A shoemaker sold two pairs of shoes at the same price. On the first pair he makes a profit of $20\%$ and on the other pair he makes a loss of $20\%$. What is his percentage of profit or loss in the whole business?

When two items are sold at the exact same selling price, one at a profit of $x\%$ and the other at a loss of $x\%$, there is always an overall net loss.
The formula to calculate the overall loss percentage is:

$$\text{Loss Percentage}=\frac{x^2}{100}$$

Given $x=20$:

$$\text{Loss Percentage}=\frac{20^2}{100}=\frac{400}{100}=4\%$$

Therefore, the trader incurs a net loss of $4\%$ in the entire transaction.