(a) Let the number of 50-cent coins Alvin and Bill each have be a and b, respectively.

Then, the value of their coins can be expressed as follows:

Alvin (value) = $2.50 + $0.50a
Bill (value) = $1.60 + $0.50b

Since they have the same number of coins,

25 + a = 16 + b (I)
Using this, we can get the following:

a = b - 9 (II)

The difference in value can be expressed as follows:

(1.60 + 0.50b) - (2.50 + 0.50a) → insert (II)

(1.60 + 0.50b) - [2.50 + 0.50 (b - 9)]
= 1.60 + 0.50b - 2.50 - 0.50b + 4.50
= 1.60 - 2.50 + 4.50
= 3.60

(b) Since Alvin has used all his fifty-cent coins, he now only has 10-cent coins left, with a total value of $2.50.

Calvin has not spent any money.
He has $2.50 + $20.00 = $22.50.

Hence, Calvin has $22.50 ÷ $0.50 = 45 fifty-cent coins.