13.012. The working day has been reduced from 8 to 7 hours. By what percentage does labor productivity need to increase to ensure that, with the same rates, the wage increases by $5 \%$?
Solution.

Let the master produce $a$ parts and earn $b$ rubles for 8 hours of work. Then the rate is $b / a$ rubles per part, and the labor productivity is $a / 8$ parts per hour. After increasing productivity by $x \%$, the master started producing $\frac{a}{8}+\frac{x a}{8 \cdot 100}$ parts per hour. Therefore, in 7 hours, he produced $\frac{7 a}{8}\left(1+\frac{x}{100}\right)$ parts and earned $\frac{7 a}{8}\left(1+\frac{x}{100}\right) \frac{b}{a}=\frac{7 b}{8}\left(1+\frac{x}{100}\right)$ rubles. According to the condition, $\frac{7 a}{8}\left(1+\frac{x}{100}\right)=1.05 b$, from which $x=20 \%$.

Answer: $20 \%$.