Question #322477

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?


1
Expert's answer
2022-04-07T17:17:53-0400

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


B loses P350 of his money to A:

a+350,b350,c;a+350=2(b350)a + 350, b - 350, c;\\ a+350=2(b-350)


Then, C loses P700 to B:

a+350,b350+700=b+350,c700;c700=13(b+350)a+350, b-350+700=b+350, c-700;\\ c-700=\cfrac{1}{3}(b+350)


A loses P210 to C:

a+350210=a+140,b+350,c700+210=c490;a+140=c490.a+350-210=a+140,b+350,c-700+210=c-490;\\ a+140=c-490.


So,

{a+350=2b7003c2100=b+350a+140=c490\begin{cases} a+350=2b-700 \\ 3c-2100=b+350\\ a+140=c-490 \end{cases}


{a2b=1050b+3c=2450ac=630\begin{cases} a&-2b&&=-1050\\ &-b&+3c&=2450\\ a&&-c&=-630 \end{cases}


{a2b=1050b=3c2450a=c630\begin{cases} a-2b=-1050 \\ b=3c-2450\\ a=c-630 \end{cases}


c6302(3c2450)=10505c=1050+63049005c=5320c=1064a=1064630=434b=310642450=742.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.


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