58$