**Question 1. Find a rational number between -3 and 1.**

**Solution:**

Between two rational numbers x and y

Rational number = (x + y) / 2

= (-3 + 1) / 2

= -2/2

= -1

So, the rational number between -3 and 1 is -1

**Question 2. Find any five rational numbers less than 2.**

**Solution:**

Five rational numbers less than 2 can be 0, 1, -2/5, 3/2, -4/5

**Question 3. Find two rational numbers between -2/9 and 5/9**

**Solution:**

The rational number between -2/9 and 5/9 is

= (-2/9 + 5/9)/2

= (3/9) / 2

3 is the common factor

= 1/3/2

= 1/3 * 1/2

= 1/6

Now the rational number between -2/9 and 1/6 is

= (-2/9 + 1/6) / 2

LCM is 18

= ((-2 × 2 + 1 × 3) /18) / 2

= (-4 + 3) / 36

= -1/36

Rational numbers between -2/9 and 5/9 are -1/36, 1/6

**Question 4. Find two rational numbers between 1/5 and 1/2**

**Solution:**

The rational number between 1/5 and 1/2 is

= (1/5 + 1/2)/2

LCM is 10

= ((1 × 2 + 1 × 5) / 10) / 2

= (2 + 5) / 20 = 7/20

Now the rational number between 1/5 and 7/20 is

= (1/5 + 7/20) / 2

LCM is 20

= ((1 × 4 + 7 × 1) / 20) / 2

= (4 + 7) / 40

= 11/40

Rational number between 1/5 and 1/2 are 7/20, 11/40

**Question 5. Find ten rational numbers between 1/4 and 1/2.**

**Solution:**

Other than average method, rational numbers can be found by converting 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

2/4 = (2 × 20 / 4 × 20) = 40/80

So, we now know that 21, 22, 23,…39 are integers between numerators 20 and 40.

Rational numbers between 1/4 and 1/2 are 21/80, 22/80, 23/80, …., 39/80

**Question 6. Find ten rational numbers between -2/5 and 1/2.**

**Solution:**

Other than average method, rational numbers can be found by converting 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

-4/10 = (-4 × 2 / 10 × 2) = -8/20

5/10 = (5 × 2 / 10 × 2) = 10/20

So, we now know that -7, -6, -5,…10 are integers between numerators -8 and 10.

Rational numbers between -2/5 and 1/2 are -7/20, -6/20, -5/20, …., 9/20

**Question 7. Find ten rational numbers between 3/5 and 3/4.**

**Solution:**

Other than average method, rational numbers can be found by converting the given rational numbers into equivalent rational numbers with same denominators.

The LCM for 5 and 4 is 20.

3/5 = 3 × 4/5 × 4 = 12/20

3/4 = 3 × 5/4 × 5 = 15/20

Making the denominator 100

12/20 = 12 × 5/20 × 5 = 60/100

15/20 = 15 × 5/20 × 5 = 75/100

Rational numbers between 3/5 and 3/4 are 61/100, 62/100, 63/100, …., 74/100