7.1 Vasya and Tolya exchanged badges. Before the exchange, Vasya had 5 more badges than Tolya. After Vasya exchanged $24\%$ of his badges for $20\%$ of Tolya's badges, Vasya had one fewer badge than Tolya. How many badges did the boys have before the exchange?
Answer. Toly had 45 badges, and Vasya had 50 badges. Solution. Let Toly have $x$ badges before the exchange, then Vasya had $(x+5)$ badges. After the exchange, Toly had $x-\frac{x}{5}+(x+5) \cdot \frac{6}{25}$, and Vasya had $x+5-(x+5) \cdot \frac{6}{25}+\frac{x}{5}$. Solving the equation

$$
x-\frac{x}{5}+(x+5) \cdot \frac{6}{25}-x-5+(x+5) \cdot \frac{6}{25}-\frac{x}{5}=1
$$

we find $x=45$.