4. In the store "Everything for School," chalk is sold in packages of three grades: regular, unusual, and superior. Initially, the ratio of the quantities by grade was 3:4:6. As a result of sales and deliveries from the warehouse, this ratio changed to 2:5:8. It is known that the number of packages of superior chalk increased by $80\%$, and the number of packages of regular chalk decreased by no more than 10 packages. How many packages of chalk were there in the store initially?

(7 points)
# Answer: 260 packages.

Solution: Let $x$ be the initial number of packages of regular chalk, then the number of packages of unusual chalk is $\frac{4 x}{3}$. Since the latter number is an integer, then $x=3 n$, where $n \in N$. Therefore, the initial quantities of all three types of packages are $3 n, 4 n, 6 n$ respectively.

After sales and deliveries, the quantity of superior chalk became $(1+0.8) 6 n=\frac{54 n}{5}$, and the quantity of unusual chalk became $\frac{54 n}{5} \cdot \frac{5}{2}=\frac{27 n}{4}$ packages. These numbers are integers, so they are divisible by 4 and 5, that is, $n=20 m$, where $m \in N$. The number of packages of regular chalk will be $\frac{27 n}{20} \cdot 2=\frac{27 n}{10}$. Considering the condition of the problem (the number of packages of regular chalk decreased by no more than 10 packages), we get $0<3 n-\frac{27 n}{10}=(60-54) m=6 m \leq 10$. The only natural number that satisfies this inequality is $m=1$. Therefore, $n=20 m=20$. Thus, the initial number of packages of chalk in the store was

$$
3 n+4 n+6 n=13 n=13 \cdot 20=260
$$