8-3. Vika has been recording her grades since the beginning of the year. At the beginning of the second quarter, she received a five, after which the proportion of fives increased by 0.15. After another grade, the proportion of fives increased by another 0.1. How many more fives does she need to get to increase their proportion by another 0.2?
Answer: 4.

Solution: Let's say Vika had $n$ grades in the first quarter, of which $k$ were fives. Then, after the first five in the second quarter, the proportion of fives increased by $\frac{k+1}{n+1}-\frac{k}{n}=0.15$. Similarly, after the second five, the increase was $\frac{k+2}{n+2}-\frac{k+1}{n+1}=0.1$. Simplifying each equation, we get the system

$$
\left\{\begin{array}{c}
n-k=0.15 n(n+1) \\
n-k=0.1(n+1)(n+2)
\end{array}\right.
$$

In particular, $0.15 n(n+1)=0.1(n+1)(n+2)$, which means $1.5 n=n+2, n=4$. Substituting this value into the first equation, we find that $k=4-0.15 \cdot 4 \cdot 5=1$. Therefore, after receiving two fives in the second quarter, the proportion of fives became $3 / 6=0.5$. Vika wants the proportion of fives to be $0.5+0.2=0.7$ after receiving another $m$ fives, which means $\frac{3+m}{6+m}=0.7$. Solving this equation, we get $m=4$.

Criteria: Answer without justification - 0 points. System of equations formulated - 2 points. Complete solution - 7 points.