R. D. Sharma | EXERCISE 1.1 PAGE NO: 1.5

 

EXERCISE 1.1 PAGE NO: 1.5

1. Add the following rational numbers:

(i) -5/7 and 3/7

(ii) -15/4 and 7/4

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

(iv) 6/13 and -9/13

Solution:

Since the denominators are of same positive numbers we can add them directly

(i) -5/7 + 3/7 = (-5+3)/7 = -2/7

(ii) -15/4 + 7/4 = (-15+7)/4 = -8/4

Further dividing by 4 we get,

-8/4 = -2

(iii) -8/11 + -4/11 = (-8 + (-4))/11 = (-8-4)/11 = -12/11

(iv) 6/13 + -9/13 = (6 + (-9))/13 = (6-9)/13 = -3/13

2. Add the following rational numbers:

(i) 3/4 and -5/8

Solution: The denominators are 4 and 8

By taking LCM for 4 and 8 is 8

We rewrite the given fraction in order to get the same denominator

3/4 = (3×2) / (4×2) = 6/8 and

-5/8 = (-5×1) / (8×1) = -5/8

Since the denominators are same we can add them directly

6/8 + -5/8 = (6 + (-5))/8 = (6-5)/8 = 1/8

(ii) 5/-9 and 7/3

Solution: Firstly we need to convert the denominators to positive numbers.

5/-9 = (5 × -1)/ (-9 × -1) = -5/9

The denominators are 9 and 3

By taking LCM for 9 and 3 is 9

We rewrite the given fraction in order to get the same denominator

-5/9 = (-5×1) / (9×1) = -5/9 and

7/3 = (7×3) / (3×3) = 21/9

Since the denominators are same we can add them directly

-5/9 + 21/9 = (-5+21)/9 = 16/9

(iii) -3 and 3/5

Solution: The denominators are 1 and 5

By taking LCM for 1 and 5 is 5

We rewrite the given fraction in order to get the same denominator

-3/1 = (-3×5) / (1×5) = -15/5 and

3/5 = (3×1) / (5×1) = 3/5

Now, the denominators are same we can add them directly

-15/5 + 3/5 = (-15+3)/5 = -12/5

(iv) -7/27 and 11/18

Solution: The denominators are 27 and 18

By taking LCM for 27 and 18 is 54

We rewrite the given fraction in order to get the same denominator

-7/27 = (-7×2) / (27×2) = -14/54 and

11/18 = (11×3) / (18×3) = 33/54

Now, the denominators are same we can add them directly

-14/54 + 33/54 = (-14+33)/54 = 19/54

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

Solution: Firstly we need to convert the denominators to positive numbers.

31/-4 = (31 × -1)/ (-4 × -1) = -31/4

The denominators are 4 and 8

By taking LCM for 4 and 8 is 8

We rewrite the given fraction in order to get the same denominator

-31/4 = (-31×2) / (4×2) = -62/8 and

-5/8 = (-5×1) / (8×1) = -5/8

Since the denominators are same we can add them directly

-62/8 + (-5)/8 = (-62 + (-5))/8 = (-62-5)/8 = -67/8

(vi) 5/36 and -7/12

Solution: The denominators are 36 and 12

By taking LCM for 36 and 12 is 36

We rewrite the given fraction in order to get the same denominator

5/36 = (5×1) / (36×1) = 5/36 and

-7/12 = (-7×3) / (12×3) = -21/36

Now, the denominators are same we can add them directly

5/36 + -21/36 = (5 + (-21))/36 = 5-21/36 = -16/36 = -4/9

(vii) -5/16 and 7/24

Solution: The denominators are 16 and 24

By taking LCM for 16 and 24 is 48

We rewrite the given fraction in order to get the same denominator

-5/16 = (-5×3) / (16×3) = -15/48 and

7/24 = (7×2) / (24×2) = 14/48

Now, the denominators are same we can add them directly

-15/48 + 14/48 = (-15 + 14)/48 = -1/48

(viii) 7/-18 and 8/27

Solution: Firstly we need to convert the denominators to positive numbers.

