I wonder if we can do some arithmetic operation, like $x+$y, within a string quote?
Yes you can. You can just let PHP caclulate the arithmetric operation and then assign it to a variable and output it.
You can also do that inside a double-quoted string (Demo):
<?php
// @link http://stackoverflow.com/a/18182233/367456
//
// Expected result is:
// 5 + 11 = 16
$x = 5;
$y = 11;
echo "$x + $y = ${0*${0}=$x + $y}"; # prints "5 + 11 = 16"
However that is probably not what you're looking for.