**Problem 1. Divide:**

**(i) 1 by 1/2**

**Solution**:

= 1 / (1 / 2)

= 1 × (2 / 1)

= 2

**ii) 5 by -5/7**

**Solution**:

= 5 / (-5 / 7)

= 5 × (-7 / 5)

5 is the common factor

= -7

**(iii) -3/4 by 9/-16**

**Solution**:

= (-3 / 4) / (-9 / 16)

= -3 / 4 × (-16 / 9)

3 and 4 are the common factor

= (-3 × -16) / (4 × 9)

= 4 / 3

**iv) -7/8 by -21/16**

**Solution**:

= (-7 / 8) / (-21 / 16)

= -7 / 8 × (-16 / 21)

= (-7 × -16) / (8 × 21)

7 and 8 are the common factor

= 2 / 3

**v) 7/-4 by 63/64**

**Solution**:

= 7 / -4 / (63 / 64)

= -7 / 4 × (64 / 63)

= (-7 × 64) / (4 × 63)

4 and 7 are common factor

= -16 / 9

**vi) 0 by -7/5**

**Solution**:

= 0 / (-7 / 5)

= 0 × (-5 / 7)

= 0

**(vii) -3/4 by -6**

**Solution**:

= -3 / 4 / (-6 / 1)

= -3 / 4 × (-1 / 6)

= (-3 × -1) / (4 × 6)

3 is the common factor

= 1 / 8

**(viii) 2/3 by -7/12**

**Solution**:

= 2 / 3 / (-7 / 12)

= 2 / 3 × (12 / -7)

= (2 × 12) / (3 × -7)

3 is the common factor

= 8 / -7

= -8 / 7

**(ix) -4 by -3/5**

**Solution**:

= -4 / (-3 / 5)

= -4 × (5 / -3)

= (-4 × -5) / 3

= 20 / 3

**(x) -3/13 by -4/65**

**Solution**:

= -3 / 13 / (-4 / 65)

= -3 / 13 × (65 / -4)

= (-3 × -65) / (13 × 4)

13 is the common factor

= (-3 × -5) / 4

= 15 / 4

**Problem 2. Find the value and express as a rational number in standard form:**

**(i) 2/5 ÷ 26/15**

**Solution**:

= 2 / 5 / (26 / 15)

= 2 / 5 × 15 / 26

= (2 × 15) / (5 × 26)

5 and 2 are the common factor

= 3 / 13

**(ii) 10/3 ÷ -35/12**

**Solution**:

= 10 / 3 / (-35 / 12)

= 10 / 3 × -12 / 35

= (10 × -12) / (3 × 35)

Common factor is 5 and 3

= (2 × -4) / (7)

= -8 / 7

**(iii) -6 ÷ -8/17**

**Solution**:

= -6 / (-8 / 17)

= -6 × (-17 / 8)

= (-6 × -17) / (8 × 1)

2 is the common factor

= (-3 × -17) / 4

= 51 / 4

**(iv) -40/99 ÷ -20**

**Solution**:

= -40 / 99 / (-20 / 1)

= -40 / 99 × (-1 / 20)

= (-40 × -1) / (99 × 20)

20 is the common factor

= 2 / 99

**(v) -22/27 ÷ -110/18**

**Solution**:

= (-22 / 27) / (-110 / 18)

= (-22 / 27) × (18 / -110)

= (-22 × 18) / (27 × -110)

9 and 22 are the common factor

= 2 / (3 × 5)

= 2 / 15

**(vi) -36/125 ÷ -3/75**

**Solution**:

= (-36 / 125) / (-3 / 75)

= (-36 / 125) × (75 / -3)

= (-12 / 25) × (15 / -1)

= (-12 × 15) / (25 × -1)

= (-12 × -3) / 5

= 36 / 5

**Problem 3. The product of two rational numbers is 15. If one of the numbers is -10, find the other.**

**Solution**:

We know that the product of two rational numbers = 15

One of the number = -10

Let the other number be x

-10x = 15

x = 15 / -10

5 is the common factor

= -3 / 2

The other number is -3 / 2

**Problem 4. The product of two rational numbers is -8/9. If one of the numbers is -4/15, find the other.**

**Solution:**

We know that the product of two rational numbers = -8 / 9

One of the number = -4 / 15

Let the other number be x

(-4 / 15) x = -8 / 9

x = (-8 / 9) / (-4 / 15)

= (-8 / 9) × (15 / -4)

3 and 4 are the common factor

= (-2 / 3) × (5 / -1)

= (-2 × 5) / (3 × -1)

= -10 / -3

= 10 / 3

The other number is 10 / 3

**Problem 5. By what number should we multiply -1/6 so that the product may be -23/9?**

**Solution**:

Let the number be x

So, x (-1 / 6) = -23 / 9

x = (-23 / 9) / (-1 / 6)

x = (-23 / 9) × (6 / -1)