7/-18 = (7 × -1)/ (-18 × -1) = -7/18

The denominators are 18 and 27

By taking LCM for 18 and 27 is 54

We rewrite the given fraction in order to get the same denominator

-7/18 = (-7×3) / (18×3) = -21/54 and

8/27 = (8×2) / (27×2) = 16/54

Since the denominators are same we can add them directly

-21/54 + 16/54 = (-21 + 16)/54 = -5/54

3.Simplify:

(i) 8/9 + -11/6

Solution: let us take the LCM for 9 and 6 which is 18

(8×2)/(9×2) + (-11×3)/(6×3)

16/18 + -33/18

Since the denominators are same we can add them directly

(16-33)/18 = -17/18

(ii) 3 + 5/-7

Solution: Firstly convert the denominator to positive number

5/-7 = (5×-1)/(-7×-1) = -5/7

3/1 + -5/7

Now let us take the LCM for 1 and 7 which is 7

(3×7)/(1×7) + (-5×1)/(7×1)

21/7 + -5/7

Since the denominators are same we can add them directly

(21-5)/7 = 16/7

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

Solution: Firstly convert the denominator to positive number

1/-12 = (1×-1)/(-12×-1) = -1/12

2/-15 = (2×-1)/(-15×-1) = -2/15

-1/12 + -2/15

Now let us take the LCM for 12 and 15 which is 60

(-1×5)/(12×5) + (-2×4)/(15×4)

-5/60 + -8/60

Since the denominators are same we can add them directly

(-5-8)/60 = -13/60

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

Solution: let us take the LCM for 19 and 57 which is 57

(-8×3)/(19×3) + (-4×1)/(57×1)

-24/57 + -4/57

Since the denominators are same we can add them directly

(-24-4)/57 = -28/57

(v) 7/9 + 3/-4

Solution: Firstly convert the denominator to positive number

3/-4 = (3×-1)/(-4×-1) = -3/4

7/9 + -3/4

Now let us take the LCM for 9 and 4 which is 36

(7×4)/(9×4) + (-3×9)/(4×9)

28/36 + -27/36

Since the denominators are same we can add them directly

(28-27)/36 = 1/36

(vi) 5/26 + 11/-39

Solution: Firstly convert the denominator to positive number

11/-39 = (11×-1)/(-39×-1) = -11/39

5/26 + -11/39

Now let us take the LCM for 26 and 39 which is 78

(5×3)/(26×3) + (-11×2)/(39×2)

15/78 + -22/78

Since the denominators are same we can add them directly

(15-22)/78 = -7/78

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

Solution: let us take the LCM for 9 and 12 which is 108

(-16×12)/(9×12) + (-5×9)/(12×9)

-192/108 + -45/108

Since the denominators are same we can add them directly

(-192-45)/108 = -237/108

Further divide the fraction by 3 we get,

-237/108 = -79/36

(viii) -13/8 + 5/36

Solution: let us take the LCM for 8 and 36 which is 72

(-13×9)/(8×9) + (5×2)/(36×2)

-117/72 + 10/72

Since the denominators are same we can add them directly

(-117+10)/72 = -107/72

(ix) 0 + -3/5

Solution: We know that anything added to 0 results in the same.

0 + -3/5 = -3/5

(x) 1 + -4/5

Solution: let us take the LCM for 1 and 5 which is 5

(1×5)/(1×5) + (-4×1)/(5×1)

5/5 + -4/5

Since the denominators are same we can add them directly

(5-4)/5 = 1/5

4. Add and express the sum as a mixed fraction:

(i) -12/5 and 43/10

Solution: let us add the given fraction

-12/5 + 43/10

let us take the LCM for 5 and 10 which is 10

(-12×2)/(5×2) + (43×1)/(10×1)

-24/10 + 43/10

Since the denominators are same we can add them directly

(-24+43)/10 = 19/10

19/10 can be written as

$\begin{array}{l}1\frac{9}{10}\end{array}$

in mixed fraction.

(ii) 24/7 and -11/4

Solution: let us add the given fraction

24/7 + -11/4

let us take the LCM for 7 and 4 which is 28

(24×4)/(7×4) + (-11×7)/(4×7)

96/28 + -77/28

Since the denominators are same we can add them directly

(96-77)/28 = 19/28

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

Solution: let us add the given fraction

-31/6 + -27/8

let us take the LCM for 6 and 8 which is 24

(-31×4)/(6×4) + (-27×3)/(8×3)

-124/24 + -81/24

Since the denominators are same we can add them directly

(-124-81)/24 = -205/24

-205/24 can be written as

$\begin{array}{l}-8\frac{13}{24}\end{array}$

in mixed fraction.

