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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