Two workers are working on identical workpieces and both are currently meeting the standard. With the application of an innovation, one completes a workpiece 5 minutes faster than the standard time, while the other completes it 3 minutes faster. As a result, their average performance increases to 137.5%. What is the standard time for completing one workpiece?
Let the sought standard time be $x$ minutes. According to the standard time, the combined productivity of two workers per minute is $\frac{1}{x}+\frac{1}{x}=\frac{2}{x}$ units. After the improvement, the combined productivity per minute is $\frac{1}{x-5}+\frac{1}{x-3}$ units. According to the problem,

$$
\frac{1}{x-5}+\frac{1}{x-3}=\frac{2}{x} \cdot 1.375=\frac{2}{x} \cdot \frac{11}{8}=\frac{11}{4 x}
$$

thus,

$$
4 x(x-3)+4 x(x-5)=11(x-3)(x-5),
$$

which simplifies to

$$
3 x^{2}-56 x+165=0
$$

from which

$$
x_{1}=15 \text { minutes. }
$$

The second root $\left[x_{2}=\frac{11}{3}\right]$ does not meet the requirements of our problem.

Almási Lajos (Bp., II., Rákóczi g. III. o. t.)