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

Here are the solutions to the problems: 2. Insert two rational numbers between (2)/(9) and (3)/(8), and arrange in descending order. Step 1: Find a common denominator for (2)/(9) and (3)/(8). The least common multiple (LCM) of 9 and 8 is 72. (2)/(9) = (2 × 8)/(9 × 8) = (16)/(72) (3)/(8) = (3 × 9)/(8 × 9) = (27)/(72) Step 2: Find two rational numbers between (16)/(72) and (27)/(72). We can choose any two fractions with a denominator of 72 and a numerator between 16 and 27. Let's choose (18)/(72) and (20)/(72). Simplify these fractions: (18)/(72) = (1)/(4) (20)/(72) = (5)/(18) Step 3: Arrange all numbers in descending order. The numbers are (2)/(9) = (16)/(72), (3)/(8) = (27)/(72), (1)/(4) = (18)/(72), and (5)/(18) = (20)/(72). Descending order means from largest to smallest: (27)/(72) > (20)/(72) > (18)/(72) > (16)/(72) Therefore, the numbers in descending order are: (3)/(8), (5)/(18), (1)/(4), (2)/(9) 3. Insert two rational numbers between -(1)/(3) and -(1)/(4), and arrange in ascending order. Step 1: Find a common denominator for -(1)/(3) and -(1)/(4). The LCM of 3 and 4 is 12. -(1)/(3) = -(1 × 4)/(3 × 4) = -(4)/(12) -(1)/(4) = -(1 × 3)/(4 × 3) = -(3)/(12) Step 2: Find two rational numbers between -(4)/(12) and -(3)/(12). To find numbers between them, we can multiply the numerator and denominator by a factor, for example, 10. -(4)/(12) = -(4 × 10)/(12 × 10) = -(40)/(120) -(3)/(12) = -(3 × 10)/(12 × 10) = -(30)/(120) Now, we need two numbers between -(40)/(120) and -(30)/(120). Let's choose -(35)/(120) and -(32)/(120). Simplify these fractions: -(35)/(120) = -(7)/(24) -(32)/(120) = -(4)/(15) Step 3: Arrange all numbers in ascending order. The numbers are -(1)/(3) = -(40)/(120), -(1)/(4) = -(30)/(120), -(7)/(24) = -(35)/(120), and -(4)/(15) = -(32)/(120). Ascending order means from smallest (most negative) to largest (least negative): -(40)/(120) < -(35)/(120) < -(32)/(120)