Question 1163137
Just going part of the way,

y of the fifty cent coins
x of the twenty cent coins


Initially, {{{y/x=2/3}}}


The exchange, {{{(y-4)/(x+4)=2/7}}}


You should be able to continue and to finish.

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The description and question as written seems to have NO WHOLE-NUMBER SOLUTION.
{{{y=9}}} and {{{x=27/2}}}.