**Question 1: Verify the property: x × y = y × x by taking:**

**(i) x = -1/3, y = 2/7**

**Solution:**

x × y = (-1/3) × 2/7

= (-1 × 2)/(3 × 7)

= -2/21

y × x = 2/7 × (-1/3)

= (2 × -1)/(7 × 3)

= -2/21

Hence, x × y = y × x, property is verified

**(ii) x = -3/5, y = -11/13**

**Solution:**

x × y = (-3/5) × (-11/3)

= (-3 × -11)/(5 × 13)

= 33/65

y × x = (-11/13) × (-3/5)

= (-11 × -3)/(13 × 5)

= 33/65

Hence, x × y = y × x, property is verified

**(iii) x = 2, y = 7/-8**

**Solution:**

x × y = 2/1 × (-7/8)

= (2 × -7)/(1 × 8)

2 is the common factor

= -7/4

y × x = (-7/8) × (2/1)

= (-7 × 2)/(8 × 1)

2 is the common factor

= -7/4

Hence, x × y = y × x, property is verified

**(iv) x = 0, y = -15/8**

**Solution:**

x × y = 0 × (-15/8)

= 0

y × x = (-15/8) × 0

= 0

Hence, x × y = y × x, property is verified

**Question 2: Verify the property: x × (y × z) = (x × y) × z by taking:**

**(i) x = -7/3, y = 12/5, z = 4/9**

**Solution:**

x × (y × z)= -7/3 × (12/5 × 4/9)

= -7/3 × ((12 × 4)/(5 × 9))

3 is the common factor of 12 and 9

= -7/3 × ((4 × 4)/(5 × 3))

= -7/3 × (16/15)

= (-7 × 16)/(3 × 15)

= -112/45

(x × y) × z= (-7/3 × 12/5) × 4/9

= ((-7 × 12)/(3 × 5)) × 4/9

= ((-7 × 4)/(5)) × 4/9

= -28/5 × 4/9

= -112/45

Hence, x × (y × z) = (x × y) × z Property is verified

**(ii) x = 0, y = -3/5, z = -9/4**

**Solution:**

x × (y × z)= 0 × (-3/5 × -9/4)

= 0 × (27/20)

= 0(Any number multiplied with zero is zero)

(x × y) × z= (0 × -3/5) × -9/4

= 0 × -9/4

= 0(Any number multiplied with zero os zero)

Hence, x × (y × z) = (x × y) × z Property is verified

**(iii) x = 1/2, y = 5/-4, z = -7/5**

**Solution:**

x × (y × z)= 1/2 × (-5/4 × -7/5)

= 1/2 × ((-5 × -7)/(4 × 5))

Common factor 5

= 1/2 × (7/4)

= 7/8

(x × y) × z= (1/2 × -5/4) × -7/5

= -5/8 × -7/5

Common factor 5

= 7/8

Hence, x × (y × z) = (x × y) × z Property is verified

**(iv) x = 5/7, y = -12/13, z = -7/18**

**Solution:**

x × (y × z)= 5/7 × (-12/13 × -7/18)

= 5/7 × ((-12 × -7)/(13 × 18))

Common factor 6 of 12 and 18

= 5/7 × ((-2 × -7)/(13 × 3))

= 5/7 × (14/39)

= (5 × 14)/(7 × 39)

7 is the common factor of 7 and 14

= 5 × 2/39

= 10/39

(x × y) × z= (5/7 × -12/13) × -7/18

= ((5 × -12)/(7 × 13)) × -7/18

= (5 × -12 × -7)/(7 × 13 × 18)

Common factor 7 and 6

= (5 × -2 × -1)/(1 × 13 × 3)

= 10/39

Hence, x × (y × z) = (x × y) × z Property is verified

**Question 3: Verify the property: x × (y + z) = x × y + x × z by taking:**

**(i) x = -3/7, y = 12/13, z = -5/6**

**Solution:**

x × (y + z)

= -3/7 × (12/13 + -5/6)

LCM of 13 and 6 is 78

= -3/7 × ((12 × 6 – 5 × 13)/78)

= -3/7 × ((72 – 65)/78)

= -3/7 × 7/78

= -3 × 7/7 × 78

Common factor 7 and 3

= -1/26

x × y + x × z

= -3/7 × 12/13 + -3/7 × -5/6

= (-3 × 12)/(7 × 13) + (-3 × -5)/(6 × 7)

