Exercise 1.1
Fill in the blanks:
(i) −19/5 lies between the integers __________ and __________. [Answer: −4 and −3]
(ii) The
decimal form of the rational number 15/-4 is __________.
(iii) The
rational numbers -8/3 and 8/3 are equidistant from
__________.
(iv) The
next rational number in the sequence −15/24, 20/−32
, −25/40
is __________.
(v) The standard
form of 58/−78 is __________.
2. Say True or False:
(i) 0 is
the smallest rational number.
(ii) −4/5 lies to the left of −3/4.
(iii) −19/5 is greater than -15/4
.
(iv) The
average of two rational numbers lies between them.
(v) There
are an unlimited number of rational numbers between 10 and 11.
3. Find the rational numbers represented
by each of the question marks marked on the following number lines.
Solution:
(i) The number lies between −3 and −4. The unit part between −3
and −4 is divided into 3 equal parts and the second part is asked.
∴ The required number is −3 (2/3) = − 11/3.
(ii) The required number lies between 0 and −1. The unit part
between 0 and −1 is divided into 5 equal parts, and the second part is taken.
∴ The required number is −2/5
(iii) The required number lies between 1 and 2. The unit part
between 1 and 2 is divided into 4 equal parts and the third part is taken.
∴ The required number is 1 (3/4) = 7/4
4. The points S, Y, N, C, R, A, T, I
and O on the number line are such that CN=NY=YS and RA=AT=TI=IO. Find the rational
numbers represented by the letters Y, N, A, T and I.
Solution:
Y = −2 + (1/3) = [−6 + 1] / 3 = −5/3
N = [−5/3] + [1/3] = [−5 + 1] / 3 = −4/3
RA = AT = TI = IO = 1/4
A = 2 + [1/4] = [8 + 1] / 4 = 9/4
T = 9/4 + 1/4 = [9 + 1] / 4 = 10/4
I = 10/4 + 1/4 = [10 + 1] / 4 = 11/4
5. Draw a number line and represent the
following rational numbers on it.
Solution:
(i) 9/4 = 2 (1/4)
∴ 9/4 lies between 2 and 3.
(ii) −8/3 = −2 (2/3)
−2 (2/3) lies between −2 and −3.
(iii) −17/−5 = 3 (2/5)
3 (2/5) lies between 3 and 4 in the number line.
(iv) 15/−4 = −3 (3/4)
−3 (3/4) lies between −3 and −4.
6. Write the decimal form of the following
rational numbers.
Solution:
(i) 1/11 = 0.0909…
(ii) 13/4 = 3.25
(iii) −18/7 = −2.571428571428…
(iv) 1 (2/5) = 7/5 = 1.4
(v) −3 (1/2) = −7/2 = −3.5
7. List any five rational numbers between
the given rational numbers.
Solution:
(i) −2 and 0
i.e. −2/1 and 0/1
−2/1 = [−2 × 10] / [1 × 10] = −20/10
0/1 = [0 × 10] / [1 × 10] = 0 / 10
∴ Five rational numbers
between −20/10 (= −2) and 0/10 (= 0) are
−20/10, −19/10, −18/10, −7/10, −6/10, −5/10, 0/10 (=0)
(ii) −1/2 and 3/5
LCM of 2 and 5 = 2 × 5 = 10
−1/2 = [−1 × 5] / [2 × 5]
= −5 / 10
3/5 = [3 × 2] / [5 × 2] = 6/10
∴ Five rational numbers between −1/2 (= −5/10) and 3/5 (= 6/10)
are −3/10, −1/10, 0, 1/10, 2/10, 5/10
(iii) 1/4 and 7/20
1/4 = [1 × 15] / [4 × 15] = 15/60
7/20 = [7 × 3] / [20 × 3] = 21/60
∴ Five rational numbers between 1/4 (=15/60) and 7/20 (=21/60) are 16/60, 17/60, 18/60,
19/60, 20/60
(iv) −6 / 4 and −23
/ 10
−6 / 4 = [−6 × 5] / [4 × 5]
= −30 / 20
−23 / 10 = [23 × 2] / [10 × 2] = −46 / 20
∴ Five rational numbers between −6/4 ( = −30/20) and −23 / 10 (= −46/20)
are −31/20, −32/20, −33/20, −34/20, −35/20.
8. Use the method of averages to write
2 rational numbers between 14/5 and 16/3.
Solution:
The average of a and b is 1/2 (a + b)
The average of 14/5 and 16/3 is C1 = 1/2 (14/5 +
16/3)
C1 = 1/2 ([42 + 80] / 15)
C1 = 122/30
C1 = 61/15
14/15 < 61/15 < 16/3 ……….(1)
The average of 14/5 and 61/15 is C2 = 1/2 ( 14/5 +
61/15 )
C2 = 1/2 × ( [42 + 61] / 15)
C2 = 1/2 × 103/15 = 103/30
14/5 < 103/30 < 61/15
…………(2)
From (1), (2) we get, 14/5 < 103/30 < 61/15 < 16/3
9. Compare the following pairs of rational
numbers.
