1. One sixth of the total quantity of a certain commodity is sold at a profit of $20 \%$, and half of the total quantity of the same commodity is sold at a loss of $10 \%$. By what percentage profit should the remainder of the commodity be sold to cover the loss?
Solution. Let $k$ be the total quantity of goods, and $c$ be the planned price. From the condition of the problem, we get the equation

$$
\frac{k}{6} \cdot 0.2 c + \frac{k}{3} \cdot x c = \frac{k}{2} \cdot 0.1 c
$$

where $x$ is the desired percentage. The solution to this equation is $x = 0.05 = 5\%$.