(iv) 101/6 and 7/8

Solution: let us add the given fraction

101/6 + 7/8

let us take the LCM for 6 and 8 which is 24

(101×4)/(6×4) + (7×3)/(8×3)

404/24 + 21/24

Since the denominators are same we can add them directly

(404+21)/24 = 425/24

425/24 can be written as

$\begin{array}{l}17\frac{17}{24}\end{array}$

in mixed fraction.

---

R. D. Sharma | EXERCISE 1.2 PAGE NO: 1.14

 

EXERCISE 1.2 PAGE NO: 1.14

1. Verify commutativity of addition of rational numbers for each of the following pairs of rational numbers:

(i) -11/5 and 4/7

Solution: By using the commutativity law, the addition of rational numbers is commutative ∴ a/b + c/d = c/d + a/b

In order to verify the above property let us consider the given fraction

-11/5 and 4/7 as

-11/5 + 4/7 and 4/7 + -11/5

The denominators are 5 and 7

By taking LCM for 5 and 7 is 35

We rewrite the given fraction in order to get the same denominator

Now, -11/5 = (-11 × 7) / (5 ×7) = -77/35

4/7 = (4 ×5) / (7 ×5) = 20/35

Since the denominators are same we can add them directly

-77/35 + 20/35 = (-77+20)/35 = -57/35

4/7 + -11/5

The denominators are 7 and 5

By taking LCM for 7 and 5 is 35

We rewrite the given fraction in order to get the same denominator

Now, 4/7 = (4 × 5) / (7 ×5) = 20/35

-11/5 = (-11 ×7) / (5 ×7) = -77/35

Since the denominators are same we can add them directly

20/35 + -77/35 = (20 + (-77))/35 = (20-77)/35 = -57/35

∴ -11/5 + 4/7 = 4/7 + -11/5 is satisfied.

(ii) 4/9 and 7/-12

Solution: Firstly we need to convert the denominators to positive numbers.

7/-12 = (7 × -1)/ (-12 × -1) = -7/12

By using the commutativity law, the addition of rational numbers is commutative.

∴ a/b + c/d = c/d + a/b

In order to verify the above property let us consider the given fraction

4/9 and -7/12 as

4/9 + -7/12 and -7/12 + 4/9

The denominators are 9 and 12

By taking LCM for 9 and 12 is 36

We rewrite the given fraction in order to get the same denominator

Now, 4/9 = (4 × 4) / (9 ×4) = 16/36

-7/12 = (-7 ×3) / (12 ×3) = -21/36

Since the denominators are same we can add them directly

16/36 + (-21)/36 = (16 + (-21))/36 = (16-21)/36 = -5/36

-7/12 + 4/9

The denominators are 12 and 9

By taking LCM for 12 and 9 is 36

We rewrite the given fraction in order to get the same denominator

Now, -7/12 = (-7 ×3) / (12 ×3) = -21/36

4/9 = (4 × 4) / (9 ×4) = 16/36

Since the denominators are same we can add them directly

-21/36 + 16/36 = (-21 + 16)/36 = -5/36

∴ 4/9 + -7/12 = -7/12 + 4/9 is satisfied.

(iii) -3/5 and -2/-15

Solution:

By using the commutativity law, the addition of rational numbers is commutative.

∴ a/b + c/d = c/d + a/b

In order to verify the above property let us consider the given fraction

-3/5 and -2/-15 as

-3/5 + -2/-15 and -2/-15 + -3/5

-2/-15 = 2/15

The denominators are 5 and 15

By taking LCM for 5 and 15 is 15

We rewrite the given fraction in order to get the same denominator

Now, -3/5 = (-3 × 3) / (5×3) = -9/15

2/15 = (2 ×1) / (15 ×1) = 2/15

Since the denominators are same we can add them directly

-9/15 + 2/15 = (-9 + 2)/15 = -7/15

-2/-15 + -3/5

-2/-15 = 2/15

The denominators are 15 and 5

By taking LCM for 15 and 5 is 15

We rewrite the given fraction in order to get the same denominator

Now, 2/15 = (2 ×1) / (15 ×1) = 2/15

-3/5 = (-3 × 3) / (5×3) = -9/15

Since the denominators are same we can add them directly

2/15 + -9/15 = (2 + (-9))/15 = (2-9)/15 = -7/15

∴ -3/5 + -2/-15 = -2/-15 + -3/5 is satisfied.

