Convert mixed numbers to improper fractions and perform the multiplication and division operations.

$$1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$$ $$1\frac{2}{9} = \frac{1 \times 9 + 2}{9} = \frac{11}{9}$$ $$1\frac{4}{37} = \frac{1 \times 37 + 4}{37} = \frac{41}{37}$$

To divide by a fraction, multiply by its reciprocal.

$$\frac{77}{36} \div \frac{41}{37} = \frac{77}{36} \times \frac{37}{41}$$

$$\frac{77 \times 37}{36 \times 41} = \frac{2849}{1476}$$

The result is an improper fraction. We can convert it to a mixed number if desired, but the improper fraction is also a valid answer.
To convert to a mixed number:
$2849 \div 1476 = 1$ with a remainder of $2849 - 1476 = 1373$.
So, $1\frac{1373}{1476}$.

$$\frac{2849}{1476}$$