Question 1205615
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Edit: I realize I made an error forming the equation. The error has been fixed.


x = number of $2 notes
3x = number of $5 notes
5x = number of $10 notes
Since x is a number of something, it can only be a positive whole number {1,2,3,4,...}


The ratio x:3x:5x reduces to 1:3:5 after dividing all parts by the GCF x.



He spends all of his $10 notes, so we ignore the 5x portion.
He spends 5/6 of his $5 notes and has 1/6 of those notes remaining.


2x = value of just the $2 notes
5*(1/6)*3x = 5*0.5x = 2.5 = value of the remaining $5 notes


2x+2.5x = 4.5x = total value of all of his remaining money


4.5x = 783
x = 783/(4.5)
x = 174


Then,
x = 174 = number of $2 notes
3x = 3*174 = 522 = number of $5 notes
5x = 5*174 = 870 = number of $10 notes


So,
2*174 + 5*522 + 10*870 = <font color=red>11658</font>
The total amount of money he started with is <font color=red size=4>$11,658</font>


Check:
He spent all 870 of the $10 notes, so he has 11658 - 10*870 = 2958 dollars remaining.
Then he spent 5/6 of the 522 copies of the $5 notes, meaning he spent 5*(5/6)*522 = 5*435 = 2175 further dollars.
He is ultimately left with 2958 - 2175 = 783 dollars which confirms the answer is correct.
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