13.047. A musical theater announced a competition for admission to the orchestra. Initially, it was planned that the number of places for violinists, cellists, and trumpeters would be distributed in the ratio $1.6: 1: 0.4$. However, it was then decided to increase the intake, and as a result, 25% more violinists and 20% fewer cellists were admitted than originally planned. How many musicians of each genre were admitted to the orchestra if a total of 32 people were admitted?
## Solution.

Let it be planned to recruit $1.6 x$ violinists, $x$ - cellists, $0.4 x$ trumpeters. Then, they recruited $1.25 \cdot 1.6 x$ violinists, $(1-0.2) x$ cellists, $0.4 x$ trumpeters. According to the condition, $1.25 \cdot 1.6 x + (1-0.2) x + 0.4 x = 32$, from which $x=10$. Therefore, they recruited $1.25 \cdot 1.6 \cdot 10=20$ violinists; $0.8 \cdot 10=8$ cellists; $0.4 \cdot 10=4$ trumpeters.

Answer: 20 violinists, 8 cellists, 4 trumpeters.