Question 1105585
.
<pre>
Let A = amount of Alan's money (shares ?)
    B = amount of Bob's  money.


The condition says that {{{(B+0.25A)/(A - 0.25A)}}} = 2.    (1)


Simplify the ratio in the left side, step by step. You will have

{{{(B+0.25A)/(A - 0.25A)}}} = {{{(B+0.25)/(0.75A)}}} = {{{B/(0.75*A)}}} + {{{(0.25*A)/(0.75*A)}}} = {{{B/(0.75*A)}}} + {{{0.25/0.75}}} = {{{B/(0.75*A)}}} + {{{1/3}}} = {{{(4/3)*(B/A)}}} + {{{1/3}}}.


So we can write (1) in an equivalent form

{{{(4/3)*(B/A)}}} + {{{1/3}}} = 2,


which implies  

{{{(4/3)*(B/A)}}} = {{{2}}} - {{{1/3}}},  ====>  {{{(4/3)*(B/A)}}} = {{{5/3}}}  ====>  {{{B/A}}} = {{{(5/3)*(3/4)}}} = {{{5/4}}}.


<U>Answer</U>.  The ratio of Alan' share to Bob' share at first was  {{{A/B}}} = {{{4/5}}}.
</pre>