How many bills of each type are there in a wallet with $\$460$?

[bergausstein]

A wallet has $\$460$, in $\$5$, $\$10$, and $\$20$ bills. the number of $\$5$ bills exceeds twice the number of $\$10$ bills by $4$, while the number of $\$20$ bills is 6 fewer than the number of $\$10$ bills. how many bills of each type are there?

i solved this problem in two ways by 1st choosing the unknown represent the number of $\$10$ bills and 2nd by choosing the unknown represent the number of $\$20$ bills.

and i got these answers from my two methods, $14$ $\$10$ bills, $8$ $\$20$ bills and $32$ $\$5$ bills.

on my third method i chose the unknown represent the number of $\$5$ bills.
and here how it goes,

let $x=$ number of $\$5$ bills,
$\frac{x}{2}-4=$ number of $\$10$ bills
$\frac{x}{2}-4-6=$ number of $\$20$ bills

$5x+10\left(\frac{x}{2}-4\right)+20\left(\frac{x}{2}-10\right)=460$
$5x+5x-40+10x-200=460$
$20x-240=460$
$20x=700$
then, $x=35$ ---> from here the number of $\$5$ bill is bigger than my previous result. can you pinpoint where is the mistake here? thanks!

p.s use only one variable.

 

[MarkFL]

If $x$ is the number of \$5 bills then:

$$\frac{x-4}{2}$$ is the number of \$10 bills.

$$\frac{x-16}{2}$$ is the number of \$20 bills.

 

[bergausstein]

If $x$ is the number of \$5 bills then:

$$\frac{x-4}{2}$$ is the number of \$10 bills.

$$\frac{x-16}{2}$$ is the number of \$20 bills.

if i use that $x=32$ which is the number of $\$5$ bill. thanks!

if some question why did you divide $x-4$ by $2$? in my first method what the number of $\$5$ bill is $2x+4$
that's why i thought of taking the opposite operation so i came up with $\frac{x}{2}-4$.

 

[MarkFL]

If $x$ is the number of \$5 bills and $y$ is the number of \$10 bills, then we have:

$$x=2y+4\implies y=\frac{x-4}{2}$$

 

[Jameson]

I think you might have solved this problem already, but just for practice can you show how to get the number of \$20 bills in terms of $x$? MarkFL showed how to get the number of \$10 bills but didn't show all the steps as to how he got $\dfrac{x-16}{2}$ for the number of \$20 bills. Can you show how to get this value? :)