将这些分数按从小到大的顺序排列。 $\boxed{2\frac{1}{2}\quad,\quad\frac{13}{5}\quad,\quad2\frac{3}{7}}$ 答：___, ___, ___
$\text{Smallest}\qquad\qquad\qquad\qquad\qquad\text{Largest}$

1. Convert the improper fraction to a mixed number to compare the whole numbers.

$\frac{13}{5} = 2\frac{3}{5}$

2. Compare the fractional parts by finding a common denominator.

The fractional parts are $\frac{1}{2}$, $\frac{3}{5}$, and $\frac{3}{7}$.
The lowest common multiple of $2$, $5$, and $7$ is $70$.
$\frac{1}{2} = \frac{35}{70}$
$\frac{3}{5} = \frac{42}{70}$
$\frac{3}{7} = \frac{30}{70}$

3. Arrange the fractions from smallest to largest based on the numerators.

Since $30 < 35 < 42$, we have $\frac{30}{70} < \frac{35}{70} < \frac{42}{70}$.
Therefore, $2\frac{3}{7} < 2\frac{1}{2} < 2\frac{3}{5}$.
The order is $2\frac{3}{7}$, $2\frac{1}{2}$, $\frac{13}{5}$.