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): $$ -\frac{40}{120} < -\frac{35}{120} < -\frac{32}{120}