Problem 6.5. Several classmates ate a cake together. Lёsha ate the most - $\frac{1}{11}$ of the entire cake, and Alёna ate the least - $\frac{1}{14}$ of the entire cake. How many classmates ate the cake? List all possible options.
Answer: 12 or 13.

Solution. If everyone in the class had eaten $\frac{1}{11}$ of the cake, like Lёsha, there would be 11 people in the class. But since everyone except Lёsha ate less than $\frac{1}{11}$, there are more than 11 people, that is, at least 12.

Similarly, we can show that there are no more than 13 people. If everyone in the class had eaten $\frac{1}{14}$ of the cake, like Alyona, there would be 14 people in the class. But since everyone except Alyona ate more than $\frac{1}{14}$, there are fewer than 14 people, that is, no more than 13.

There are two possible options left: 12 and 13. We will show that both are possible.

After Lёsha took $\frac{1}{11}$ of the cake and Alyona took $\frac{1}{14}$ of the cake, we will distribute the remaining cake equally among the rest (10 or 11 people). We need to understand why each of them will get less than $\frac{1}{11}$ of the cake but more than $\frac{1}{14}$ of the cake.

Suppose each of the rest got at least $\frac{1}{11}$ of the cake. Besides Alyona, there are at least 11 people in the class, each of whom ate at least $\frac{1}{11}$ of the cake. Then, in total, everyone except Alyona ate at least $\frac{11}{11}=1$, that is, a whole cake. But then Alyona would not have gotten anything, a contradiction. Therefore, each of the rest got less than $\frac{1}{11}$ of the cake. Suppose each of the rest got no more than $\frac{1}{14}$ of the cake. Besides Lёsha, there are no more than 12 people in the class, each of whom ate no more than $\frac{1}{14}$ of the cake. Then, in total, everyone except Lёsha ate no more than $\frac{12}{14}=\frac{6}{7}$ of the cake. Then the remaining at least $\frac{1}{7}$ of the cake should be eaten by Lёsha, but he only ate $\frac{1}{11}<\frac{1}{7}$, a contradiction. Therefore, each of the rest got more than $\frac{1}{14}$ of the cake.