Compare mixed numbers and fractions to determine the correct inequality symbol.

To solve these problems, we need to compare the two mixed numbers in each question and decide which symbol goes in the box:
- > means "greater than" (the left number is bigger)
- < means "less than" (the left number is smaller)
- = means "equal to" (both numbers are the same)

Here is the step-by-step solution for each problem:

1. Compare $2\frac{2}{3}$ and $1\frac{1}{3}$
Look at the whole numbers first. 2 is greater than 1.
So, $2\frac{2}{3} > 1\frac{1}{3}$.

2. Compare $1\frac{5}{6}$ and $1\frac{3}{8}$
The whole numbers are both 1, so we look at the fractions $\frac{5}{6}$ and $\frac{3}{8}$.
Let's find a common denominator for 6 and 8, which is 24.
$\frac{5}{6} = \frac{20}{24}$
$\frac{3}{8} = \frac{9}{24}$
Since 20 is greater than 9, $\frac{5}{6}$ is bigger.
So, $1\frac{5}{6} > 1\frac{3}{8}$.

3. Compare $2\frac{3}{5}$ and $2\frac{3}{4}$
The whole numbers are both 2. Compare $\frac{3}{5}$ and $\frac{3}{4}$.
When numerators are the same, the fraction with the smaller denominator is bigger. Since 4 is smaller than 5, $\frac{3}{4}$ is bigger.
Alternatively, convert to decimals: $\frac{3}{5} = 0.6$ and $\frac{3}{4} = 0.75$.
So, $2\frac{3}{5} < 2\frac{3}{4}$.

6. Compare $1\frac{1}{4}$ and $2\frac{1}{2}$
Look at the whole numbers. 1 is less than 2.
So, $1\frac{1}{4} < 2\frac{1}{2}$.

7. Compare $1\frac{7}{8}$ and $1\frac{1}{2}$
Whole numbers are both 1. Compare $\frac{7}{8}$ and $\frac{1}{2}$.
Convert $\frac{1}{2}$ to eighths: $\frac{1}{2} = \frac{4}{8}$.
Since $\frac{7}{8}$ is greater than $\frac{4}{8}$, the first number is bigger.
So, $1\frac{7}{8} > 1\frac{1}{2}$.

8. Compare $2\frac{1}{5}$ and $1\frac{5}{8}$
Look at the whole numbers. 2 is greater than 1.
So, $2\frac{1}{5} > 1\frac{5}{8}$.

11. Compare $1\frac{5}{6}$ and $1\frac{5}{9}$
Whole numbers are both 1. Compare $\frac{5}{6}$ and $\frac{5}{9}$.
Same numerator rule: smaller denominator means bigger fraction. 6 is smaller than 9, so $\frac{5}{6}$ is bigger.
So, $1\frac{5}{6} > 1\frac{5}{9}$.

12. Compare $1\frac{1}{8}$ and $1\frac{1}{3}$
Whole numbers are both 1. Compare $\frac{1}{8}$ and $\frac{1}{3}$.
Same numerator rule: 3 is smaller than 8, so $\frac{1}{3}$ is bigger. This means the left side is smaller.
So, $1\frac{1}{8} < 1\frac{1}{3}$.

13. Compare $2\frac{3}{5}$ and $1\frac{1}{2}$
Look at the whole numbers. 2 is greater than 1.
So, $2\frac{3}{5} > 1\frac{1}{2}$.

Final Answer:
1. >
2. >
3. <
6. <
7. >
8. >
11. >
12. <
13. >