## Subject IV. (20 points)

Emil is waiting in line at a ticket booth, along with other people, standing in a row. Andrei, who is right in front of Emil, says: "Behind me, there are 5 times as many people as in front of me." Mihai, who is right behind Emil, says: "Behind me, there are 3 times as many people as in front of me." How many people are waiting at the ticket booth?

Prof. Sorin Borodi, "Alexandru Papiu Ilarian" Theoretical High School, Dej

## Grading Scale for 5th Grade (OLM 2014 - local stage)

## Official $10 \mathrm{p}$
## Subject IV.

Let $n$ be the number of people in front of Andrei. According to the data, there are $5 n$ people behind Andrei. The total number of people at the counter is $n+1+5 n=6 n+1$. (1)

In front of Mihai, there are $n+2$ people, and behind Mihai, there are $3(n+2)$ people. The total number of people at the counter can be written as $(n+2)+1+3(n+2)=4 n+9$ (2)

By equating (1) and (2), we get the equation $6 n+1=4 n+9$, with the solution $n=4$.

Substituting into (1) or (2) gives that the number of people at the counter is 25.