**1. Using appropriate properties find.**

**(i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6**

**Solution:**

Given: -2/3 × 3/5 + 5/2 – 3/5 × 1/6

Now,

= -2/3 × 3/5– 3/5 × 1/6+ 5/2 (by commutative property)

= 3/5 (-2/3 – 1/6)+ 5/2

= 3/5 ((- 4 – 1)/6)+ 5/2

= 3/5 ((–5)/6)+ 5/2 (by distributive property)

= – 15 /30 + 5/2

= – 1 /2 + 5/2

= 4/2

= 2

Thus, the required value is 2.

**(ii) 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5**

**Solution:**

Given: 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5

= 2/5 × (- 3/7) + 1/14 × 2/5 – (1/6 × 3/2) (by commutative property)

= 2/5 × (- 3/7 + 1/14) – 3/12

= 2/5 × ((- 6 + 1)/14) – 3/12

= 2/5 × ((- 5)/14)) – 1/4

= (-10/70) – 1/4

= – 1/7 – 1/4

= (– 4– 7)/28

= – 11/28

Thus, the required value is – 11/28.

**2. Write the additive inverse of each of the following**

**(i) 2/8**

**Solution:**

Additive inverse of 2/8 = – 2/8

**(ii) -5/9**

**Solution:**

Additive inverse of -5/9 = 5/9

**(iii) -6/-5**

**Solution:**

Standard form of -6/-5 = 6/5

Additive inverse of 6/5 = -6/5

**(iv) 2/-9**

**Solution:**

Standard form of 2/-9 = -2/9

Additive inverse of -2/9 = 2/9

**(v) 19/-16**

**Solution:**

Standard form of 19/-16 = -19/16

Additive inverse of -19/16 = 19/16

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

**(i) x = 11/15**

**Solution:**

We have, x = 11/15

– x = -11/15

Now,

-(-x) = -(-11/15) [ – x – = +]

-(-x)= 11/15

i.e., -(-x) = x [verified]

**(ii) -13/17**

**Solution:**

We have, x = -13/17

– x = -(-13/17) = 13/17

Now,

-(-x) = -13/17

i.e., -(-x) = x

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

**(i) -13**

**Solution:**

Multiplicative inverse of -13 = -1/13

**(ii) -13/19**

**Solution:**

Multiplicative inverse of -13/19 = -19/13

**(iii) 1/5**

**Solution:**

Multiplicative inverse of 1/5 = 5

**(iv) -5/8 × (-3/7) = 15/56**

**Solution:**

Multiplicative inverse of 15/56 = 56/15

**(v) -1 × (-2/5) = 2/5**

**Solution:**

Multiplicative inverse of 2/5 = 5/2

**(vi) -1**

**Solution:**

Multiplicative inverse of -1 = -1

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

**(i) -4/5 × 1 = 1 × (-4/5) = -4/5**

**Solution:**

Commutative property of multiplication

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

**Solution:**

Commutative property of multiplication

**(iii) -19/29 × 29/-19 = 1**

**Solution:**

Multiplicative inverse property

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

**Solution:**

Reciprocal of -7/16 = 16/-7 = -16/7

According to the question,

= 6/13 × -16/7 = -96/91

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

**Solution:**

We know that, a × (b × c) = (a × b) × c shows the associative property of multiplications.

1/3 × (6 × 4/3) = (1/3 × 6) × 4/3

Hence, the Associative Property is used here.

**8. Is 8/9 the multiplication inverse of -1 ^{1}/_{8} ? Why or why not?**

**Solution:**

-1 ^{1}/_{8} = -9/8

According to the question,

8/9 × (-9/8) = -1 ≠ 1

Hence, 8/9 is not the multiplicative inverse of -1 ^{1}/_{8 .}

**9. If 0.3 the multiplicative inverse of 3 ^{1}/_{3}? Why or why not?**

**Solution:**

3 ^{1}/_{3 }= 10/3

0.3 = 3/10

According to the question,

3/10 × 10/3 = 1

Hence, 0.3 is the multiplicative inverse of 3 ^{1}/_{3.}

**10. Write**

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

**Solution: **

=0 is the rational number that does not have a reciprocal.

Reason:

Reciprocal of 0 = 1/0, which is not defined.

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

**Solution: **

1 and -1 are the rational numbers that are equal to their reciprocals.

i.e.

1 = 1/1

Reciprocal of 1 = 1/1 = 1

Similarly, Reciprocal of -1 = – 1

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

**Solution: **

0 is the rational number that is equal to its negative.

i.e.

Negative of 0=-0=0

**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 _________.

**Solution:**

(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 number.

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

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