## Chapter 9 Rational Numbers Exercise 9.1

Question 1: List five rational numbers between:
(i) -1 and 0

= -1/1 and 0/1
= -1/1 x 10/10 = -10/10
= 0/1 x 10/10 = 0/10
= -9/10, -8/10, -7/10, -6/10, -5/10

(ii) -2 and -1

= -2/1 and -1/1
= -2/1 x 10/10 = -20/10
= -1/1 x 10/10 = -10/10
= -19/10, -18/10, -17/10, -16/10, -15/10

(iii) -4/5 and -2/3

= -4/5 and -2/3
= -4/5 x 10/10 = -40/50
= 0/1 x 10/10 = 0/10
= -9/10, -8/10, -7/10, -6/10, -5/10

(iv) -1/2 and 2/3

LCM of 2, 3 = 6
= -1/2 x 3/3 = -3/6
= 2/3 x 2/2 = 4/6
= -3/6 x 10/10 = -30/60
= 4/6 x 10/10 = 40/60
= -29/60, -28/60, -27/60, -26/60, -25/60

Question 2: Write four more rational numbers in each of the following patterns:
(i) -3/5, -6/10, -9/15, -12/20, …..

The numerator and denominator are multiples of 3, 5.
= -3 × 1/5 × 1 = -3/5
= -3 × 2/5 × 2 = -6/10
= -3 × 3/5 × 3 = -9/15
= -3 × 4/5 × 4 = -12/20
Next four rational numbers:
= -3 × 5/5 × 5 = -15/25
= -3 × 6/5 × 6 = -18/30
= -3 × 7/5 × 7 = -21/35
= -3 × 8/5 × 8 = -24/40

(ii) -1/4, -2/8, -3/12, …..

The numerator and denominator are multiples of 1, 4.
= -1 × 1/4 × 1 = -1/4
= -1 × 2/4 × 2 = -2/8
= -1 × 3/1 × 3 = -3/12
Next four rational numbers:
= -1 × 4/4 × 4 = -4/16
= -1 × 5/4 × 5 = -5/20
= -1 × 6/4 × 6 = -6/24
= -1 × 7/4 × 7 = -7/28

(iii) -1/6, 2/-12, 3/-18, 4/-24 …..

The numerator and denominator are multiples of 1, 6.
= -1 × 1/6 × 1 = -1/6
= 1 × 2/-6 × 2 = 2/-12
= 1 × 3/-6 × 3 = 3/-18
= 1 × 4/-6 × 4 = 4/-24
Next four rational numbers:
= 1 × 5/-6 × 5 = 5/-30
= 1 × 6/-6 × 6 = 6/-36
= 1 × 7/-6 × 7 = 7/-42
= 1 × 8/-6 × 8 = 8/-48

(iv) -2/3, 2/-3, 4/-6, 6/-9 …..

The numerator and denominator are multiples of 2, 3.
= -2 × 1/3 × 1 = -2/3
= 2 × 1/-3 × 1 = 2/-3
= 2 × 2/-3 × 2 = 4/-6
= 2 × 3/-3 × 3 = 6/-9
Next four rational numbers:
= 2 × 4/-3 × 4 = 8/-12
= 2 × 5/-3 × 5 = 10/-15
= 2 × 6/-3 × 6 = 12/-18
= 2 × 7/-3 × 7 = 14/-21

Question 3: Give four rational numbers equivalent to:
(i) -2/7

Four rational numbers equivalent to -2/7:
= -2 × 2/7 × 2 = -4/14
= -2 × 3/7 × 3 = -6/21
= -2 × 4/7 × 4 = -8/28
= -2 × 5/7× 5 = -10/35

(ii) 5/-3

Four rational numbers equivalent to 5/-3:
= 5 × 2/-3 × 2 = 10/-6
= 5 × 3/-3 × 3 = 15/-9
= 5 × 4/-3 × 4 = 20/-12
= 5 × 5/-3× 5 = 25/-15

(iii) 4/9

Four rational numbers equivalent to 4/9:
= 4 × 2/9 × 2 = 8/18
= 4 × 3/9 × 3 = 12/27
= 4 × 4/9 × 4 = 16/36
= 4 × 5/9× 5 = 20/45

Question 4: Draw the number line and represent the following rational numbers on it:
(i) 3/4

(ii) -5/8

(iii) -7/4

(iv) 7/8

Question 5: The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.

The distance between A and B = 1 unit
= AP = PQ = QB = 1/3
= P = 2 + (1/3)
= (6 + 1)/ 3
= 7/3
Q = 2 + (2/3)
= (6 + 2)/ 3
= 8/3
The distance between U and T = 1 unit
= TR = RS = SU = 1/3
R = -1 - (1/3)
= (- 3 - 1)/ 3
= -4/3
S = -1 - (2/3)
= – 3 - 2)/ 3
= -5/3

Question 6: Which of the following pairs represent the same rational number?
(i) (-7/21) and (3/9)

= -7/21 = 3/9
= -1/3 = 1/3
Since, -1/3 ≠ 1/3
Therefore, -7/21 ≠ 3/9
So, the given pair doesn't represents the same rational number.