(iv) 2/-7 and 12/-35

Solution: Firstly we need to convert the denominators to positive numbers.

2/-7 = (2 × -1)/ (-7 × -1) = -2/7

12/-35 = (12 × -1)/ (-35 × -1) = -12/35

By using the commutativity law, the addition of rational numbers is commutative.

∴ a/b + c/d = c/d + a/b

In order to verify the above property let us consider the given fraction

-2/7 and -12/35 as

-2/7 + -12/35 and -12/35 + -2/7

The denominators are 7 and 35

By taking LCM for 7 and 35 is 35

We rewrite the given fraction in order to get the same denominator

Now, -2/7 = (-2 × 5) / (7 ×5) = -10/35

-12/35 = (-12 ×1) / (35 ×1) = -12/35

Since the denominators are same we can add them directly

-10/35 + (-12)/35 = (-10 + (-12))/35 = (-10-12)/35 = -22/35

-12/35 + -2/7

The denominators are 35 and 7

By taking LCM for 35 and 7 is 35

We rewrite the given fraction in order to get the same denominator

Now, -12/35 = (-12 ×1) / (35 ×1) = -12/35

-2/7 = (-2 × 5) / (7 ×5) = -10/35

Since the denominators are same we can add them directly

-12/35 + -10/35 = (-12 + (-10))/35 = (-12-10)/35 = -22/35

∴ -2/7 + -12/35 = -12/35 + -2/7 is satisfied.

(v) 4 and -3/5

Solution: By using the commutativity law, the addition of rational numbers is commutative.

∴ a/b + c/d = c/d + a/b

In order to verify the above property let us consider the given fraction

4/1 and -3/5 as

4/1 + -3/5 and -3/5 + 4/1

The denominators are 1 and 5

By taking LCM for 1 and 5 is 5

We rewrite the given fraction in order to get the same denominator

Now, 4/1 = (4 × 5) / (1×5) = 20/5

-3/5 = (-3 ×1) / (5 ×1) = -3/5

Since the denominators are same we can add them directly

20/5 + -3/5 = (20 + (-3))/5 = (20-3)/5 = 17/5

-3/5 + 4/1

The denominators are 5 and 1

By taking LCM for 5 and 1 is 5

We rewrite the given fraction in order to get the same denominator

Now, -3/5 = (-3 ×1) / (5 ×1) = -3/5

4/1 = (4 × 5) / (1×5) = 20/5

Since the denominators are same we can add them directly

-3/5 + 20/5 = (-3 + 20)/5 = 17/5

∴ 4/1 + -3/5 = -3/5 + 4/1 is satisfied.

(vi) -4 and 4/-7

Solution: Firstly we need to convert the denominators to positive numbers.

4/-7 = (4 × -1)/ (-7 × -1) = -4/7

By using the commutativity law, the addition of rational numbers is commutative.

∴ a/b + c/d = c/d + a/b

In order to verify the above property let us consider the given fraction

-4/1 and -4/7 as

-4/1 + -4/7 and -4/7 + -4/1

The denominators are 1 and 7

By taking LCM for 1 and 7 is 7

We rewrite the given fraction in order to get the same denominator

Now, -4/1 = (-4 × 7) / (1×7) = -28/7

-4/7 = (-4 ×1) / (7 ×1) = -4/7

Since the denominators are same we can add them directly

-28/7 + -4/7 = (-28 + (-4))/7 = (-28-4)/7 = -32/7

-4/7 + -4/1

The denominators are 7 and 1

By taking LCM for 7 and 1 is 7

We rewrite the given fraction in order to get the same denominator

Now, -4/7 = (-4 ×1) / (7 ×1) = -4/7

-4/1 = (-4 × 7) / (1×7) = -28/7

Since the denominators are same we can add them directly

-4/7 + -28/7 = (-4 + (-28))/7 = (-4-28)/7 = -32/7

∴ -4/1 + -4/7 = -4/7 + -4/1 is satisfied.

