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RATIONAL NUMBERS
Exercise 1.1
Q.1. Using appropriate properties find:
2 3 5 3 1
(i) – × + – ×
3 5 2 5 6
2 ⎛ 3⎞ 1 3 1 2
(ii) × ⎜ – ⎟ – × + ×
5 ⎝ 7 ⎠ 6 2 14 5
Ans. (i)
2 3 5 3 1
× + – ×
3 5 2 5 6
2 3 3 1 5
= – × – × +
3 5 5 6 2
–
=
3⎛ 2 1 ⎞ 5
⎜– – ⎟ +
5⎝ 3 6 ⎠ 2
= –
3 ⎡2 1⎤ 5
+
+
5 ⎢⎣ 3 6 ⎥⎦ 2
= –
3 ⎡ 4 + 1⎤ 5
+
⎢
⎥
5⎣ 6 ⎦ 2
= –
3 ⎡5⎤ 5
+
5 ⎢⎣ 6 ⎥⎦ 2
(by commutativity)
(by distributivity)
3 5 5
1 5
= – +
× +
5 6 2
2 2
–1+5 4
=
= =2
2
2
= –
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2 ⎛ 3⎞ 1 3
1
2
× ⎜– ⎟ – × +
×
5 ⎝ 7 ⎠ 6 2 14 5
(ii)
=
2 ⎛ 3⎞ 1
1
2
× ⎜– ⎟ – +
×
5 ⎝ 7 ⎠ 4 14 5
=
2 ⎛ 3⎞ 2
1
1
× ⎜– ⎟ + ×
–
(by commutativity)
5 ⎝ 7 ⎠ 5 14 4
=
2⎛ 3
1 ⎞ 1
–
+
⎜
⎟–
5 ⎝ 7 14 ⎠ 4
=
2 ⎛ – 6 + 1⎞ 1
⎜
⎟–
5 ⎝ 14 ⎠ 4
=
2⎛ – 5⎞ 1
⎜
⎟–
5 ⎝ 14 ⎠ 4
(by distributivity)
–2 1
–
14
4
1 1
= – –
7 4
–4–7
=
28
11
=
28
=
Q.2. Write the additive inverse of each of the following.
2
–5
–6
2
19
(i)
(ii)
(iii)
(iv)
(v)
8
9
–5
–9
–6
2
2
is the additive inverse of
8
8
2 2 –2 + 2 0
because – + =
= =0
8
8
8 8
Ans. (i) –
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5
–5
is the additive inverse of
because
9
9
–5 5
–5 + 5
0
=
=
= 0
+
9
9
9
9
–6
6
(iii)
=
–5
5
6
6
– is the additive inverse of because
5
5
6 6
–6+6
0
=
=
= 0
– +
5 5
5
5
2
2
(iv)
is the additive inverse of
because
–9
9
2 2
–2 + 2
=
= 0
– +
9 9
9
19
19
is the additive inverse of
because
(v)
–6
6
19 19
–19 + 19
=
= 0
–
+
6
6
6
(ii)
Q.3. Verify that – (– x) = x for:
11
(i) x =
(ii)
15
Ans. (i) We have, x =
x= –
11
15
The additive inverse of x =
Since
13
17
11
–11
is – x =
15
15
11 ⎛ –11 ⎞
+⎜
⎟ = 0.
15 ⎝ 15 ⎠
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The same equality
11 ⎛ –11 ⎞
+⎜
⎟ = 0,
15 ⎝ 15 ⎠
shows that the additive inverse of
or
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–11 11
is
15
15
⎛ –11 ⎞ 11
, i.e., – (– x) = x.
–⎜
⎟=
15
15
⎝
⎠
(ii) We have,
x =
–13
17
The additive inverse of x =
–13
13
is – x =
17
17
–13 13
=0
+
17
17
–13 13
= 0,
The same equality
+
17
17
Since
shows that the additive inverse of
or
13
–13
is
,
17
17
⎛ 13 ⎞ –13
, i.e. – (– x) = x
–⎜ ⎟=
17
⎝ 17 ⎠
Q.4. Find the multiplicative inverse of the following.
–13
1
–5 –3
(i) – 13
(ii)
(iii)
(iv)
×
8
7
17
5
–2
(v) – 1 ×
(vi) – 1
5
–1
is the multiplicative inverse of –13
13
–17
–13
is the multiplicative inverse of
(ii)
13
17
Ans. (i)
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(iii)
(iv)
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5
1
is the multiplicative inverse of
1
5
–5
–3
15
=
×
8
7
56
56
15
is the multiplicative inverse of
15
56
(v) –1 ×
–2
2
=
5
5
5
2
is the multiplicative inverse of
2
5
(vi) – 1 is the multiplicative inverse of – 1
Q.5. Name the property under multiplication used in each of
the following.
–4
–4
4
13 –2 –2 –13
(i)
(ii) – ×
×1 = 1×
=–
=
×
5
5
17 7
7
17
5
–19 29
(iii)
×
=1
29 –19
Ans. (i)
–4
–4
–4
=
×1 = 1×
5
5
5
1 is the multiplicative identity.
So, the property used is property of multiplication.
(ii)
(iii)
–13 –2
–2
–13
=
,
i.e. a × b = b × a
×
×
17
7
7
17
Here, the property used is commutative property
–19
29
= 1
×
29
–19
Here, property used is multiplicative inverse property.
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Q.6. Multiply
Ans.
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6
–7
by the reciprocal of
.
13
16
16
–7
is the reciprocal of
, because the
–7
16
16
–7
product of
and
is 1.
