A bowl contains $30$ coins consisting of nickels, dimes, and quarters. The number of dimes is $4$ times the number of quarters. If the total value of the coins is $2.60$ dollar how many coins of each type are there?
I'm tutoring someone and I'm not quite sure why I keep getting this problem wrong. I set it up like this
$n = $number of nickels
$5n = $value of nickels
$d = $number of dimes
$10d= $value of dimes
$q = $number of quarters
$25q = $value of quarters
$d = 4q$
Then I set up two equations and had
$5n+10d +25q= 260$
which I rewrote as: $5n+65q=260 \tag1$
And then I had $n+q+d=30$
which I rewrote as $n+5q=30\tag 2$
Then I solved the system of equations and got $q$ to be approximately $2.75$ . I know this is wrong because it has to represent the number of quarters. I then rounded it off to 3 quarters which would mean I have $15$ nickels and $12$ dimes but with that number the total comes out to $2.70$ dollars. What am I doing wrong?