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From Mahavira (Posted on 2017-07-18)

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.

No Solution Yet Submitted by Jer

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Analytical Solution (spoiler)

| Comment 1 of 11

Let the merchants be a, b, c and the purse p

Then a + p = 2b + 2c

b + p = 3a + 3c

c + p = 5a + 5b

Eliminating p,

a + (5a + 5b - c) = 2b + 2c

b + (5a + 5b - c) = 3a + 3c

Simplifying

6a + 3b = 3c

2a + 6b = 4c

Eliminating b

10a = 2c

Simplifying

c = 5a

Then b = 3a

p = 15a

The minimum is when a = 1

(a,b,c,p) = (1,3,5,15), which checks out.

Posted by Steve Herman on 2017-07-18 10:18:14

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