**Problem 1. Simplify each of the following and write as a rational number in the form of p/q:**

**(i) 3/4 + 5/6 + -7/8**

**Solution:**

2 = 2 × 2

6 = 2 × 3

8 = 2 × 2 × 2

LCM is 2 × 2 × 2 × 3 = 24

= (3 × 6 + 5 × 4 + (-7 × 3)) / 24

= (18 + 20 – 21) / 24

= (38 – 21) / 24

= 17 / 24

**(ii) 2/3 + -5/6 + -7/9**

**Solution:**

LCM of 3, 6 and 9 is 18

= (2 × 6 + (-5 × 3) + (-7 × 2)) / 18

= (12 – 15 – 14) / 18

= (12 – 29) / 18

= -17 / 18

**(iii) -11/2 + 7/6 + -5/8**

**Solution:**

2 = 2 × 1

6 = 2 × 3

8 = 2 × 2 × 2

LCM is 2 × 2 × 2 × 3 = 24

= (-11 × 12 + 7 × 4 + (-5 × 3)) / 24

= (-132 + 28 – 15) / 24

= (-147 + 28) / 24

= -119 / 24

**(iv) -4/5 + -7/10 + -8/15**

**Solution**:

10 = 5 × 2

15 = 3 × 5

LCM is 5 × 2 × 3 = 30

= (-4 × 6 + (-7 × 3) + (-8 × 2)) / 30

= (-24 – 21 – 16) / 30

= -61 / 30

**(v) -9/10 + 22/15 + 13/-20**

**Solution**:

This can be written as

-9/10 + 22/15 + -13/20

10 = 2 × 5

15 = 3 × 5

20 = 2 × 2 × 5

LCM is 2 × 2 × 3 × 5 = 60

= (-9 × 6 + 22 × 4 + (-13 × 3)) / 60

= (-54 + 88 – 39) / 60

= (-93 + 88) / 60

= -5 / 60

= -1 / 12

**(vi) 5/3 + 3/-2 + -7/3 + 3**

**Solution**:

This can be written as

5 / 3 + -3 / 2 + -7 / 3 + 3 / 1

LCM is 6

= (5 × 2 + (-3 × 3) + (-7 × 2) + 3 × 6) / 6

= (10 – 9 -14 + 18) / 6

= (28 – 23) / 6

= 5 / 6

**Problem 2. Express each of the following as **a **rational number of the form p/q:**

**(i) -8/3 + -1/4 + -11/6 + 3/8 + -3**

**Solution**:

4 = 2 × 2

6 = 2 × 3

8 = 2 × 2 × 2

LCM is 2 × 2 × 2 × 3 = 24

= (-8 × 8 + (-1 × 6) + (-11 × 4) + 3 × 3 + (-3 × 24)) / 24

= (-64 – 6 – 44 + 9 – 72) / 24

= (-186 + 9) / 24

= -177 / 24

= -59 / 8

**(ii) 6/7 + 1 + -7/9 + 19/21 + -12/7**

**Solution**:

(6 / 7 + -12 / 7) + (-7 / 9) + 19 / 21 + 1(Taking numbers with same denominators together)

= (6 – 12) / 7 + (-7 / 9) + 19 / 21+1

= -6 / 7 + -7 / 9 + 19 / 21 + 1 / 1

9 = 3 × 3

21 = 3 × 7

LCM of 7, 1, 9 and 21 is 63

= (-6 × 9 + (-7 × 7) + 19 × 3 + 1 × 63) / 63

= (-54 – 49 + 57 + 63) / 63

= (-103 + 120) / 63

= 17 / 63

**(iii) 15/2 + 9/8 + -11/3 + 6 + -7/6**

**Solution**:

15 / 2 + 9/8 + (-11 / 3) + 6 / 1 + (-7 / 6)

LCM of 2, 8, 3, 1 and 6 is 24

= (15 × 12 + 9 × 3 + (-11 × 8) + 6 × 24 + (-7 × 4)) / 24

= (180 + 27 – 88 + 144 – 28) / 24

= (351 – 116) / 24

= (235) / 24

**(iv) -7/4 + 0 + -9/5 + 19/10 + 11/14**

**Solution**:

4 = 2 × 2

5 = 5 × 1

10 = 2 × 5

14 = 2 × 7

LCM is 2 × 2 × 5 × 7 is 140

= (-7 × 35 + (-9 × 28) + 19 × 14 + 11 × 10) / 140

= (-245 – 252 + 266 + 110) / 140

= (-497 + 376) / 140

= (-121) / 140

**(v) -7/4 + 5/3 + -1/2 + -5/6 + 2**

**Solution**:

LCM of 4, 3, 2 and 6 is 12

= (-7 × 3 + 5 × 4 + (-1 × 6) + (-5 × 2) + 2 × 12) / 12

= (-21 + 20 – 6 – 10 + 24) / 12

= (-37 + 44) / 12

= 7 / 12

**Problem 3. Simplify:**

**(i) -3/2 + 5/4 + -7/4**

**Solution:**

Taking numbers with the same denominators together

= -3 / 2 + (5 – 7) / 4

= -3 / 2 – 2 / 4

LCM of 2 and 4 is 4

= (-3 × 2 – 2 × 1) / 4

= (-6 – 2) / 4

= (-8) / 4

= -2

**(ii) 5/3 + -7/6 + -2/3**

**Solution**:

Taking numbers with same denominators together

(5 / 3 + -2 / 3) + -7 / 6

= (5 – 2) / 3 + -7 / 6

= 3 / 3 + (-7 / 6)

LCM of 3 and 6 is 6

= (3 × 2 + (-7 × 1)) / 6

= (6 – 7) / 6

= -1 / 6

**(iii) 5/4 – 7/6 – (-2/3)**

**Solution**:

This can be written as

5 / 4 – 7 / 6 + 2 / 3

LCM of 4,6 and 3 is 12

= (5 × 3 – 7 × 2 + 2 × 4) / 12

= (15 – 14 + 8) / 12

= (23 – 14) / 12

= 9 / 12

= 3 / 4

**(iv) -2/5 – (-3/10) – (-4/7)**

**Solution**:

This can be written as:

-2 / 5 + 3 / 10 + 4 / 7

LCM of 5,10 and 7 is 70

= (-2 × 14 + 3 × 7 + 4 × 10) / 70

= (-28 + 21 + 40) / 70

= (-28 + 61) / 70

= 33 / 70

**(v) 5/6 + -2/5 – (-2/15)**

**Solution**:

This can be written as

5 / 6 + -2 / 5 + 2 / 15

6 = 2 × 3

5 = 5 × 1

15 = 3 × 5

LCM is 2 × 3 × 5 = 30

= (5 × 5 + (-2 × 6) + 2 × 2) / 30

= (25 – 12 + 4) / 30

= (29 – 12) / 30

= 17 / 30

**(vi) 3/8 – (-2/9) + (-5/36)**

**Solution**:

This can be written as

3 / 8 + 2 / 9 – 5 / 36

8 = 2 × 2 × 2

9 = 3 × 3

36 = 2 × 2 × 3 × 3

LCM is 2 × 2 × 2 × 3 × 3 = 72

= (3 × 9 + 2 × 8 – 5 × 2) / 72

= (27 + 16 – 10) / 72

= (43 – 10) / 72

= 33 / 72

= 11 / 24