Determine with as few calculations as possible which of the following two fractions is larger:

$\frac{33333333331}{33333333334}, \quad \frac{22222222221}{22222222223}$.
I. solution: Transform the first fraction as follows:

$$
\begin{gathered}
t_{1}=\frac{33333333331}{33333333334}=\frac{33333333334-3}{33333333334}= \\
=1-\frac{3}{33333333333+1}=1-\frac{1}{11111111111+\frac{1}{3}}
\end{gathered}
$$

Similarly,

$$
t_{2}=\frac{22222222221}{22222222223}=1-\frac{1}{11111111111+\frac{1}{2}}
$$

The two differences thus obtained differ only in the denominator of the subtrahend. The denominator in \( t_{2} \) is larger than the denominator in \( t_{1} \), so the subtrahend in \( t_{2} \) is smaller than the subtrahend in \( t_{1} \), and thus \( t_{2} \) is larger than \( t_{1} \). The second fraction is the larger one.

II. solution: By extending the fractions by 2 and 3, respectively,

$$
\frac{66666666662}{66666666668} \text { and } \frac{66666666663}{66666666669}
$$

we need to compare. The second fraction is obtained from the first by increasing both the numerator and the denominator by 1. With this change, a larger proper fraction is formed from positive proper fractions, and a smaller improper fraction is formed from positive improper fractions. If \( 0 < a < b \) and \( c > d > 0 \), then

$$
\frac{a+1}{b+1} > \frac{a}{b} \quad \text { and } \quad \frac{c+1}{d+1} < \frac{c}{d}
$$

In our problem, we are dealing with proper fractions, so the second fraction is larger.

Corradi Gábor (Győr, Czuczor G. g. I. o. t.)

Note. The direction of change in the improper fraction \( \frac{c}{d} \) can also be determined from the fact that the reciprocal of an improper fraction is a proper fraction, - if the number 1 is not considered an improper fraction.

III. solution: Our fractions can be simply written if we denote the number 11111111111 by \( c \):

$$
\frac{3 c-2}{3 c+1} \text { and } \frac{2 c-1}{2 c+1}
$$

After bringing to a common denominator, the numerators are:

$$
6 c^{2}-c-2 \quad \text { and } \quad 6 c^{2}-c-1
$$

the latter is larger, so the second fraction is larger.

Tapody György (Budapest, Bem J. g. II. о. .t.)