# 11.2. (7 points)

The master makes a whole number of parts in one hour, more than 5, and the apprentice makes 2 parts less. The master completes the order in a whole number of hours, and two apprentices together - one hour faster. How many parts does the order consist of?
Answer: 24.

Solution: Let $x$ be the number of parts in the order; $y$ (parts per hour) be the master's productivity $(y>5)$; $y-2$ (parts per hour) be the apprentice's productivity; $2 y$ - 4 (parts per hour) be the productivity of two apprentices.

Then $\frac{x}{y}$ is the time for the master to complete the order, $\frac{x}{2 y-4}$ is the time for the apprentices to complete the order.

According to the condition, $\frac{x}{y}=\frac{x}{2 y-4}+1$.

Expressing $x$ from this equation, we get: $x=\frac{2 y^{2}-4 y}{y-4}, \frac{x}{y}=\frac{2 y-4}{y-4}$.

Consider $2 y-4=2 y-8+4=2(y-4)+4$.

Then $\frac{x}{y}=\frac{2(y-4)+4}{y-4}=2+\frac{4}{y-4}$.

According to the condition $\frac{x}{y} \in Z, 2 \in Z$. Therefore, $\frac{4}{y-4} \in Z$.

$y>5$, then $y-4=2, y=6$ and $y-4=4, y=8$.

Find $x: \frac{x}{6}=4, x=24$ or $\frac{x}{8}=3, x=24$.