(ii) (-16/20) and (20/-25)

= -16/20 = 20/-25
= -4/5 = 4/-5
Since, -4/5 = -4/5
Therefore, -16/20 = 20/-25
So, the given pair is represents the same rational number.

(iii) (-2/-3) and (2/3)

= -2/-3 = 2/3
= 2/3= 2/3
Since, 2/3 = 2/3
Therefore, -2/-3 = 2/3
So, the given pair is represents the same rational number.

(iv) (-3/5) and (-12/20)

= -3/5 = – 12/20
= -3/5 = -3/5
Since, -3/5 = -3/5
Therefore, -3/5= -12/20
So, the given pair is represents the same rational number.

(v) (8/-5) and (-24/15)

= 8/-5 = -24/15
= 8/-5 = -8/5
Since, -8/5 = -8/5
Therefore, 8/-5 = -24/15
So, the given pair is represents the same rational number.

(vi) (1/3) and (-1/9)

= 1/3 = -1/9
Since, 1/3 ≠ -1/9
Therefore, 1/3 ≠ -1/9
So, the given pair doesn't represents the same rational number.

(vii) (-5/-9) and (5/-9)
= -5/-9 = 5/-9
Since, 5/9 ≠ -5/9
Therefore, -5/-9 ≠ 5/-9
So, the given pair doesn't represents the same rational number.

Question 7: Rewrite the following rational numbers in the simplest form:
(i) -8/6
-4/3

(ii) 25/45
5/9

(iii) -44/72
-11/18

(iv) -8/10
-4/5

Question 8: Fill in the boxes with the correct symbol out of >, <, and =.
(i) -5/7 [ ] 2/3

LCM of 3, 7 = 21
= -5/7 = -5 × 3/7 × 3 = -15/21
= 2/3 = 2 × 7/3 × 7 = 14/21
-15 < 14
So, -15/21 < 14/21
Therefore, -5/7 [<] 2/3

(ii) -4/5 [ ] -5/7

LCM of 5, 7 = 35
= -4/5 = -4 × 7/5 × 7 = -28/35
= -5/7 = -5 × 5/7 × 5 = -25/35
-28 < -25
So, -28/35 < - 25/35
Therefore, -4/5 [<] -5/7

(iii) -7/8 [ ] 14/-16

= 7/-8
So, -7/8 = -7/8
Therefore, -7/8 [=] 14/-16

(iv) -8/5 [ ] -7/4

LCM of 4, 5 = 20
= -8/5 = -8 × 4/5 × 4 = -32/20
= -7/4 = -7 × 5/4 × 5 = -35/20
-32 > – 35
So, -32/20 > -35/20
Therefore, -8/5 [>] -7/4

(v) 1/-3 [ ] -1/4

LCM of 3, 4 = 12
= -1/3 = -1 × 4/3 × 4 = -4/12
= -1/4 = -1 × 3/4 × 3 = -3/12
-4 < -3
So, -4/12 < -3/12
Therefore, 1/-3 [<] -1/4

(vi) 5/-11 [ ] -5/11

= -5/11 = -5/11
Therefore, 5/-11 [=] -5/11

(vii) 0 [ ] -7/6

= 0 [>] -7/6

Question 9: Which is greater in each of the following:
(i) 2/3, 5/2

LCM of 2, 3 = 6
= 2/3 = 2 × 2/3 × 2 = 4/6
= 5/2 = 5 × 3/2 × 3 = 15/6
4 < 15
So, 4/6 < 15/6
= 2/3 < 5/2
Therefore, 5/2 is greater.

(ii) -5/6, -4/3

LCM of 3, 6 = 6
= -5/6 = -5 × 1/6 × 1 = -5/6
= -4/3 = -4 × 2/3 × 2 = -12/6
-5 > -12
So, -5/6 > -12/6
So, -5/6 > -12/6
Therefore, -5/6 is greater.

(iii) -3/4, 2/-3

LCM of 3, 4 = 12
= -3/4 = -3 × 3/4 × 3 = -9/12
= -2/3 = -2 × 4/3 × 4 = -8/12
-9 < -8
So, (-9/12) < (- 8/12)
So, -3/4 < 2/-3
Therefore, 2/-3 is greater.

(iv) -1/4, 1/4

So, -1/4 < 1/4
Therefore, 1/4 is greater.

(v) -3 2/7, -3 4/5

-3 2/7 = -23/7
-3 4/5 = -19/5
LCM of 5, 7 = 35
= -23/7 = -23 × 5/7 × 5 = -115/35
= -19/5 = -19 × 7/5 × 7 = (133/35
-115 > -133
So, -115/35 > - 133/35
= -3 2/7 > -3 4/5
Therefore, -3 2/7 is greater.

Question 10: Write the following rational numbers in ascending order:
(i) -3/5, -2/5, -1/5

= -3/5 < -2/5 < -1/5

(ii) -1/3, -2/9, -4/3

LCM of 3, 3, 9 = 9
= -1/3 = -1 × 3/3 × 9 = -3/9
= -2/9 = -2 × 1/9 × 1 = -2/9
= -4/3 = -4 × 3/3 × 3 = -12/9
= -12/9 < -3/9 < -2/9
= -4/3 < -1/3 < -2/9

(iii) -3/7, -3/2, -3/4