**Problem 1 :**

Verify the closure property for addition and multiplication of the rational numbers −5/7 and 8/9.

**Problem 2 :**

Verify the associative property for addition and multiplication of the rational numbers.

−10/11, 5/6, -4/3

**Problem 3 :**

Check the commutative property for addition and multiplication of the rational numbers −10/11 and −8/33.

**Problem 1 :**

Verify the closure property for addition and multiplication of the rational numbers −5/7 and 8/9.

**Solution :**

Let a = -5/7 and b = 8/9

Closure property for addition :

a + b = b + a

Closure property for multiplication :

a x b = b x a

(i) a + b = (-5/7) + (8/9)

= (-45 + 56)/63

= 11/63 ----(1)

b + a = (8/9) + (-5/7)

= (56 - 45)/63

= 11/63 ----(2)

(1) = (2)

(ii) a x b = (-5/7) x (8/9)

= -40/63 -----(1)

b x a = (8/9) x (-5/7)

= -40/63 ----(2)

(1) = (2)

**Problem 2 :**

Verify the associative property for addition and multiplication of the rational numbers.

−10/11, 5/6, -4/3

**Solution :**

Addition and multiplication are associative for rational numbers.That is, for any three rational numbers a,b and c,

(i) (a + b) + c = a + (b + c) and

(ii) (a × b) × c = a × (b × c)

a = -10/11, b = 5/6 and c = -4/3

(i)

L.H.S :

a + b = (-10/11) + (5/6) = (-60 + 55)/66 = -5/66

(a + b) + c = (-5/66) + (-4/3) = (-5-88)/66

= 93/66 = 31/22 ---(1)

R.H.S :

b + c = (5/6) + (-4/3) = (5 - 8)/6 = -3/6 = -1/2

a + (b + c) = (-10/11) + (-1/2) = (-20-11)/22

= 31/22 ---(2)

(1) = (2)

Hence associative property for addition is true.

(ii) (a × b) × c = a × (b × c)

L.H.S :

a x b = (-10/11) x (5/6) = -25/33

(a x b) x c = (-25/33) x (-4/3) = 100/99 -----(1)

R.H.S :

(b × c) = (5/6) x (-4/3) = -10/9

a x (b x c) = (-10/11) x (-10/9) = 100/99 ----(2)

(1) = (2)

Hence associative property for multiplication is true.

**Problem 3 :**

Check the commutative property for addition and multiplication of the rational numbers −10/11 and −8/33.

**Solution :**

(i) Commutative property for addition :

a + b = b + a

(ii) Commutative property for multiplication :

a x b = b x a

(i) a = -10/11 and b = -8/33

a + b = (-10/11) + (-8/33)

= (-30 - 8)/33

= -38/33 ----(1)

b + a = (-8/33) + (-10/11)

= (-8 - 30)/33

= -38/33 ----(2)

(1) = (2)

Hence proved.

(ii)

a x b = (-10/11) x (-8/33)

= 80/363-----(1)

b x a = (-8/33) x (-10/11)

= 80/363 ------(2)

(1) = (2)

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