From the year c. 850 the book Ganita-Sara-Sangraha
contains the following:
Three merchants saw in the road a purse. 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 three times as rich."
Then the third said, "I shall become five times as rich."
What is the value of the money in the purse, as also the money on hand?
There are an infinite number of solutions. Find the smallest whole number amounts the merchants could have.
For brevity, I will refer to the three merchants as Alex (1st), Bert (2nd), and Carl (3rd).
Start with Bert: In order for Bert to triple his wealth after acquiring the purse, the purse must start with two times his wealth. So Bert:Purse = 1:2
Similarly from Carl's statement Carl:Purse = 1:4. This can be combined with Bert's 1:2 ration to make a compound ratio Bert:Carl:Purse = 2:1:4
Now Alex: If he is twice as rich as the union of Bert and Carl then he would have 6 times as much wealth as Bert, calculated from 2*(1+2). But this happens after Alex gets the purse, so he starts with 6-4=2 times as much wealth as Bert. Then the final ratio is:
Alex:Bert:Carl:Purse = 2:2:1:4