1. In the periodic decimal fraction $0.242424 \ldots$, the first digit after the decimal point is replaced by 4. How many times greater is the resulting number than the original?
Solution. The original fraction is found as the sum of an infinite decreasing geometric progression: $\frac{24}{100\left(1-\frac{1}{100}\right)}=\frac{24}{99}$ (another way: let the original fraction be $x$, then $100 x=24+x$, so $\left.x=\frac{24}{99}\right)$. After replacing the digit, the number became $\frac{24}{99}+\frac{2}{10}=\frac{438}{99 \cdot 10}$. The desired ratio: $\frac{438}{99 \cdot 10} \cdot \frac{99}{24}=\frac{438}{240}=\frac{73}{40}$.

Answers in different versions: by $\frac{73}{40}$ times; by $\frac{16}{5}$ times; by $\frac{53}{20}$ times; by $\frac{20}{9}$ times.