Roger earns 40 per day plus 4.50 commission for each pair of shoes he sells. How many pairs must he sell to earn $112 in a day?

Roger's total daily earnings are the sum of his daily wage and his commission from selling shoes.
Let $p$ be the number of pairs of shoes Roger sells.
His daily wage is $40. His commission is$4.50 per pair of shoes, so for $p$ pairs, it is $4.50p$. His daily earnings goal is $112.

The equation is:

$$40 + 4.50p = 112$$

The equation to determine how many pairs of shoes, $p$, Roger must sell is:

$$\boxed{40 + 4.50p = 112}$$

Subtract $40 from both sides of the equation:

$$4.50p = 112 - 40$$ $$4.50p = 72$$

Divide both sides by 4.50:

$$p = \frac{72}{4.50}$$ $$p = 16$$

Roger must sell 16 pairs of shoes to meet his daily earnings goal.

The number of pairs of shoes he must sell is:

$$\boxed{16 \text{ pairs of shoes}}$$