= -36/91 + 15/42

= (-36 × 6 + 15 × 13)/546

= 196 – 216/546

= -21/546

= -1/26

Hence, the property x × (y + z) = x × y + x × z is verified

**(ii) x = -12/5, y = -15/4, z = 8/3**

**Solution:**

x × (y + z)

= -12/5 × (-15/4 + 8/3)

LCM is 12

= -12/5 × ((-15 × 3 + 8 × 4)/12)

= -12/5 × ((-45 + 32)/12)

= -12/5 × (-13)/12

= (-12 × -13)/(5 × 12)

12 is the common factor

= 13/5

x × y + x × z

= -12/5 × -15/4 + -12/5 × 8/3

= (-12 × -15)/(5 × 4) + (-12 × 8)/(5 × 3)

Common factor 4 and 5, 3

= 9 + -32/5

LCM is 5

= (9 × 5 – 32)/5

= 45 – 32/5

= 13/5

Hence, the property x × (y + z) = x × y + x × z is verified

**(iii) x = -8/3, y = 5/6, z = -13/12**

**Solution:**

x × (y + z)

= -8/3 × (5/6 + -13/12)

LCM is 12

= -8/3 × (5 × 2 – 13)/(12)

= -8/3 × (10 – 13)/12

= -8/3 × (-3/12)

= (-8 × -3)/(3 × 12)

Common factor 4 and 3

= 2/3

x × y + x × z

= -8/3 × 5/6 + -8/3 × -13/12

= (-8 × 5)/(3 × 6) + (-8 × -13)/(3 × 12)

Common factor 2 and 4

= (-4 × 5)/(3 × 3) + (-2 × -13)/(3 × 3)

= -20/9 + 26/9

= (-20 + 26)/9

= 6/9

Common factor is 3

= 2/3

Hence, the property x × (y + z) = x × y + x × z is verified

**iv) x = -3/4, y = -5/2, z = 7/6**

**Solution:**

x × (y + z)

= -3/4 × (-5/2 + 7/6)

LCM is 6

= -3/4(-5 × 3 + 7)/6

= -3/4 × (-15 + 7)/6

= -3/4 × -8/6

= (-3 × -8)/(4 × 6)

Common factor 3 and 4

= 1

x × y × + x × z

= -3/4 × -5/2 + -3/4 × 7/6

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

= 15/8 + -7/8

= (15 – 7)/8

= 8/8

= 1

Hence, Hence, the property x × (y + z) = x × y + x × z is verified

**Question 4: Use the distributivity of multiplication of rational numbers over their addition to simplify:**

**(i) 3/5 × ((35/24) + (10/1))**

**Solution:**

= 3/5 × 35/24 + 3/5 × 10/1

= (3 × 35)/(5 × 24) + (3 × 10)/(5 × 1)

Common factor is 5 and 3

= (1 × 7)/(1 × 8) + (3 × 2)/(1)

= 7/8 + 6/1

LCM is 8

= (7 + 6 × 8)/8

= 7 + 48/8

= 55/8

**(ii) -5/4 × ((8/5) + (16/5))**

**Solution:**

= -5/4×8/5+-5/4×16/5

= (-5×8)/(4×5)+(-5×16)/(4×5)

Common factor is 4, 5

= -2/1+-4/1

= -6

**(iii) 2/7 × ((7/16) — (21/4))**

**Solution:**

= 2/7×7/16-2/7×21/4

= (2×7)/(7×16)-(2×21)/(7×4)

Common factor 2 and 7

= 1/8-3/2

LCM is 8

= (1-3×4)/8

= (1-12)/8

= -11/8

**(iv) 3/4 × ((8/9) — 40)**

**Solution:**

= 3/4×8/9-3/4×40

= (3×8)/(4×9)-(3×40)/(4×1)

Common factor 3 and 4

= 2/3-30/1

LCM is 3

= (2-90)/3

= -88/3

**Question 5: Find the multiplicative inverse (reciprocal) of each of the following rational numbers:**

**(i) 9**

**Solution:**

Multiplicative inverse of 9/1 is 1/9

**(ii) -7**

**Solution:**

Multiplicative inverse of -7/1 is 1/-7 or -1/7

**(iii) 12/5**

**Solution:**

Multiplicative inverse of 12/5 is 5/12

**(iv) -7/9**

**Solution:**

Multiplicative inverse of -7/9 is 9/-7 or -9/7

**(v) -3/-5**

**Solution:**

Multiplicative inverse of -3/-5 is -5/-3 or 5/3

**(vi) 2/3 × 9/4**

**Solution:**

(2×9)/(3×4)

