Find a Rational number between a and b if, a = - 5 6 , b = - 2 5

We know, if a b and c d are two rational numbers, such that a b < c d , then 1 2 a b + c d is a rational number lying between a b and c d . So, Required Rational number: = 1 2 - 5 6 + - 2 5 = 1 2 - 25 - 12 30 = - 37 60

Find five rational numbers between - 3 5 and - 1 2 .

Given, − 3 5 , − 1 2 L.C.M of 5 , 2 is 10 . − 3 × 2 5 × 2 = − 6 × 4 10 × 4 = − 24 × 2 40 × 2 = − 48 80 − 1 × 5 2 × 5 = − 5 × 4 10 × 4 = − 20 × 2 40 × 2 = − 40 80 So, the rational numbers between − 48 80 , − 40 80 are − 45 80 , − 44 80 , − 43 80 , − 42 80 , − 41 80 .

Find six rational numbers between - 4 8 and 3 8 .

A rational number is a number that can be expressed as a fraction where both the numerator and the denominator in the fraction are integers. The denominator in a rational number cannot be zero. We have, - 4 8 and 3 8 . If the denominators are same of two rational numbers then, we can find some rational numbers between them. We know that - 4 < - 3 < - 2 < - 1 < 0 < 1 < 2 < 3 ⇒ - 4 8 < - 3 8 < - 2 8 < - 1 8 < 0 8 < 1 8 < 2 8 < 3 8 Therefore, the 6 rational numbers that lie between - 4 8 and 3 8 are - 3 8 , - 2 8 , - 1 8 , 0 8 , 1 8 , 2 8 . [ Note: This is an open-ended problem. You can write in solution, 'the answer may vary as there can be many rational numbers between two given numbers' ]

If x = - 2 11 and y = 5 13 , find three rational numbers lying between | x - y | and x - | y |

Given: x = - 2 11 , y = 5 13 To find: Three rational numbers between | x - y | and x - | y | | x - y | = - 2 11 - 5 13 Now, denominators 11 , 13 are unequal and which are prime numbers. Therefore, LCM of 11 , 13 is 11 × 13 = 143 x + y = - 2 11 - 5 13 = - 2 11 × 13 13 - 5 13 × 11 11 = - 26 143 - 55 143 = - 26 143 + - 55 143 = - 81 143 We know that, absolute value of an integer is its numerical value without taking the sign into consideration. x - y = 81 143 x = - 2 11 = 2 11 and y = 5 13 = 5 13 x - y = 2 11 - 5 13 = 2 × 13 11 × 13 - 5 × 11 13 × 11 = 26 143 - 55 143 = 26 143 + - 55 143 So, x - y = - 29 143 Three rational numbers between - 29 143 and 81 143 are - 28 143 , - 27 143 and - 26 143 . Note: There exist infinitely many rational numbers between any two rational numbers.

Find three rational numbers between - 2 7 & 3 - 5 .

Taking LCM of denominators 7 , 5 = 35 Converting these rational numbers into multiples of LCM, - 2 7 = - 2 × 5 7 × 5 = - 10 35 3 - 5 = 3 × 7 - 5 × 7 = 21 - 35 = - 21 35 So, - 10 35 > - 11 35 > - 12 35 > … > - 21 35 Hence, the three rational numbers in between the given rational numbers can be - 11 35 , - 12 35 , - 13 35 . Note: There are many rational numbers in between the given rational numbers. So answer is not unique.

Find 10 rational numbers between 7 13 and - 4 13 .

A rational number is a number that can be expressed as a fraction where both the numerator and the denominator in the fraction are integers. The denominator in a rational number cannot be zero. We have, - 4 13 and 7 13 . If the denominators are same of two rational numbers then, we can find some rational numbers between them u. We know that - 4 < - 3 < - 2 < - 1 < 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 ⇒ - 4 13 < - 3 13 < - 2 13 < - 1 13 < 0 13 < 1 13 < 2 13 < 3 13 < 4 13 < 5 13 < 6 13 < 7 13 Therefore, the 10 rational numbers that lie between - 4 13 and 7 13 are - 3 13 , - 2 13 , - 1 13 , 0 13 , 1 13 , 2 13 , 3 13 , 4 13 , 5 13 , 6 13 . [ Note: This is an open-ended problem. You can write in solution, 'the answer may vary as there can be many rational numbers between two given numbers' ]

If x = - 2 11 and y = 5 13 , find three rational numbers lying between | x + y | and x + | y |

Given: x = - 2 11 , y = 5 13 To find: Three rational numbers between | x + y | and x + | y | | x + y | = - 2 11 + 5 13 Now, denominators 11 , 13 are unequal and which are prime numbers. Therefore, L.C.M of 11 , 13 is 11 × 13 = 143 x + y = - 2 11 + 5 13 = - 2 11 × 13 13 + 5 13 × 11 11 = - 26 143 + 55 143 = - 26 + 55 143 = 29 143 We know that, absolute value of an integer is its numerical value without taking the sign into consideration. x + y = 29 143 x = - 2 11 = 2 11 and y = 5 13 = 5 13 x + y = 2 11 + 5 13 = 2 × 13 11 × 13 + 5 × 11 13 × 11 = 26 143 + 55 143 So, x + y = 81 143 Three rational numbers between 29 143 and 81 143 are 30 143 , 31 143 and 32 143 . Note: There exist infinitely many rational numbers between any two rational numbers.

Find 10 rational numbers between - 3 4 and 5 6 .

We have to find out 10 rational numbers between - 3 4 and 5 6 . LCM of 4 and 6 is 12 . Now, - 3 4 = - 3 × 3 4 × 3 = - 9 12 and 5 6 = 5 × 2 6 × 2 = 10 12 . Rational numbers lying between - 9 12 and 10 12 are - 8 12 , - 7 12 , - 6 12 , - 5 12 , - 4 12 , - 3 12 , - 2 12 , - 1 12 , 0 12 , 1 12 , 2 12 , 3 12 , 4 12 . Out of these any 10 can be taken. Therefore, 10 rational numbers between - 3 4 and 5 6 are - 8 12 , - 7 12 , - 6 12 , - 5 12 , - 4 12 , - 3 12 , - 2 12 , - 1 12 , 0 and 1 12 .