Task 3. (15 points) At the quality control department of an oil refinery, Engineer Pavel Pavlovich received a research object consisting of about 100 oil samples (a container designed for 100 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low-sulfur or high-sulfur, and density - either light or heavy. The relative frequency (statistical probability) that a randomly selected sample will be a heavy oil sample is $\frac{1}{7}$. The relative frequency that a randomly selected sample will be a light low-sulfur oil sample is $\frac{9}{14}$. How many high-sulfur oil samples does the object contain, if there were no low-sulfur samples among the heavy oil samples?
# Solution.

Let's determine the exact number of oil samples. It is known that the relative frequency of a sample being a heavy oil sample is $\frac{1}{7}$. The number closest to 100 that is divisible by $7$ is $98$. Therefore, there are 98 samples in total in the container. The samples of high-sulfur oil include all the heavy oil samples ($98: 7=14$) and a portion of the light oil samples. Let's find out how many high-sulfur oil samples are among the light oil samples. The total number of light oil samples is $98:7 \cdot 6=84$. Among them, the samples of low-sulfur oil constitute $\frac{9}{14}$ of the total, which is 63. Therefore, the number of high-sulfur light oil samples is 21. Thus, the total number of high-sulfur oil samples is: $14+21=35$.

Answer. 35.