# List five rational numbers between: (1.) -1 and 0 (2.) -2 and -1 (3.) -4/5 and -2/3 (4.) -1/2 and 2/3

Rational numbers are in the form of p/q, where p and q can be any integer and q ≠ 0

## Answer: Five rational numbers between each pair of given rational numbers are (1) -1/6, -2/6, -3/6, -4/6, -5/6 (2) -7/6, -8/6, -9/6, -10/6, -11/6 (3) -31/45, -32/45, -11/15, -34/45, -7/9 (4) -11/36, -1/9, 1/12, 5/18, 17/36

Let's understand the calculation to find rational numbers between any two rational numbers

**Explanation:**

To find 'n' number of rational numbers between 'a' and 'b', a < b, find a + (b-a)k/(n+1), where k = 1, 2, 3,..., n

We will use this fact to find all the rational numbers.

(1) Rational numbers between -1 and 0

Substituting a = -1, b = 0, n = 5 in a + (b-a)k/(n+1)

we get, -1+k/6, where k = 1, 2, 3, 4, 5

Substituting k = 1, 2, 3, 4, 5 in -k/6 we get the following rational numbers

-1/6, -2/6, -3/6, -4/6, -5/6

(2) Rational numbers between -2 and -1

Substituting a = -2, b = -1, n = 5 in a + (b-a)k/(n+1)

we get, -2+(k/6), where k = 1, 2, 3, 4, 5

Substituting k = 1, 2, 3, 4, 5 in -2+(k/6) we get the following rational numbers

-7/6, -8/6, -9/6, -10/6, -11/6

(3) Rational numbers between -4/5 and -2/3

Substituting a = -4/5, b = -2/3, n = 5 in a + (b-a)k/(n+1)

we get, -4/5+(k/45), where k = 1, 2, 3, 4, 5

Substituting k = 1, 2, 3, 4, 5 in -4/5+(k/45) we get the following rational numbers

-31/45, -32/45, -11/15, -34/45, -7/9

(4) Rational numbers between -1/2 and 2/3

Substituting a = -1/2, b = 2/3, n = 5 in a + (b-a)k/(n+1)

we get, -1/2+(7k/36), where k = 1, 2, 3, 4, 5

Substituting k = 1, 2, 3, 4, 5 in -1/2+(7k/36) we get the following rational numbers

-11/36, -1/9, 1/12, 5/18, 17/36

### Thus, the rational numbers are (1) -1/6, -2/6, -3/6, -4/6, -5/6 (2) -7/6, -8/6, -9/6, -10/6, -11/6 (3) -31/45, -32/45, -11/15, -34/45, -7/9 (4) -11/36, -1/9, 1/12, 5/18, 17/36

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