The equation is:

5n + 10d + 25q = 45

It can be divided through by 5
 
n + 2d + 5q = 9

We will consider two cases

Case 1.  Using no quarters, i.e., q=0, then

n + 2d + 5(0) = 9

n + 2d = 9

d = 

n must be an odd number to make the numerator even so it
can be divided evenly by the denominator 2.

1.  n=1 nickel, 4 dimes
2.  n=3 nickels, 3 dimes.
3.  n=5 nickels, 2 dimes.
4.  n=7 nickels, 1 dime.
5.  n=9 nickels, 0 dimes.

Case 2. q=1

n + 2d + 5(1) = 9

n + 2d + 5 = 9

n + 2d = 4

d = 

n must be an even number to make the numerator even so it
can be divided evenly by the denominator 2.  (Don't forget
that 0 is an even number!)

6.  n=0 nickels, 2 dimes
7.  n=2 nickels, 1 dime.
8.  n=4 nickels, 0 dimes.

So there are 8 ways to make 45 cents using nickels and/or dimes and/or 
quarters. 

Edwin