Rs Aggarwal Solutions: Exercise 1A - Rational Numbers | Mathematics (Maths) Class 8 PDF Download

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Q.1. Express -3/5 as a rational number with denominator
(a) 20
(b) -30
(c) 35
(d) -40
Solution: If a/b is a fraction and m is a non-zero integer, then a/b =
Now,



Q.2. Express -42/98 as a rational number with denominator 7.
Solution: If a/b is a rational number and m is a common divisor of a and b, then
∴

Q.3. Express -48/60 as a rational number with denominator 5.
Solution: If a/b is a rational integer and m is a common divisor of a and b, then
∴

Q.4. Express each of the following rational numbers in standard form:
(a) -12/30
(b) -14/49
(c) 24/-64
(d)  -36/-63
Solution: A rational number a/b is said to be in the standard form if a and b have no common divisor other than unity and b > 0.
Thus,
(a) The greatest common divisor of 12 and 30 is 6.
∴  In the standard form
(b) The greatest common divisor of 14 and 49 is 7.
∴  In the standard form
(c)
The greatest common divisor of 24 and 64 is 8.
∴  In the standard form
(d)
The greatest common divisor of 36 and 63 is 9.
∴  In the standard form

Q.5. Which of the two rational numbers is greater in the given pair?
(a) 3/8 or 0
(b) -2/9 or 0
(c) -3/4 or 1/4
(d) -5/7 or -4/7
(e) 2/3 or 3/4
(f) -1/2 or -1
Solution: We know:
(i) Every positive rational number is greater than 0.
(ii) Every negative rational number is less than 0.
Thus, we have:
(a) 3/8 is a positive rational number.
∴ 3/8 > 0
(b) -2/9 is a negative rational number.
∴ -2/9 < 0
(c) -3/4 is a negative rational number.
∴ -3/4 < 0
Also,
1/4 is a positive rational number.
∴ 1/4 > 0
Combining the two inequalities, we get:

(d) Both -5/7 and -4/7 have the same denominator, that is, 7.
So, we can directly compare the numerators.
∴
(e) The two rational numbers are 2/3 and 3/4.
The LCM of the denominators 3 and 4 is 12.
Now,

Also,

Further

∴
(f) The two rational numbers are -1/2 and -1.
We can write -1 = -1/1.
The LCM of the denominators 2 and 1 is 2.
Now,

Also,


Q.6. Which of the two rational numbers is greater in the given pair?
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
Solution: 
(a) The two rational numbers are -4/3 and -8/7.
The LCM of the denominators 3 and 7 is 21.
Now,

Also,

Further,

∴
(b) The two rational numbers are 7/-9 and -5/8.
The first fraction can be expressed as
The LCM of the denominators 9 and 8 is 72.
Now,

Also,

Further,

∴
(c) The two rational numbers are -1/3 and 4/-5.

The LCM of the denominators 3 and 5 is 15.
Now,

Also,

Further,

∴
(d) The two rational numbers are 9/-13 and 7/-12.
Now,
The LCM of the denominators 13 and 12 is 156.
Now,

Also,

Further,

∴
(e) The two rational numbers are 4/-5 and -7/10.
∴
The LCM of the denominators 5 and 10 is 10.
Now,

Also,

Further,

∴
(f) The two rational numbers are -12/5 and -3.
-3 can be written as -3/1.
The LCM of the denominators is 5.
Now,

Because  we can conclude that

Q.7. Fill in the blanks with the correct symbol out of >, = and <:
(a)
(b)
(c)
(d)
(e)
(f)
Solution: (a) We will write each of the given numbers with positive denominators.
One number = -3/7
Other number =
LCM of 7 and 13 = 91
∴
And,

Clearly,
-39 > - 41
∴
Thus,

(ii) We will write each of the given numbers with positive denominators.
One number
Other number = -35/91
LCM of 13 and 91 = 91
∴
Clearly,
-35 = - 35
∴
Thus,

(c) We will write each of the given numbers with positive denominators.
One number = -2
We can write -2 as -2/1.
Other number = -13/5
LCM of 1 and 5 = 5
∴
Clearly,
-10 > - 13
∴
Thus,


(d) We will write each of the given numbers with positive denominators.
One number = -2/3
Other number =
LCM of 3 and 8 = 24
∴
Clearly,
-16 < - 15
∴
Thus,


(e)
3/5 is a positive number.
Because every positive rational number is greater than 0,
(f) We will write each of the given numbers with positive denominators.
One number = -8/9
Other number = -9/10
LCM of 9 and 10 = 90
∴
Clearly,
-81 < - 80
∴
Thus,


Q.8. Arrange the following rational numbers in ascending order:
(a) 
(b)
(c)
(d) 
Solution: 
(a) We will write each of the given numbers with positive denominators.
We have:

Thus, the given numbers are
LCM of 9, 12, 18 and 3 is 36.
Now,




Clearly,

∴
That is

(b) We will write each of the given numbers with positive denominators.
We have:

Thus, the given numbers are
LCM of  4, 12, 16 and 24 is 48.
Now,




Clearly,

∴
That is

(c) We will write each of the given numbers with positive denominators.
We have:

Thus, the given numbers are
LCM of 5, 10, 15 and 20 is 60.
Now,




Clearly,

∴
That is

(d) We will write each of the given numbers with positive denominators.
We have:

Thus, the given numbers are
LCM of 7, 14, 28 and 42 is 84.
Now,




Clearly,

∴
That is


Q.9. Arrange the following rational numbers in descending order:
(a)
(b)
(c)
(d)
Solution:
(a) We will first write each of the given numbers with positive denominators. We have:

Thus, the given numbers are
LCM of 1, 6, 3 and 3 is 6
Now,

and

Clearly, Thus,

∴
(b) We will first write each of the given numbers with positive denominators. We have:

Thus, the given numbers are
LCM of 10, 15, 20 and 30 is 60
Now,



and

Clearly,

∴
(c) We will first write each of the given numbers with positive denominators. We have:

Thus, the given numbers are
LCM of 6, 12, 18 and 24 is 72
Now,



and

Clearly,

∴
(d) The given numbers are
LCM of 11, 22, 33 and 44 is 132
Now,



and

Clearly,

∴

Q.10. Which of the following statements are true and which are false?
(a) Every whole number is a rational number.
(b) Every integer is a rational number.
(c) 0 is a whole number but it is not a rational number.
Solution:
(a) True
A whole number can be expressed as a/b, with b = 1 and a ≥ 0. Thus, every whole number is rational.
(b) True
Every integer is a rational number because any integer can be expressed as a/b, with b = 1 and 0 > a ≥ 0. Thus, every integer is a rational number.
(c) 0 = a/b, for a = 0 and b ≠ 0. Thus, 0 is a rational and whole number.