31. Compare the fractions:

$$
\text { a) } \frac{373737}{777777} \text { and } \frac{37}{77} ; \text { b) } \frac{41}{61} \text { and } \frac{411}{611}
$$
31. a) Since $\frac{373737}{777777}=\frac{37 \cdot 10101}{77 \cdot 10101}=\frac{37}{77}$, the fractions are equal.

b) Let's find the fractions that complement the given ones to 1, and compare them.

$1-\frac{41}{61}=\frac{20}{61} ; \quad 1-\frac{411}{611}=\frac{200}{611} ; \quad \frac{20}{61}=\frac{200}{610}$.

Since $\frac{200}{610}>\frac{200}{611}$, then $\frac{20}{61}>\frac{200}{611}$, and therefore $\frac{41}{61}<\frac{411}{611}$.