1. A bridge over a river connects two different regions of a country. Once, one of the regions repainted the part of the bridge that belonged to it. If the freshly painted part of the bridge had been $30 \%$ larger, then the unpainted part would have been $50 \%$ smaller. Could the painted part of the bridge be exactly half of the bridge? What part of the bridge needs to be repainted (or vice versa) so that exactly half of the bridge is painted?
# Solution

Let $x$ and $y$ be the fractions of the painted and unpainted parts, respectively. Clearly, $x + y = 1$. According to the condition, $1.3x + 0.5y = 1$. We obtain the equation

$$
x + y = 1.3x + 0.5y
$$

from which we can find the ratio

$$
\frac{x}{y} = \frac{5}{3}
$$

Now we can find the fraction of the painted part

$$
\frac{x}{x + y} = \frac{1}{1 + y / x} = \frac{1}{1 + 3 / 5} = \frac{5}{8} = 0.625 = 62.5\%
$$

Thus, the painted part is more than half of the bridge by $12.5\%$.

Answer: The painted part is more than half of the bridge by $12.5\%$.