2. Verify associativity of addition of rational numbers i.e., (x + y) + z = x + (y + z), when:

(i) x = ½, y = 2/3, z = -1/5

Solution: As the property states (x + y) + z = x + (y + z)

Use the values as such,

(1/2 + 2/3) + (-1/5) = 1/2 + (2/3 + (-1/5))

Let us consider LHS (1/2 + 2/3) + (-1/5)

Taking LCM for 2 and 3 is 6

(1× 3)/(2×3) + (2×2)/(3×2)

3/6 + 4/6

Since the denominators are same we can add them directly,

3/6 + 4/6 = 7/6

7/6 + (-1/5)

Taking LCM for 6 and 5 is 30

(7×5)/(6×5) + (-1×6)/(5×6)

35/30 + (-6)/30

Since the denominators are same we can add them directly,

(35+(-6))/30 = (35-6)/30 = 29/30

Let us consider RHS 1/2 + (2/3 + (-1/5))

Taking LCM for 3 and 5 is 15

(2/3 + (-1/5)) = (2×5)/(3×5) + (-1×3)/(5×3)

= 10/15 + (-3)/15

Since the denominators are same we can add them directly,

10/15 + (-3)/15 = (10-3)/15 = 7/15

1/2 + 7/15

Taking LCM for 2 and 15 is 30

1/2 + 7/15 = (1×15)/(2×15) + (7×2)/(15×2)

= 15/30 + 14/30

Since the denominators are same we can add them directly,

= (15 + 14)/30 = 29/30

∴ LHS = RHS associativity of addition of rational numbers is verified.

(ii) x = -2/5, y = 4/3, z = -7/10

Solution: As the property states (x + y) + z = x + (y + z)

Use the values as such,

(-2/5 + 4/3) + (-7/10) = -2/5 + (4/3 + (-7/10))

Let us consider LHS (-2/5 + 4/3) + (-7/10)

Taking LCM for 5 and 3 is 15

(-2× 3)/(5×3) + (4×5)/(3×5)

-6/15 + 20/15

Since the denominators are same we can add them directly,

-6/15 + 20/15= (-6+20)/15 = 14/15

14/15 + (-7/10)

Taking LCM for 15 and 10 is 30

(14×2)/(15×2) + (-7×3)/(10×3)

28/30 + (-21)/30

Since the denominators are same we can add them directly,

(28+(-21))/30 = (28-21)/30 = 7/30

Let us consider RHS -2/5 + (4/3 + (-7/10))

Taking LCM for 3 and 10 is 30

(4/3 + (-7/10)) = (4×10)/(3×10) + (-7×3)/(10×3)

= 40/30 + (-21)/30

Since the denominators are same we can add them directly,

40/30 + (-21)/30 = (40-21)/30 = 19/30

-2/5 + 19/30

Taking LCM for 5 and 30 is 30

-2/5 + 19/30 = (-2×6)/(5×6) + (19×1)/(30×1)

= -12/30 + 19/30

Since the denominators are same we can add them directly,

= (-12 + 19)/30 = 7/30

∴ LHS = RHS associativity of addition of rational numbers is verified.

(iii) x = -7/11, y = 2/-5, z = -3/22

Solution: Firstly convert the denominators to positive numbers

2/-5 = (2×-1)/ (-5×-1) = -2/5

As the property states (x + y) + z = x + (y + z)

Use the values as such,

(-7/11 + -2/5) + (-3/22) = -7/11 + (-2/5 + (-3/22))

Let us consider LHS (-7/11 + -2/5) + (-3/22)

Taking LCM for 11 and 5 is 55

(-7×5)/(11×5) + (-2×11)/(5×11)

-35/55 + -22/55

Since the denominators are same we can add them directly,

-35/55 + -22/55 = (-35-22)/55 = -57/55

-57/55 + (-3/22)

Taking LCM for 55 and 22 is 110

(-57×2)/(55×2) + (-3×5)/(22×5)

-114/110 + (-15)/110

Since the denominators are same we can add them directly,

(-114+(-15))/110 = (-114-15)/110 = -129/110

Let us consider RHS -7/11 + (-2/5 + (-3/22))

Taking LCM for 5 and 22 is 110

(-2/5 + (-3/22))= (-2×22)/(5×22) + (-3×5)/(22×5)

= -44/110 + (-15)/110

Since the denominators are same we can add them directly,

-44/110 + (-15)/110 = (-44-15)/110 = -59/110

-7/11 + -59/110

Taking LCM for 11 and 110 is 110

-7/11 + -59/110 = (-7×10)/(11×10) + (-59×1)/(110×1)

= -70/110 + -59/110

Since the denominators are same we can add them directly,

= (-70 -59)/110 = -129/110

∴ LHS = RHS associativity of addition of rational numbers is verified.

