Without using algebra:
Those 18 extra nickels are worth 18×$0.05 = $0.90. So if he took out the 18
extra nickels, he'd have only $5.40-$0.90 = $4.50 left with the same number
of nickels as quarters. If he then were to pair them up by taping a nickel
to each quarter, he'd have a bunch of quarter-nickel pairs each worth $0.30.
$4.50÷$.30 = 15 quarter-nickel pairs.  So there are 15 quarters and 18
additional nickels, or 15+18 = 33 nickels.

Using algebra:
Let the number of quarters = q
Then the number of nickels = q+18

   $0.25q + $0.05(q+18) = $5.40
$0.25q + $0.05q + $0.90 = $5.40 
         $0.30q + $0.90 = $5.40
                 $0.30q = $4.50
                      q = $4.50/$0.30
                      q = 15

The number of quarters = q = 15
The number of nickels = q+18 = 15+18 = 33

Edwin