To solve this problem, we need to use the formula:

total cost = price per unit x number of units

We know that Bill paid $59.20 for 12 1/3 pieces of yarn, so we need to find the price per unit.

To do this, we can divide the total cost by the number of units:

price per unit = $59.20 ÷ 12 1/3

We can convert the mixed number to an improper fraction by multiplying the whole number (12) by the denominator (3) and adding the numerator (1):

price per unit = $59.20 ÷ (12 + 1/3)

price per unit = $59.20 ÷ (37/3)

price per unit = $59.20 x (3/37)

price per unit = $4.80

Therefore, Bill paid $4.80 per piece of yarn.