(iv) x = -2, y = 3/5, z = -4/3

Solution: As the property states (x + y) + z = x + (y + z)

Use the values as such,

(-2/1 + 3/5) + (-4/3) = -2/1 + (3/5 + (-4/3))

Let us consider LHS (-2/1 + 3/5) + (-4/3)

Taking LCM for 1 and 5 is 5

(-2×5)/(1×5) + (3×1)/(5×1)

-10/5 + 3/5

Since the denominators are same we can add them directly,

-10/5 + 3/5= (-10+3)/5 = -7/5

-7/5 + (-4/3)

Taking LCM for 5 and 3 is 15

(-7×3)/(5×3) + (-4×5)/(3×5)

-21/15 + (-20)/15

Since the denominators are same we can add them directly,

(-21+(-20))/15 = (-21-20)/15 = -41/15

Let us consider RHS -2/1 + (3/5 + (-4/3))

Taking LCM for 5 and 3 is 15

(3/5 + (-4/3)) = (3×3)/(5×3) + (-4×5)/(3×5)

= 9/15 + (-20)/15

Since the denominators are same we can add them directly,

9/15 + (-20)/15 = (9-20)/15 = -11/15

-2/1 + -11/15

Taking LCM for 1 and 15 is 15

-2/1 + -11/15 = (-2×15)/(1×15) + (-11×1)/(15×1)

= -30/15 + -11/15

Since the denominators are same we can add them directly,

= (-30 -11)/15 = -41/15

∴ LHS = RHS associativity of addition of rational numbers is verified.

3. Write the additive of each of the following rational numbers:

(i) -2/17

(ii) 3/-11

(iii) -17/5

(iv) -11/-25

Solution:

(i) The additive inverse of -2/17 is 2/17

(ii) The additive inverse of 3/-11 is 3/11

(iii) The additive inverse of -17/5 is 17/5

(iv) The additive inverse of -11/-25 is -11/25

4. Write the negative(additive) inverse of each of the following:

(i) -2/5

(ii) 7/-9

(iii) -16/13

(iv) -5/1

(v) 0

(vi) 1

(vii) – 1

Solution:

(i) The negative (additive) inverse of -2/5 is 2/5

(ii) The negative (additive) inverse of 7/-9 is 7/9

(iii) The negative (additive) inverse of -16/13 is 16/13

(iv) The negative (additive) inverse of -5/1 is 5

(v) The negative (additive) inverse of 0 is 0

(vi) The negative (additive) inverse of 1 is -1

(vii) The negative (additive) inverse of -1 is 1

5. Using commutativity and associativity of addition of rational numbers, express each of the following as a rational number:

(i) 2/5 + 7/3 + -4/5 + -1/3

Solution: Firstly group the rational numbers with same denominators

2/5 + -4/5 + 7/3 + -1/3

Now the denominators which are same can be added directly.

(2+(-4))/5 + (7+(-1))/3

(2-4)/5 + (7-1)/3

-2/5 + 6/3

By taking LCM for 5 and 3 we get, 15

(-2×3)/(5×3) + (6×5)/(3×5)

-6/15 + 30/15

Since the denominators are same can be added directly

(-6+30)/15 = 24/15

Further can be divided by 3 we get,

24/15 = 8/5

(ii) 3/7 + -4/9 + -11/7 + 7/9

Solution: Firstly group the rational numbers with same denominators

3/7 + -11/7 + -4/9 + 7/9

Now the denominators which are same can be added directly.

(3+ (-11))/7 + (-4+ 7)/9

(3-11)/7 + (-4+7)/9

-8/7 + 3/9

-8/7 + 1/3

By taking LCM for 7 and 3 we get, 21

(-8×3)/ (7×3) + (1×7)/ (3×7)

-24/21 + 7/21

Since the denominators are same can be added directly

(-24+7)/21 = -17/21

(iii) 2/5 + 8/3 + -11/15 + 4/5 + -2/3

Solution: Firstly group the rational numbers with same denominators

2/5 + 4/5 + 8/3 + -2/3 + -11/15

Now the denominators which are same can be added directly.