= (-23 / 3) × (2 × -1)

= (-23 × -2) / (3 × 1)

= 46 / 3

It should be multiplied by 46 / 3

**Problem 6. By what number should we multiply -15/28 so that the product may be -5/7?**

**Solution**:

Let the number be x

So, x (-15 / 28) = -5 / 7

x = (-5 / 7) / (-15 / 28)

x = (-5 / 7) × (28 / -15)

= (-5 ×28) / (7 × -15)

5 and 7 are the common factor

= -4 / -3

= 4 / 3

**Problem 7. By what number should we multiply -8/13 so that the product may be 24?**

**Solution**:

Let the number be x

So, x (-8 / 13) = 24

x = (24) / (-8 / 13)

= (24) × (13 / -8)

= (24 × 13) / (-8)

8 is the common factor

= -3 × 13

= -39

It should be multiplied by -39

**Problem 8. By what number should -3/4 be multiplied in order to produce 2/3?**

**Solution**:

Let the number be x

x (-3 / 4) = 2 / 3

x = (2 / 3) / (-3 / 4)

= (2 / 3) × (4 / -3)

= -8 / 9

It should be multiplied by -8 / 9

**Problem 9. Find (x+y) ÷ (x-y), if**

**(i) x = 2/3, y = 3/2**

**Solution**:

x + y = 2 / 3 + 3 / 2

LCM is 6

= (2 × 2 + 3 × 3) / 6

= (4 + 9) / 6

= 13 / 6

x – y = 2 / 3 – 3 / 2

LCM is 6

= (2 × 2 – 3 × 3) / 6

= (4 – 9) / 6

= -5 / 6

(x + y) ÷ (x – y) = (13 / 6) / (-5 / 6)

= (13 / 6) × (6 / -5)

= (13 × -6) / (6 × 5)

6 is the common factor

= -13 / 5

**(ii) x = 2/5, y = 1/2**

**Solution**:

x + y = 2 / 5 + 1 / 2

LCM is 10

= (2 × 2 + 1 × 5) / 10

= (4 + 5) / 10

= 9 / 10

x – y = 2 / 5 – 1 / 2

LCM is 10

(2 × 2 – 1 × 5) / 10

= (4 – 5) / 10

= -1 / 10

(x + y) ÷ (x – y) = (9 / 10) / (-1 / 10)

= (9 / 10) × (10 / -1)

= (9 × 10) / (10 × -1)

10 is the common factor

= -9

**(iii) x = 5/4, y = -1/3**

**Solution**:

x + y = 5 / 4 + -1 / 3

LCM is 12

= (5 × 3 – 1 × 4) / 12

= (15 – 4) / 12

= 11 / 12

x – y = 5 / 4 – (-1/3)

= 5 / 4 + 1 / 3

LCM is 12

= (5 × 3 + 1 × 4) / 12

= 19 / 12

(x + y) ÷ (x – y) = (11 / 12) / (19 / 12)

= (11 / 12) × (12 / 19)

= (11 × 12) / (12 × 19)

Common factor is 12

= 11 / 19

**(iv) x = 2/7, y = 4/3**

**Solution**:

x + y = 2 / 7 + 4 / 3

LCM is 21

= (2 × 3 + 4 × 7) / 21

= (6 + 28) / 21

= 34 / 21

x – y = 2 / 7 – 4 / 3

LCM is 21

= (2 × 3 – 4 × 7) / 21

= (6 – 28) / 21

= -22 / 21

(x + y) ÷ (x – y) = (34 / 21) / (-22 / 21)

= (34 / 21) × (21 / -22)

21 is the common factor

= -34 / 22

= -17 / 11

**(v) x = 1/4, y = 3/2**

**Solution**:

x + y = 1 / 4 + 3 / 2

LCM is 4

= (1 + 3 × 2) / 4

= 7 / 4

x – y = 1 / 4 – 3 / 2

LCM is 4

= (1 – 3 × 2) / 4

= -5 / 4

(x + y) ÷ (x – y) = (7 / 4) / (-5 / 4)

= (7 / 4) × (4 / -5)

4 is the common factor

= -7 / 5

**Problem 10. The cost of 723 meters of rope is Rs 12 ¾. Find the cost per meter.**

**Solution**:

23 / 3 meters of rope = Rs 51 / 4

Let us consider a number = x

So, x (23 / 3) = 51 / 4

x = (51 / 4) / (23 / 3)

= (51 / 4) × (3 / 23)

= (51 × 3) / (4 × 23)

= 153 / 92

= 1 61 / 92

Cost per meter is Rs 1 61 / 92