Problem 5. All students from $V I^{a}, V I^{b}, V^{c}, V I^{d}$ and $V^{e}$, along with their class advisors, went to the cinema. In $V I^{b}$ there is one more student than in $V I^{a}$, and in $V I^{c}$ there are 2 more students than in $V I^{b}$, in $V I^{d}$ there are 3 more students than in $V I^{c}$, and in $V I^{e}$ there are 4 more students than in $V I^{d}$. The cinema has 12 rows with 18 seats each. How many students are there in each of $V^{a}, V^{b}, V^{c}, V I^{d}$ and $V I^{e}$, if 96 seats in the cinema remained empty?
Solution. The cinema has a total of $12 \cdot 18=216$ seats. Out of this number of seats, 96 remained empty, so the total number of people present in the cinema is 
$216-96=120$, and the number of students is $120-5=115$. Let there be $x$ students in $V I^{a}$. Then in $V I^{b}$ there are $x+1$, in $V I^{c}$ there are $x+1+2=x+3$, in $V I^{d}$ there are $x+1+2+3=x+6$, and in $V I^{e}$ there are $x+1+2+3+4=x+10$. We have,

$$
x+x+1+x+3+x+6+x+10=115
$$

which simplifies to $5 x+20=115$, from which $5 x=95$, so $x=19$. Therefore, in $V I^{a}$ there are 19 students, in $V I^{b}$ there are 20 students, in $V I^{c}$ there are 22 students, in $V I^{d}$ there are 25 students, and in $V I^{e}$ there are 29 students.

## VII Department