**Question 1. **Add the following rational numbers:

**(i) -5 / 7 and 3 / 7**

**Solution:**

(-5 / 7) + 3 / 7

Since denominators are the same hence numerators will be directly added considering their sign.

Therefore, (-5 + 3) / 7 = (-2) / 7

**(ii) -15 / 4 and 7 / 4**

**Solution:**

Denominators are same so numerators are directly added.

= (-15) / 4 + 7 / 4

= (-15 + 7) / 4

= (-8) / 4

= (4 * (-2)) / 4

= (-2)

**(iii) -8 / 11 and -4 / 11**

**Solution:**

As denominators are the same numerators are added along with their sign.

= (-8) / 11 + (-4) / 11

= (-8 – 4) / 11 (integers with same sign are added)

= (-12) / 11

**(iv) 6 / 13 and -9 / 13**

**Solution:**

As, denominators are same numerators are added with their sign.

= 6 / 13 + (-9) / 13

= (6 – 9) / 13 (integers with opposite sign)

= (-3) / 13

**Question 2. Add the following rational numbers:**

**(i) 3 / 4 and -5 / 8**

**Solution:**

Denominators are different, so we need to take the LCM of denominators to make them into like fractions.

LCM of 4 and 8 = 8

= 3 / 4 + (-5) / 8

= (3 × 2 + (-5)) / 8

= (6 – 5) / 8

= 1 / 8

**(ii) 5 / -9 and 7 / 3**

**Solution:**

Denominators are different, so we need to take the LCM of denominators to make them into like fractions.

LCM of 9 and 3 = 9

= (-5 / 9) + 7 / 3

= (-5 + 7 × 3) / 9

= (-5 + 21) / 9

= 16 / 9

**(iii) -3 and 3 / 5**

**Solution:**

Denominators are different, so we need to take the LCM of denominators to make them into like fractions.

= (-3) / 1 + 3 / 5

LCM of 1 and 5 = 5

= ((-3) × 5 + 3) / 5

= (-15 + 3) / 5

= (-12) / 5

**(iv) -7 / 27 and 11 / 18**

**Solution:**

LCM of 27 and 18

27 = 3 × 3 × 3

18 = 2 × 3 × 3

LCM = 3 × 3 × 3 × 2 = 54

Therefore,

= (-7) / 27 + 11 / 18

= ((-7 × 2 + 11 × 3)) / 54

= (-14 + 33) / 54

= 19 / 54

**(v) 31 / -4 and -5 / 8**

**Solution:**

LCM of 4 and 8 = 8

= ((-31 × 2) + (-5)) / 8

= (-62 – 5) / 8

= (-67) / 8

**(vi) 5 / 36 and -7 / 12**

**Solution:**

LCM of 36 and 12 is 36

= 5 / 36 + (-7) / 12

= (5 + (-7 × 3)) / 36

= (5 + (-21)) / 36

= (-16) / 36

4 is the common factor theta can be canceled

= (-4) / 9

**(vii) -5 / 16 and 7 / 24**

**Solution:**

LCM of 16 and 24

16 = 2 × 2 × 2 × 2

24 = 2 × 2 × 2 × 3

LCM = 2 × 2 × 2 × 2 × 3 = 48

= (-5) / 16 + 7 / 24

= ((-5 × 3) + 7 × 2) / 48

= (-15 + 14) / 48

= (-1) / 48

**(viii) 7 / -18 and 8 / 27**

**Solution:**

LCM of 18 and 27

18 = 2 × 3 × 3

27 = 3 × 3 × 3

LCM = 3 × 3 × 3 × 2 = 54

= ((-7 × 3) + 8 × 2) / 54

= (-21 + 16) / 54

= (-5) / 54

**Question 3. Simplify:**

**(i) 8 / 9 + -11 / 6**

**Solution:**

LCM of 9 and 6

9 = 3 × 3

6 = 2 × 3

LCM = 2 × 3 × 3 = 18

= (8 × 2 + (-11 × 3)) / 18

= (16 – 33) / 18

= (-17) / 18

**(ii) 3 + 5 / -7**

LCM of 1 and 7 is 7

= (3 × 7 + (-5)) / 7

= (21 – 5) / 7

= 16 / 7

**(iii) 1 / -12 + 2 / -15**

**Solution:**

LCM of 12 and 15

12 = 2 × 2 × 3

15 = 3 × 5

LCM = 2 × 2 × 3 × 5 = 60

= ((-1 × 5) + (-2 × 4)) / 60

= (-5 – 8) / 60

= (-13) / 60

**(iv) -8 / 19 + -4 / 57**

**Solution:**

LCM of 19 and 57 is 57

= ((-8 × 3) + (-4)) / 57

= (-24 – 4) / 57

= (-28) / 57

(**v) 7 / 9 + 3 / -4**

**Solution:**

LCM of 9 and 4 is 36

= (7 × 4 + (-3 × 9)) / 36

= (28 – 27) / 36

= 1 / 36

**(vi) 5 / 26 + 11 / -39**

**Solution:**

LCM of 26 and 39

26 = 13×2

39 = 13×3

LCM = 13 × 2 × 3 = 78

= (5 × 3 + (-11 × 2)) / 78

= (15 – 22) / 78

= (-7) / 78

**(vii) -16 / 9 + -5 / 12**

**Solution:**

LCM of 16 and 12

9 = 3×3

12 = 2 × 2 × 3

LCM = 3 × 3 × 2 × 2 = 36

= ((-16 × 4) + (-5 × 3)) / 36

= (-64 – 15) / 36

= (-79) / 36

= (-79) / 36

**(viii) -13 / 8 + 5 / 36**

**Solution:**

LCM of 8 and 36

8 = 2 × 2 × 2

36 = 2 × 2 × 3 × 3

LCM = 2 × 2 × 2 × 3 × 3 = 72

= ((-13 × 9) + 5 × 2) / 72

= (-117 + 10) / 72

= (-107) / 72

**(ix) 0 + -3 / 5**

**Solution:**

0 is the additive identity, if added to any number gives the same number

= (-3) / 5

**(x) 1 + -4 / 5**

**Solution:**

LCM of 1 and 5 is 5

= (1 × 5 + (-4)) / 5

= (5 – 4) / 5

= 1 / 5

**Question 4. Add and express the sum as mixed fraction:**

**(i) -12 / 5 and 43 / 10**

**Solution:**

LCM is 10

= ((-12 × 2 + 43)) / 10

= (-24 + 43) / 10

= 19 / 10

**(ii) 24 / 7 and -11 / 4**

**Solution:**

LCM of 7 and 4 is 28

= (24 × 4 + (-11 × 7)) / 28

= (96 – 77) / 28

= 19 / 28

Proper fraction cannot be converted to mixed fraction

**(iii) -31 / 6 and -27 / 8**

**Solution:**

LCM of 8 and 6 is 24

= ((-31 × 4) + (-27 × 3)) / 16

= (-124 – 81) / 24

= (-205) / 24

**(iv) 101 / 6 and 7 / 8**

**Solution:**

LCM of 8 and 6 is 24

101 / 6 + 7 / 8

= (101 × 4 + 7 × 3) / 24

= (404 + 21) / 24

= 425 / 24