Find a common denominator for (2)/(9) and (3)/(8).

Here are the solutions to the problems:

2. Insert two rational numbers between $\frac{2}{9}$ and $\frac{3}{8}$, and arrange in descending order.

Step 1: Find a common denominator for $\frac{2}{9}$ and $\frac{3}{8}$.
The least common multiple (LCM) of 9 and 8 is 72.

$$\frac{2}{9} = \frac{2 \times 8}{9 \times 8} = \frac{16}{72}$$ $$\frac{3}{8} = \frac{3 \times 9}{8 \times 9} = \frac{27}{72}$$

Step 2: Find two rational numbers between $\frac{16}{72}$ and $\frac{27}{72}$.
We can choose any two fractions with a denominator of 72 and a numerator between 16 and 27. Let's choose $\frac{18}{72}$ and $\frac{20}{72}$.
Simplify these fractions:

$$\frac{18}{72} = \frac{1}{4}$$ $$\frac{20}{72} = \frac{5}{18}$$

Step 3: Arrange all numbers in descending order.
The numbers are $\frac{2}{9} = \frac{16}{72}$, $\frac{3}{8} = \frac{27}{72}$, $\frac{1}{4} = \frac{18}{72}$, and $\frac{5}{18} = \frac{20}{72}$.
Descending order means from largest to smallest:

$$\frac{27}{72} > \frac{20}{72} > \frac{18}{72} > \frac{16}{72}$$

Therefore, the numbers in descending order are:

$$\boxed{\frac{3}{8}, \frac{5}{18}, \frac{1}{4}, \frac{2}{9}}$$

3. Insert two rational numbers between $-\frac{1}{3}$ and $-\frac{1}{4}$, and arrange in ascending order.

Step 1: Find a common denominator for $-\frac{1}{3}$ and $-\frac{1}{4}$.
The LCM of 3 and 4 is 12.

$$-\frac{1}{3} = -\frac{1 \times 4}{3 \times 4} = -\frac{4}{12}$$ $$-\frac{1}{4} = -\frac{1 \times 3}{4 \times 3} = -\frac{3}{12}$$

Step 2: Find two rational numbers between $-\frac{4}{12}$ and $-\frac{3}{12}$.
To find numbers between them, we can multiply the numerator and denominator by a factor, for example, 10.

$$-\frac{4}{12} = -\frac{4 \times 10}{12 \times 10} = -\frac{40}{120}$$ $$-\frac{3}{12} = -\frac{3 \times 10}{12 \times 10} = -\frac{30}{120}$$

Now, we need two numbers between $-\frac{40}{120}$ and $-\frac{30}{120}$. Let's choose $-\frac{35}{120}$ and $-\frac{32}{120}$.
Simplify these fractions:

$$-\frac{35}{120} = -\frac{7}{24}$$ $$-\frac{32}{120} = -\frac{4}{15}$$

Step 3: Arrange all numbers in ascending order.
The numbers are $-\frac{1}{3} = -\frac{40}{120}$, $-\frac{1}{4} = -\frac{30}{120}$, $-\frac{7}{24} = -\frac{35}{120}$, and $-\frac{4}{15} = -\frac{32}{120}$.
Ascending order means from smallest (most negative) to largest (least negative):

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