SOLUTION: How many ways can you make $10.75 using halves and quarters?

You can use from 0 up to 21 halves and the rest quarters.

Counting no halves that makes 22 ways.

You don't need any algebra, but if your teacher insiste that
you show some algebra, then:

0.25q + 0.50h = 10.75

    25q + 50h = 1075

       q + 2h = 43

            q = 43 - 2h

there must be at least 1 quarter and no more than 21 halves

          1 ≤ q ≤ 43

       1 ≤ 43 - 2h ≤ 43

Subtract 43 from all 3 sides

     -42 ≤ -2h ≤ 0

Divide all three sides by -2, reversing the inequality symbols,

       21 ≥ h ≥ 0 

There are 22 integers inclusively from 0 to 21.

Answer: 22 ways.

Here they all are:

1 halves and 41 quarters.
2 halves and 39 quarters.
3 halves and 37 quarters.
4 halves and 35 quarters.
5 halves and 33 quarters.
6 halves and 31 quarters.
7 halves and 29 quarters.
8 halves and 27 quarters.
9 halves and 25 quarters.
10 halves and 23 quarters.
11 halves and 21 quarters.
12 halves and 19 quarters.
13 halves and 17 quarters.
14 halves and 15 quarters.
15 halves and 13 quarters.
16 halves and 11 quarters.
17 halves and 9 quarters.
18 halves and 7 quarters.
19 halves and 5 quarters.
20 halves and 3 quarters.
21 halves and 1 quarters.

Edwin