Find two rational numbers between - 2 9 and 5 9 .

We know that between two rational numbers x and y where x < y there is a rational number ( x + y ) 2 x < ( x + y ) 2 < y Here x = - 2 9 , y = 5 9 The rational numbers between - 2 9 and 5 9 is ( - 2 9 + 5 9 ) ÷ 2 = 3 9 ÷ 2 = 1 3 ÷ 2 = 1 6 Here x = - 2 9 , y = 1 6 The rational numbers between - 2 9 and 1 6 is ( - 2 9 + 1 6 ) ÷ 2 ( - 2 × 2 + 1 × 3 ) 18 ÷ 2 = ( - 4 + 3 ) 36 = - 1 36 ∴ the rational numbers between - 2 9 and 5 9 are - 1 36 , 1 6 .

Find two rational numbers between 1 5 and 1 2 .

We know that between two rational numbers x and y where x < y there is a rational number ( x + y ) 2 , where x < ( x + y ) 2 < y Here x = 1 5 , y = 1 2 The rational numbers between 1 5 and 1 2 is ( 1 5 + 1 2 ) ÷ 2 1 × 2 + 1 × 5 10 ÷ 2 ( 2 + 5 ) 20 = 7 20 Now find the rational number between previous result and one of the end value The rational numbers between 1 5 and 7 20 is ( 1 5 + 7 20 ) ÷ 2 1 × 4 + 7 × 1 20 ÷ 2 ( 4 + 7 ) 40 11 40 ∴ the rational numbers between 1 5 and 1 2 are 7 20 , 11 40

Find ten rational numbers between 1 4 and 1 2 .

Firstly convert the given rational numbers into equivalent rational numbers with same denominators. The LCM for 4 and 2 is 4 1 4 = 1 4 1 2 = ( 1 × 2 ) ( 2 × 2 ) = 2 4 1 4 = ( 1 × 20 4 × 20 ) = 20 80 1 2 = ( 2 × 20 4 × 20 ) = 40 80 So, we now know that 21 , 22 , 23 , … 39 are integers between numerators 20 and 40 ∴ The rational numbers between 1 4 and 1 2 are 21 80 , 22 80 , 23 80 , 24 80 , 25 80 , 26 80 , 27 80 , 28 80 , 29 80 , 30 80 . Note: Answer is not unique as there are many rational numbers possible.

Find ten rational numbers between - 2 5 and 1 2 .

Firstly convert the given rational numbers into equivalent rational numbers with same denominators. The LCM for 5 and 2 is 10 - 2 5 = ( - 2 × 2 ) 10 = - 4 10 1 2 = ( 1 × 5 ) 10 = 5 10 - 2 5 = - 4 × 2 10 × 2 = - 8 20 1 2 = 5 × 2 10 × 2 = 10 20 So, we now know that - 7 , - 6 , - 5 , … 10 are integers between numerators - 8 and 10 ∴ The rational numbers between - 2 5 and 1 2 are - 7 20 , - 6 20 , - 5 20 , … , 9 20 . Hence, required ten rational numbers are - 7 20 , - 6 20 , - 5 20 , - 4 20 , - 3 20 , - 2 20 , - 1 20 , 1 20 , 2 20 , 3 20 .

Find ten rational numbers between 3 5 and 3 4 .

Firstly convert the given rational numbers into equivalent rational numbers with same denominators. The LCM for 5 and 4 is 20 . 3 5 = 3 × 20 5 × 20 = 60 100 3 4 = 3 × 25 4 × 25 = 75 100 So, we now know that 61 , 62 , 63 , . . 74 are integers between numerators 60 and 75 ∴ The rational numbers between 3 5 and 3 4 are 61 100 , 62 100 , 63 100 , … . , 74 100 . Hence, ten rational numbers in between are 61 100 , 62 100 , 63 100 , 64 100 , 65 100 , 66 100 67 100 , 68 100 , 69 100 70 100 . Note: Answer is not unique as there are many rational numbers possible.