The product of two fractions is

Let the other fraction be x.

Now, we have:

$9 \frac{3}{7} \times x=9 \frac{3}{5}$

$\Rightarrow x=9 \frac{3}{5} \div 9 \frac{3}{7}$

$=\left(9+\frac{3}{5}\right) \div\left(9+\frac{3}{7}\right)$

$=\frac{48}{5} \div \frac{66}{7}$

$=\frac{48}{5} \times \frac{7}{66}$

$=\frac{48 \times 7}{5 \times 66}$

$=\frac{336}{330}$

$=\frac{56}{55}$

$=1 \frac{1}{55}$

Hence, the other fraction is $1 \frac{1}{55}$.