(2 + 4)/5 + (8 + (-2))/3 + -11/15

6/5 + (8-2)/3 + -11/15

6/5 + 6/3 + -11/15

6/5 + 2/1 + -11/15

By taking LCM for 5, 1 and 15 we get, 15

(6×3)/ (5×3) + (2×15)/ (1×15) + (-11×1)/ (15×1)

18/15 + 30/15 + -11/15

Since the denominators are same can be added directly

(18+30+ (-11))/15 = (18+30-11)/15 = 37/15

(iv) 4/7 + 0 + -8/9 + -13/7 + 17/21

Solution: Firstly group the rational numbers with same denominators

4/7 + -13/7 + -8/9 + 17/21

Now the denominators which are same can be added directly.

(4 + (-13))/7 + -8/9 + 17/21

(4-13)/7 + -8/9 + 17/21

-9/7 + -8/9 + 17/21

By taking LCM for 7, 9 and 21 we get, 63

(-9×9)/ (7×9) + (-8×7)/ (9×7) + (17×3)/ (21×3)

-81/63 + -56/63 + 51/63

Since the denominators are same can be added directly

(-81+(-56)+ 51)/63 = (-81-56+51)/63 = -86/63

6. Re-arrange suitably and find the sum in each of the following:

(i) 11/12 + -17/3 + 11/2 + -25/2

Solution: Firstly group the rational numbers with same denominators

11/12 + -17/3 + (11-25)/2

11/12 + -17/3 + -14/2

By taking LCM for 12, 3 and 2 we get, 12

(11×1)/(12×1) + (-17×4)/(3×4) + (-14×6)/(2×6)

11/12 + -68/12 + -84/12

Since the denominators are same can be added directly

(11-68-84)/12 = -141/12

(ii)-6/7 + -5/6 + -4/9 + -15/7

Solution: Firstly group the rational numbers with same denominators

-6/7 + -15/7 + -5/6 + -4/9

(-6 -15)/7 + -5/6 + -4/9

-21/7 + -5/6 + -4/9

-3/1 + -5/6 + -4/9

By taking LCM for 1, 6 and 9 we get, 18

(-3×18)/(1×18) + (-5×3)/(6×3) + (-4×2)/(9×2)

-54/18 + -15/18 + -8/18

Since the denominators are same can be added directly

(-54-15-8)/18 = -77/18

(iii) 3/5 + 7/3 + 9/ 5+ -13/15 + -7/3

Solution: Firstly group the rational numbers with same denominators

3/5 + 9/5 + 7/3 + -7/3 + -13/15

(3+9)/5 + -13/15

12/5 + -13/15

By taking LCM for 5 and 15 we get, 15

(12×3)/(5×3) + (-13×1)/(15×1)

36/15 + -13/15

Since the denominators are same can be added directly

(36-13)/15 = 23/15

(iv) 4/13 + -5/8 + -8/13 + 9/13

Solution: Firstly group the rational numbers with same denominators

4/13 + -8/13 + 9/13 + -5/8

(4-8+9)/13 + -5/8

5/13 + -5/8

By taking LCM for 13 and 8 we get, 104

(5×8)/(13×8) + (-5×13)/(8×13)

40/104 + -65/104

Since the denominators are same can be added directly

(40-65)/104 = -25/104

(v) 2/3 + -4/5 + 1/3 + 2/5

Solution: Firstly group the rational numbers with same denominators

2/3 + 1/3 + -4/5 + 2/5

(2+1)/3 + (-4+2)/5

3/3 + -2/5

1/1 + -2/5

By taking LCM for 1 and 5 we get, 5

(1×5)/(1×5) + (-2×1)/(5×1)

5/5 + -2/5

Since the denominators are same can be added directly

(5-2)/5 = 3/5

(vi) 1/8 + 5/12 + 2/7 + 7/12 + 9/7 + -5/16

Solution: Firstly group the rational numbers with same denominators

1/8 + 5/12 + 7/12 + 2/7 + 9/7 + -5/16

1/8 + (5+7)/12 + (2+9)/7 + -5/16

1/8 + 12/12 + 11/7 + -5/16

1/8 + 1/1 + 11/7 + -5/16

By taking LCM for 8, 1, 7 and 16 we get, 112

(1×14)/(8×14) + (1×112)/(1×112) + (11×16)/(7×16) + (-5×7)/(16×7)

14/112 + 112/112 + 176/112 + -35/112

Since the denominators are same can be added directly

(14+112+176-35)/112 = 267/112

---