Solution:
(i) −11/5, −21/8
LCM of 5, 8 is 40
−11/5 = [−11 × 8] / [5 × 8] = −88/40
−21/8 = [−21 × 5] / [8 × 5] = −105/40
−105/40 < −88/40
∴ −21/8 < −11/5
(ii) 3/−4, −1/2
LCM of 4 and 2 = 4
3/−4 = −3/4
−1/2 = [−1 × 2] / [2 × 2] = −2/4
3/−4 < −2/4
−3/4 < −1/2
(iii) 2/3, 4/5
LCM of 3 and 5 is 15.
2/3 = [2 × 5] / [3 × 5] = 10 / 15
4/5 = [4 × 3] / [5 × 3] = 12 / 15
10/15 < 12/15
∴ 2/3 < 4/5
10. Arrange the following rational numbers
in ascending and descending order.
(i) −5/−12, −11/8, −15/24,
−7/−9, 12/36
LCM of 12, 8, 24, 9, 36 is 4 × 3 × 2 × 3 = 72
−5/−12 = [−5 × 6] / [12 × 6] = −30/72
−11/8 = [−11 × 9] / [8 × 9] = −99/72
−15/24 = [−15 × 3] / [24 × 3] = −45/72
−7/−9 = [7 × 8] / [9 × 8] = 56/72
12/36 = [12 × 2] / [36 × 2] = 24/72
Now comparing the numerators −30, −99, −45, 56, 24 we get 56
> 24 > −30 > −45 > −99
i.e 56/72 > 24/72 > −30/72 > −45/72 > −99/72 and
so −7/−9 > 12/36 > −5/12 > −15/24
> −11/8
∴ Descending order −7/−9 > 12/36 > −5/12 > −15/24 > −11/8
Ascending order −11/8 <
−15/24 < −5/12 < 12/36 < −7/−9
(ii) −17/10, −7/5, 0, −2/4, −19/20
LCM of 10, 5, 4, 20 is 5 × 2 × 2 = 20
−17/10 = [−17 × 2] / [10 × 2] = −34/20
−7/5 = [−7 × 4] / [5 × 4] = −28/20
−2/4 = [−2 × 5] / [4 × 5] = −10/20
−19/20 = −19/20
Negative numbers are less then zero.
∴ Arranging the numerators we get −34 < − 28 < −19 < −10
< 0
∴ −34/20 < −28/20 < −19/20 < −10/20 < 0
Ascending order = −17/10 < −7/5 < −19/20 < −2/4 < 0
Descending order 0 > −2/4 > −19/20 > −7/5 > −17/10
Objective
Type Questions
11. The number which is subtracted from
−6/11 to get 8/9 is __________.
[Ans. (B) −142/99 ]
Solution:
Let x be the number to
be subtracted
(−6/11) – x = 8/9
(−6 /11) – 8/9 = x
x = [(−6 × 9) + (−8
× 11)] / [11 × 9] = [−54+ (−88)] / [99]
= −142 / 99
12. Which of the following pairs is equivalent?
[Answer: (B) 16/−30, −8/15]
Solution:
−20/12 = [−20 ÷ 4] / [12 ÷
4] = −5/3 ≠ 5/3
16/−30 = [−16 ÷ 2] / [30
÷ 2] = −8/15
−18/36 = [−18 ÷ 9] / [36
÷ 9] = −2/4 = [−2 × 11] / [4 × 11] = −22/44
≠ −20/44
∴ 16/−30 and −8/15 are equivalent fraction.
13. -5/4 is a rational number which lies
between __________ .
[Answer: (C) −l and −2]
Solution:
−5 /4 = − 1 (1/4)
∴ −5/4 lies between −1 and −2.
14. Which of the following rational numbers
is the greatest?
[Answer: (A) −17/24]
Solution:
LCM of 24, 16, 8, 32 = 8 × 2 × 3 × 2 = 96
−17/24 = [−17 × 4] / [24
× 4] = −68/96
−13/16 = [−13 × 6] / [16 × 6] = −78/96
7/−8 = [−7 × 12] / [8 × 12] = −84/96
−31/32 = [−31 × 3] / [32 × 3] = −93/96
−93/96 < −84/96 < −78/96 < −68/96
−31/32 < 7/−8 < −13/16 < −17/24
∴ −17/24 is the greatest number.
15.The sum of the digits of the denominator
in the simplest form of 112 / 528 is _________ .
(A) 4
(B) 5
(C) 6
(D) 7
[Answer: (C) 6]
Solution:
112/528 = [112 ÷ 8] / [528 ÷ 8] = 14/66 = [14 ÷ 2] / [66 ÷ 2] =
7/33
Sum of digits in the denominator = 3 + 3 = 6
Answer:
Exercise 1.1
1. (i) –4
and –3 (ii) –3.75 (iii ) 0 (iv) 30/−48 ( v) −29/39
2.(i) False
(ii) True (iii) False (iv) True (v) True
3. (i) –
11/3 (ii) −2/5 (iii) 7/4
4. Y = -5/3
, N=-4/3 , A =9/4 , T =10/4 , I =11/4
5. (i)
(ii)
(iii)
(iv)
6. (i) 0.0909...
(ii) 3.25 (iii) –2.5714285714 (iv) 1.4
(v) –3.5
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