Question 1163137
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            If you take a labor to solve the system of equations,  which @josgarithmetic prepared for you in his post,


            you will discover at the end that the system  HAS  NO  integer solution.


            If so,  it means that something is going wrong.


            So, I read the problem attentively,  and I came to the conclusion that the problem should be read and interpreted  DIFFERENTLY.


            Below is my interpretation and the full solution.



<pre>
According to the condition, we can assume that initially there were 2x of the 50-cent coins and 3x of the 20-cent coins in the box.


When 4 of the 50-cent coins were taken out, the number of the 50-cent coins became (2x-4).


When these 4 50-cent coins were exchanged for 20-cent coins, 10 of 20-cent coins were returned to the box,
making the number of the 20-cent coins in the box equal to (3x+10).


So, your proportion is now


    {{{(2x-4)/(3x+10)}}} = {{{2/7}}}.


From the proportion,


    7*(2x-4) = 2*(3x+10)

    14x - 28 = 6x + 20

    14x - 6x = 20 + 28

     8x      = 48

      x      = 48/8 = 6.


So, initially in the box there were  2*6 = 12 of the 50-cent coins  and  

                                     3*6 = 18 of the 20-cent coins.


Therefore, the total money in the box was  12*50 + 18*20 = 960 cents,  or  9.60 dollars.     <U>ANSWER</U>
</pre>

Solved.