Find three rational numbers between 1 2 and 3 2 .

We are given to find three rational numbers between 1 2 and 3 2 . We shall find equivalent rational numbers with the same denominators of the given rational numbers such that the difference between their numerators is 3 or more. Thus, we have, 1 2 = 1 × 2 2 × 2 = 2 4 and 3 2 = 3 × 2 2 × 2 = 6 4 . Hence, three rational numbers between 2 4 = 1 2 and 6 4 = 3 2 are 3 4 , 4 4 = 1 and 5 4 . Therefore, the required three rational numbers are 3 4 , 1 and 5 4 .

Find eight rational numbers between - 3 7 and - 5 7

We are given to find three rational numbers between - 3 7 and - 5 7 . We shall find equivalent rational numbers with the same denominators of the given rational numbers such that the difference between their numerators is 8 or more. Thus, we have, - 3 7 = - 3 × 5 7 × 5 = - 15 35 and - 5 7 = - 5 × 5 7 × 5 = - 25 35 . Hence, eight rational numbers between - 15 35 = - 3 7 and - 5 7 = - 25 35 are - 16 35 , - 17 35 , - 18 35 , - 19 35 , - 20 35 = - 4 7 , - 21 35 , - 22 35 and - 23 35 . Therefore, the required three rational numbers are - 16 35 , - 17 35 , - 18 35 , - 19 35 , - 4 7 , - 21 35 , - 22 35 and - 23 35 .