Linear Algebra & Number Theory

Question

Three men came across a purse and picked it up. The first man, after looking at its contents, said: "If I get that purse, I'll have twice as much as your combined money", the second, after having done the same, said: "If I keep the purse, I'll have thrice as much as both of you have". The third, said: "If I'll get that purse, I'll have five times as much as your combined money. How much money was in the purse? How much money does each have?

Answer 1

Suppose the men have x,y,z and the purse has p

Then

x + p = 2y + 2z

y + p = 3x + 3z

z + p = 5x + 5y

----------------------------------------------------------------------------------------------------------------------------

-x + 2y + 2z = p (1)

3x - y + 3z = p (2)

5x + 5y - z = p (3)

-----------------------------------------------------------------------------------------------------------------------------

Here we have 3 equations in 4 unknowns, so we cannot solve uniquely for x, y, z, p.

You can solve for x, y, z in terms of p. ( just use elimination).

from (1), x = 2y + 2z - p

substitute for x in (2) and (3) giving two linear equations in y, z, and p.

Then solve for y,z and go back and find x