TOPICS

Exercise - 1.1

Rational Numbers

**Question-1 :-** Using appropriate properties find.

(i) -2/3 x 3/5 + 5/2 - 3/5 x 1/6

(ii) 2/5 x (-3/7) - 1/6 x 3/2 + 1/14 x 2/5

(i) -2/3 x 3/5 + 5/2 - 3/5 x 1/6 = -2/3 x 3/5 - 3/5 x 1/6 + 5/2 [by commutativity] = (-2/3 x 3/5) - (3/5 x 1/6) + 5/2 = 3/5 [-2/3 - 1/6] + 5/2 [by distributivity] = -3/5 [2/3 + 1/6] + 5/2 = -3/5 [(4 + 1)/6] + 5/2 = -3/5 x 5/6 + 5/2 = -3/6 + 5/2 = -1/2 + 5/2 = (-1 + 5)/2 = 4/2 = 2

(ii) 2/5 x (-3/7) - 1/6 x 3/2 + 1/14 x 2/5 = 2/5 x (-3/7) + 1/14 x 2/5 - 1/6 x 3/2 [by commutativity] = [2/5 x (-3/7)] + [1/14 x 2/5] - 1/6 x 3/2 = 2/5 [-3/7 + 1/14] - 3/12 [by distributivity] = 2/5 [(-6 + 1)/14] - 1/4 = 2/5 x (-5/14) - 1/4 = -2/14 - 1/4 = (-4 - 7)/28 = -11/28

**Question-2 :-** Write the additive inverse of each of the following.

(i) 2/8, (ii) -5/9, (iii) -6/(-5), (iv) 2/(-9), (v) 19/(-6)

(i) 2/8, Here, -2/8 is the additive inverse of 2/8 because -2/8 + 2/8 = 0 (ii) -5/9, Here, 5/9 is the additive inverse of -5/9 because -5/9 + 5/9 = 0 (iii) -6/(-5), Here, 6/(-5) is the additive inverse of -6/(-5) because -6/(-5) + [-6/(-5)] = 0 (iv) 2/(-9), Here, -2/(-9) is the additive inverse of 2/(-9) because -2/(-9) + 2/(-9) = 0 (v) 19/(-6) Here, -19/(-6) is the additive inverse of 19/(-6) because -19/(-6) + 19/(-6) = 0

**Question-3 :-** Verify that – (– x) = x for.

(i) x = 11/15, (ii) x = -13/17

(i) x = 11/15 The additive inverse of x = 11/15 is -x = -11/15 Then -11/15 + 11/15 = 0 Now, - (- x) = - (- 11/15) = 11/15 (ii) x = -13/17 The additive inverse of x = -13/17 is -x = 13/17 Then -13/17 + 13/17 = 0 Now, - (- x) = - [- (-13/17)] = -13/17

**Question-4 :-** Find the multiplicative inverse of the following.

(i) -13, (ii) -13/19, (iii) 1/5, (iv) -5/8 × (-3/7), (v) -1 × (-2/5), (vi) -1

(i) -13 Here, 1/(-13) is the mutiplicative inverse of -13 because -13 x 1/(-13) = 1. [a/b x c/d = 1] (ii) -13/19 Here, 19/(-13) is the mutiplicative inverse of -13/19 because -13/19 x 19/(-13) = 1. [a/b x c/d = 1] (iii) 1/5 Here, 5/1 is the mutiplicative inverse of 1/5 because 1/5 x 5/1 = 1. [a/b x c/d = 1] (iv) -5/8 × (-3/7) = 15/56 Here, 56/15 is the mutiplicative inverse of 15/56 because 15/56 x 56/15 = 1. [a/b x c/d = 1] (v) -1 × (-2/5) = 2/5 Here, 5/2 is the mutiplicative inverse of 2/5 because 2/5 x 5/2 = 1. [a/b x c/d = 1] (vi) -1 Here, 1/(-1) is the mutiplicative inverse of -1 because -1 x 1/(-1) = 1. [a/b x c/d = 1]

**Question-5 :-** Name the property under multiplication used in each of the following.

(i) -4/5 × 1 = 1 × (-4/5) = -4/5

(ii) -13/17 × (-2/7) = -2/7 × (-13/17)

(iii) -19/29 × 29/(-19) = 1

(i) -4/5 × 1 = 1 × (-4/5) = -4/5 Here, a x 1 = 1 x a = a in the form. So, this is multiplicative identity. (ii) -13/17 × (-2/7) = -2/7 × (-13/17) Here, a x b = b x a in the form. So, this is commutative property. (iii) -19/29 × 29/(-19) = 1 Here, a x 1/a = 1 in the form. So, this is multiplicative inverse property.

**Question-6 :-** Multiply 6/13 by the reciprocal of -7/16.

(i) x = 11/15 The additive inverse of x = 11/15 is -x = -11/15 Then -11/15 + 11/15 = 0 Now, - (- x) = - (- 11/15) = 11/15 (ii) x = -13/17 The additive inverse of x = -13/17 is -x = 13/17 Then -13/17 + 13/17 = 0 Now, - (- x) = - [- (-13/17)] = -13/17

**Question-7 :-** Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3.

Here, a x (b x c) = (a x b) x c in the form. So, this is associative property.

**Question-8 :-** Is 8/9 the multiplicative inverse of -1^{1}⁄_{8} .

If it will be the multiplicative inverse then their product will be 1. -1^{1}⁄_{8}= -7/8 8/9 × -7/8 = -7/9 ≠ 1 [a/b x c/d = 1] Hence, 8/9 is not the multiplicative inverse.

**Question-9 :-** Is 0.3 the multiplicative inverse of 3^{1}⁄_{3} ? Why or why not?

If it will be the multiplicative inverse then their product will be 1. 3^{1}⁄_{3}= 10/3 also, 0.3 = 3/10 3/10 × 10/3 = 1 [a/b x c/d = 1] Hence, 0.3 is the multiplicative inverse.

**Question-10 :-** Write.

(i) The rational number that does not have a reciprocal.

(ii) The rational numbers that are equal to their reciprocals.

(iii) The rational number that is equal to its negative.

(i) 0 is the rational number that does not have a reciprocal. (ii) 1 and -1 are the rational numbers that are equal to their reciprocals. (iii) 0 is the rational number that is equal to its negative.

**Question-11 :-** Fill in the blanks.

(i) Zero has ________ reciprocal.

(ii) The numbers ________ and ________ are their own reciprocals.

(iii) The reciprocal of –5 is ________.

(iv) Reciprocal of 1/x, where x ≠ 0 is ________.

(v) The product of two rational numbers is always a _______.

(vi) The reciprocal of a positive rational number is ________.

(i) Zero has no reciprocal. (ii) The numbers 1 and -1 are their own reciprocals (iii) The reciprocal of -5 is -1/5. (iv) Reciprocal of 1/x, where x ≠ 0 is x. (v) The product of two rational numbers is always a rational numbers. (vi) The reciprocal of a positive rational number is positive rational numbers.

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