3. By reducing the price of a notebook by half of its price, for 800 denars, one could buy one more notebook than for 900 denars before the reduction. What is the price of one notebook?
Solution. Let the price of one notebook be $x$. Before the price reduction, for 900 denars, a total of $\frac{900}{x}$ notebooks could be bought. After the price reduction by one fifth, for 800 denars, $\frac{800}{x-\frac{1}{5} x}$ notebooks could be bought. According to the condition of the problem, the number of notebooks in both cases differs by one, so $\frac{800}{x-\frac{1}{5} x}=\frac{900}{x}+1$. From here $\frac{800}{\frac{4}{5} x}=\frac{900+x}{x}$, or $\frac{800 \cdot 5}{4 x}=\frac{900+x}{x}$, which is $\frac{4000}{4 x}=\frac{900+x}{x}$. Now $\frac{1000}{x}=\frac{900+x}{x}$, or $1000=900+x$. From this, the price of one notebook is $x=100$ denars.