2 is common factor of 2 and 4, 3 is common factor of 3 and 9

=3/2

Multiplicative inverse is 2/3

**(vii) -5/8 × 16/15**

**Solution:**

(-5×16)/(8×15)

5 is the common factor of 5 and 15, 8 is the common factor of 8 and 16

=-2/3

Multiplicative inverse is 3/-2 or -3/2

**(viii) -2 × -3/5**

**Solution:**

=(-2×-3)/(1×5)

=6/5

Multiplicative inverse is 5/6

**(ix) -1**

**Solution:**

Multiplicative inverse is -1

**(x) 0/3**

**Solution:**

Multiplicative inverse is 3/0 which does not exist

**(xi) 1**

**Solution:**

Multiplicative inverse is 1

**Question 6: Name the property of multiplication of rational numbers illustrated by the following statements:**

**(i) -5/16 × 8/15 = 8/15 × -5/16**

**Solution:**

a×b=b×a

This is commutative property

**(ii) -17/5 ×9 = 9 × -17/5**

**Solution:**

a×b=b×a

This is commutative property

**(iii) 7/4 × (-8/3 + -13/12) = 7/4 × -8/3 + 7/4 × -13/12**

**Solution:**

a×(b+c)=a×b+a×c

This is distributive property of multiplication over addition

**(iv) -5/9 × (4/15 × -9/8) = (-5/9 × 4/15) × -9/8**

**Solution:**

a×(b×c)=(a×b)×c

This is associative property of multiplication

**(v) 13/-17 × 1 = 13/-17 = 1 × 13/-17**

**Solution:**

a×1=a=1×a

This is multiplicative identity

**(vi) -11/16 × 16/-11 = 1**

**Solution:**

a×1/a=1

This is multiplicative inverse

**(vii) 2/13 × 0 = 0 = 0 × 2/13**

**Solution:**

a×0=0=0×a

Any number multiplied with 0 is 0

**(viii) -3/2 × 5/4 + -3/2 × -7/6 = -3/2 × (5/4 + -7/6)**

**Solution:**

a×b+a×c=a×(b+c)

This is distributive law of multiplication over addition

**Question 7: Fill in the blanks:**

**(i) The product of two positive rational numbers is always…**

**Solution:**

The product of two positive rational numbers is always positive.

**(ii) The product of a positive rational number and a negative rational number is always….**

**Solution:**

The product of a positive rational number and a negative rational number is always negative

**(iii) The product of two negative rational numbers is always…**

**Solution:**

The product of two negative rational numbers is always positive

**(iv) The reciprocal of a positive rational number is…**

**Solution:**

The reciprocal of a positive rational numbers is positive

**(v) The reciprocal of a negative rational number is…**

**Solution:**

The reciprocal of a negative rational numbers is negative

**(vi) Zero has …. Reciprocal.**

**Solution:**

Zero has no reciprocal.

**(vii) The product of a rational number and its reciprocal is…**

**Solution:**

The product of a rational number and its reciprocal is 1

**(viii) The numbers … and … are their own reciprocals.**

**Solution:**

The numbers 1 and -1 are their own reciprocals.

**(ix) If a is reciprocal of b, then the reciprocal of b is.**

**Solution:**

If a is reciprocal of b, then the reciprocal of b is a

**(x) The number 0 is … the reciprocal of any number.**

**Solution:**

The number 0 is not the reciprocal of any number.

**(xi) reciprocal of 1/a, a ≠ 0 is …**

**Solution:**

Reciprocal of 1/a, a ≠ 0 is a

**(xii) (17×12) ^{-1} = 17^{-1} × …**

**Solution:**

(17×12)

^{-1}= 17^{-1}× 12^{-1}

**Question 8: Fill in the blanks:**

**(i) -4 × 79 = 79 × …**

**Solution:**

-4 × 79= 79 × -4

By using commutative property.

**(ii) 5/11 × -3/8 = -3/8 × …**

**Solution:**

5/11 × -3/8 = -3/8 × 5/11

By using commutative property.

**(iii) 1/2 × (3/4 + -5/12) = 1/2 × … + … × -5/12**

**Solution:**

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

By using distributive property.

**(iv) -4/5 × (5/7 + -8/9) = (-4/5 × …) + -4/5 × -8/9**

**Solution:**

-4/5 × (5/7 + -8/9) = (-4/5 × 5/7) + -4/5 × -8/9

By using distributive property.