Answer to Question #322477 in Linear Algebra for Kenth Smith

Question

A, B, and C are on a betting game. B loses P350 of his money to A. As a result, A now has twice as much as what is left with B. Then, C loses P700 to B. As a consequence, C now has only one-third as much money as B would then have. If A loses P210 to C, C will have as much money as A would have left. How much did each have at the start?

Answer 1

Let's A, B, C had a, b, c at the start accordingly.

B loses P350 of his money to A:

$a + 350, b - 350, c;\\ a+350=2(b-350)$

Then, C loses P700 to B:

$a+350, b-350+700=b+350, c-700;\\ c-700=\cfrac{1}{3}(b+350)$

A loses P210 to C:

$a+350-210=a+140,b+350,c-700+210=c-490;\\ a+140=c-490.$

So,

$\begin{cases} a+350=2b-700 \\ 3c-2100=b+350\\ a+140=c-490 \end{cases}$

$\begin{cases} a&-2b&&=-1050\\ &-b&+3c&=2450\\ a&&-c&=-630 \end{cases}$

$\begin{cases} a-2b=-1050 \\ b=3c-2450\\ a=c-630 \end{cases}$

$c-630-2\cdot(3c-2450)=-1050\\ -5c=-1050+630-4900\\ -5c=-5320\\c=1064\\ a=1064-630=434\\ b=3\cdot1064-2450=742.$

At the start A had P434, B had P742, C had 1064.