–7
16
6
16
–96
So, the required product =
=
×
13 –7
91
Q.7. Tell what property allows you to compute
.
1 ⎛
4⎞
×⎜6× ⎟
3 ⎝
3⎠
⎛1
⎞ 4
as ⎜ × 6 ⎟ ×
⎝3
⎠ 3
Ans. By associative property of multiplication
We have, a × (b × c) = (a × b) × c.
So by associative property of multiplication. We can
1 ⎛
4⎞ ⎛1
⎞ 4
complete × ⎜ 6 × ⎟ = ⎜ × 6 ⎟ ×
3 ⎝
3⎠ ⎝3
⎠ 3
Where
1
,
3
4
c =
3
a =
b = 6,
8
1
the multiplicative inverse of –1 ? Why or
9
8
why not?
Q.8. Is
Ans. No, because the product of
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6
8
1 8
–9
and –1 = ×
= –1 ≠ 1
8
9
8 9
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Q.9. Is 0.3 the multiplicative inverse of 3 ? Why or
3
why not?
1
Ans. Yes, 0.3 is the multiplicative inverse of 3 , because
3
1
3
10
=1
0.3 × 3 =
×
3 10
3
Q.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.
Ans. (i) Zero is a rational number which has no reciprocal.
(ii) 1 and (–1) are the numbers which are equal to their
reciprocals.
(iii) Zero is the rational number which is equal to its
negative.
Q.11. Fill in the blanks.
(i) Zero has __________ reciprocal.
(ii) The numbers ___________ and _________ are their
own reciprocals.
(iii) The reciprocal of –5 is ___________.
1
(iv) Reciprocal of , where x ≠ 0 is _________.
x
(v) The product of two rational numbers is always a
_________.
(vi) The reciprocal of a positive rational number is
__________.
Ans. (i) Zero has no reciprocal.
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(ii) The numbers 1 and –1 are their own reciprocals.
(iii) The reciprocal of –5 is
–1
.
5
1
, where x ≠ 0 is x .
x
(v) The product of two rational numbers is always a
rational number.
(iv) Reciprocal of
(vi) The reciprocal of a positive rational number is positive.
Exercise 1.2
Q.1. Represent these numbers on the number line.
7
–5
(ii)
(i)
4
6
Ans. (i)
(ii)
7
means seven of four equal parts on the right of 0 on
4
the number line.
–5
means five of six equal parts on the left of 0 on
6
the number line.
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Q.2.Represent
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–2 –5 –9
on the number line.
, ,
11 11 11
–2 –5 –9
on the number
, ,
11 11 11
line are P, Q and R respectively.
Ans. Location of rational numbers
–2
means, two of eleven equal parts on the left of 0.
11
–5
means, five of eleven equal parts on the left of 0.
11
–9
means, nine of eleven equal parts on the left of 0.
11
Q.3. Write five rational numbers which are smaller than 2.
1
–1
, 0, –1,
.
2
2
(Any number on the left of 2)
Ans. Five rational numbers less than 2 are 1,
Because, number on the left side is always less than the
number on the right side.
Q.4. Find ten rational numbers between
Ans. First convert
–2
1
and .
5
2
–2
1
and to rational numbers with the same
5
2
denominator.
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–2 × 4
–8
1 × 10 10
=
and
=
5× 4
20
2 × 10 20
–7 –6 –5 –4
1
9
,
,
,
---,-,
20 20 20 20
20
20
–8
10
as the rational numbers between
.
and
20
20
We can take any ten of these.
Thus, we have
Q.5. Find five rational numbers between.
2
4
–3
5
1
1
(i) and
(ii)
(iii)
and
and
3
5
2
3
4
2
2
4
and to rational numbers with the
5
3
same denominator.
Ans. (i) First convert
2 2 × 20 40
=
=
3 3 × 20 60
4 4 × 12
48
=
=
and
5 5 × 12
60
Hence, the five rational numbers between
2
4
and are: any
3
5
five numbers between
41 42 43 44 45 46 47
.
,
,
,
,
,
60 60 60 60 60 60 60
–3
5
(ii) First convert
and to rational numbers with the
2
3
same denominator.
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–3 –3 × 3 –9
=
=
2
2×3
6
5 5 × 2 10
=
=
and
3 3×2
6
–3
5
and are:
2
3
–8 –7 –6 –5 –4
–9
10
. (Any five from
)
,
,
,
,
and
6 6 6 6 6
6
6
1
1
(iii) First convert and to rational numbers with the
4
2
same denominator.
Hence, the five rational numbers between
1 1×8
8
=
=
4 4 × 8 32
1 1 × 16 16
=
=
and
2 2 × 16 32
1
1
and are :
4
2
9 10 11 12 13
8
16
. (Any five from
)
,
,
,
,
and
32 32 32 32 32
32
32
Hence, the five rational numbers between
Q.6. Write five rational numbers greater than –2.
Ans. On a number line right side number is always greater than
number on the left side.
∴ So five rational numbers greater than –2 are
–1
1
3
, 0, , 1 and . (Any five numbers on the right side
2
2
2
of –2)
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Q.7. Find ten rational numbers between
Ans. First convert
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3
3
and .
5
4
3
3
and to rational numbers with the same
5
4
denominator
L.C.M of 5 and 4 = 20.
Convert the denominator with multiples of 20 i.e. 20 × 4 = 80
3 × 16 48
3 × 20 60
=
and
=
5 × 16 80
4 × 20 80
3
3
and are any
5
4
49 50 51 52 53 54
59
ten from
,
,
,
,
,
..........
.
80 80 80 80 80 80
80
Hence, the ten rational numbers between
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