4.1. To the fraction $\frac{1}{6}$, some fraction was added, and the result turned out to be a proper fraction with a denominator less than 6. What is the largest fraction that could have been added?

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Answer: $\frac{19}{30}$

Solution. Since the result of adding fractions is a proper fraction with a denominator less than 6, the maximum value of the sum is $\frac{4}{5}$. To obtain it, it is necessary to add $\frac{4}{5}-\frac{1}{6}=\frac{19}{30}$.