Mahaviracharya/Ganita Sara Samgraha/Chapter VI/236-237

: Chapter $\text {VI}$: Mixed Problems: Problem $236 \text - 237$
$3$ merchants saw in the road a purse containing money.

One said:
 * If I secure this purse, I shall become twice as rich as both of you together.

Then the second said:
 * I shall become $3$ times as rich.

Then the third said:
 * I shall become $5$ times as rich.

What is the value of the money in the purse, as also the money on hand with each of the $3$ merchants?

Solution
The solution is not unique.


 * The second has $3$ times as much as the first.


 * The third has $5$ times as much as the first.


 * The purse holds $15$ times as much as the first owns.

Proof
Let $x$, $y$ and $z$ denote the amount of money owned by the $1$st, $2$nd and $3$rd merchant respectively.

Let $u$ denote the amount of money in the purse.

Let $a$, $b$ and $c$ denote the factor by which $x$, $y$ and $z$ respectively will be richer, were they to get the purse, than the other two combined.

We have:

Let $T = u + x + y + z$.

Then:

Setting $a = 2$, $b = 3$, $c = 5$ we have:

Hence:

and:

from which it is seen that:
 * $x : y : z = 1 : 3 : 5$

and the solution in smallest integers is: