Rational Numbers (Exercise 4.5) RD Sharma Solutions | Mathematics (Maths) Class 7 PDF Download

 Page 1


 
Exercise 4.5 page no: 4.20 
1. Which of the following rational numbers are equal?
(i) (-9/12) and (8/-12)
(ii) (-16/20) and (20/-25)
(iii) (-7/21) and (3/-9)
(iv) (-8/-14) and (13/21)
Solution: 
(i) Given (-9/12) and (8/-12)
The standard form of (-9/12) is (-3/4) [on diving the numerator and denominator of
given number by their HCF i.e. by 3]
The standard form of (8/-12) = (-2/3) [on diving the numerator and denominator of
given number by their HCF i.e. by 4]
Since, the standard forms of two rational numbers are not same. Hence, they are not
equal.
(ii) Given (-16/20) and (20/-25)
Multiplying numerator and denominator of (-16/20) by the denominator of (20/-25)
i.e. -25.
(-16/20) × (-25/-25) = (400/-500)
Now multiply the numerator and denominator of (20/-25) by the denominator of
(-16/20) i.e. 20
(20/-25) × (20/20) = (400/-500)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iii) Given (-7/21) and (3/-9)
Multiplying numerator and denominator of (-7/21) by the denominator of (3/-9)
i.e. -9.
(-7/21) × (-9/-9) = (63/-189)
Now multiply the numerator and denominator of (3/-9) by the denominator of
(-7/21) i.e. 21
(3/-9) × (21/21) = (63/-189)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
Page 2


 
Exercise 4.5 page no: 4.20 
1. Which of the following rational numbers are equal?
(i) (-9/12) and (8/-12)
(ii) (-16/20) and (20/-25)
(iii) (-7/21) and (3/-9)
(iv) (-8/-14) and (13/21)
Solution: 
(i) Given (-9/12) and (8/-12)
The standard form of (-9/12) is (-3/4) [on diving the numerator and denominator of
given number by their HCF i.e. by 3]
The standard form of (8/-12) = (-2/3) [on diving the numerator and denominator of
given number by their HCF i.e. by 4]
Since, the standard forms of two rational numbers are not same. Hence, they are not
equal.
(ii) Given (-16/20) and (20/-25)
Multiplying numerator and denominator of (-16/20) by the denominator of (20/-25)
i.e. -25.
(-16/20) × (-25/-25) = (400/-500)
Now multiply the numerator and denominator of (20/-25) by the denominator of
(-16/20) i.e. 20
(20/-25) × (20/20) = (400/-500)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iii) Given (-7/21) and (3/-9)
Multiplying numerator and denominator of (-7/21) by the denominator of (3/-9)
i.e. -9.
(-7/21) × (-9/-9) = (63/-189)
Now multiply the numerator and denominator of (3/-9) by the denominator of
(-7/21) i.e. 21
(3/-9) × (21/21) = (63/-189)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
 
 
 
 
 
 
(iv) Given (-8/-14) and (13/21) 
Multiplying numerator and denominator of (-8/-14) by the denominator of (13/21) 
i.e. 21 
(-8/-14) × (21/21) = (-168/-294) 
Now multiply the numerator and denominator of (13/21) by the denominator of  
(-8/-14) i.e. -14 
(13/21) × (-14/-14) = (-182/-294) 
Clearly, the numerators of the above obtained rational numbers are not equal.  
Hence, the given rational numbers are also not equal 
 
2. In each of the following pairs represent a pair of equivalent rational numbers, find 
the values of x. 
(i) (2/3) and (5/x) 
(ii) (-3/7) and (x/4) 
(iii) (3/5) and (x/-25) 
(iv) (13/6) and (-65/x) 
 
Solution: 
(i) Given (2/3) and (5/x) 
Also given that they are equivalent rational number so (2/3) = (5/x) 
x = (5 × 3)/2 
x = (15/2) 
 
(ii) Given (-3/7) and (x/4) 
Also given that they are equivalent rational number so (-3/7) = (x/4) 
x = (-3 × 4)/7 
x = (-12/7) 
 
(iii) Given (3/5) and (x/-25) 
Also given that they are equivalent rational number so (3/5) = (x/-25) 
x = (3 × -25)/5 
x = (-75)/5 
x = -15 
 
(iv) Given (13/6) and (-65/x) 
Also given that they are equivalent rational number so (13/6) = (-65/x) 
x = 6/13 x (- 65)  
Page 3


 
Exercise 4.5 page no: 4.20 
1. Which of the following rational numbers are equal?
(i) (-9/12) and (8/-12)
(ii) (-16/20) and (20/-25)
(iii) (-7/21) and (3/-9)
(iv) (-8/-14) and (13/21)
Solution: 
(i) Given (-9/12) and (8/-12)
The standard form of (-9/12) is (-3/4) [on diving the numerator and denominator of
given number by their HCF i.e. by 3]
The standard form of (8/-12) = (-2/3) [on diving the numerator and denominator of
given number by their HCF i.e. by 4]
Since, the standard forms of two rational numbers are not same. Hence, they are not
equal.
(ii) Given (-16/20) and (20/-25)
Multiplying numerator and denominator of (-16/20) by the denominator of (20/-25)
i.e. -25.
(-16/20) × (-25/-25) = (400/-500)
Now multiply the numerator and denominator of (20/-25) by the denominator of
(-16/20) i.e. 20
(20/-25) × (20/20) = (400/-500)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iii) Given (-7/21) and (3/-9)
Multiplying numerator and denominator of (-7/21) by the denominator of (3/-9)
i.e. -9.
(-7/21) × (-9/-9) = (63/-189)
Now multiply the numerator and denominator of (3/-9) by the denominator of
(-7/21) i.e. 21
(3/-9) × (21/21) = (63/-189)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
 
 
 
 
 
 
(iv) Given (-8/-14) and (13/21) 
Multiplying numerator and denominator of (-8/-14) by the denominator of (13/21) 
i.e. 21 
(-8/-14) × (21/21) = (-168/-294) 
Now multiply the numerator and denominator of (13/21) by the denominator of  
(-8/-14) i.e. -14 
(13/21) × (-14/-14) = (-182/-294) 
Clearly, the numerators of the above obtained rational numbers are not equal.  
Hence, the given rational numbers are also not equal 
 
2. In each of the following pairs represent a pair of equivalent rational numbers, find 
the values of x. 
(i) (2/3) and (5/x) 
(ii) (-3/7) and (x/4) 
(iii) (3/5) and (x/-25) 
(iv) (13/6) and (-65/x) 
 
Solution: 
(i) Given (2/3) and (5/x) 
Also given that they are equivalent rational number so (2/3) = (5/x) 
x = (5 × 3)/2 
x = (15/2) 
 
(ii) Given (-3/7) and (x/4) 
Also given that they are equivalent rational number so (-3/7) = (x/4) 
x = (-3 × 4)/7 
x = (-12/7) 
 
(iii) Given (3/5) and (x/-25) 
Also given that they are equivalent rational number so (3/5) = (x/-25) 
x = (3 × -25)/5 
x = (-75)/5 
x = -15 
 
(iv) Given (13/6) and (-65/x) 
Also given that they are equivalent rational number so (13/6) = (-65/x) 
x = 6/13 x (- 65)  
 
 
 
 
 
 
x = 6 x (-5)  
x = -30 
 
3. In each of the following, fill in the blanks so as to make the statement true: 
(i)  A number which can be expressed in the form p/q, where p and q are integers and 
q is not equal to zero, is called a ……….. 
(ii)  If the integers p and q have no common divisor other than 1 and q is positive, then 
the rational number (p/q) is said to be in the …. 
(iii) Two rational numbers are said to be equal, if they have the same …. form 
(iv) If m is a common divisor of a and b, then (a/b) = (a ÷ m)/….. 
(v)  If p and q are positive Integers, then p/q is a ….. rational number and (p/-q) is a 
…… rational number. 
(vi) The standard form of -1 is … 
(vii)  If (p/q) is a rational number, then q cannot be …. 
(viii)  Two rational numbers with different numerators are equal, if their numerators 
are in the same …. as their denominators. 
 
Solution: 
(i) Rational number 
(ii) Standard form 
(iii) Standard 
(iv) b ÷ m 
(v) Positive, negative 
(vi) (-1/1) 
(vii) Zero 
(viii) Ratio 
 
4. In each of the following state if the statement is true (T) or false (F): 
(i) The quotient of two integers is always an integer. 
(ii) Every integer is a rational number. 
(iii) Every rational number is an integer. 
(iv) Every traction is a rational number. 
(v)  Every rational number is a fraction. 
(vi) If a/b is a rational number and m any integer, then (a/b) = (a x m)/ (b x m) 
(vii) Two rational numbers with different numerators cannot be equal. 
(viii) 8 can be written as a rational number with any integer as denominator. 
(ix) 8 can be written as a rational number with any integer as numerator. 
Page 4


 
Exercise 4.5 page no: 4.20 
1. Which of the following rational numbers are equal?
(i) (-9/12) and (8/-12)
(ii) (-16/20) and (20/-25)
(iii) (-7/21) and (3/-9)
(iv) (-8/-14) and (13/21)
Solution: 
(i) Given (-9/12) and (8/-12)
The standard form of (-9/12) is (-3/4) [on diving the numerator and denominator of
given number by their HCF i.e. by 3]
The standard form of (8/-12) = (-2/3) [on diving the numerator and denominator of
given number by their HCF i.e. by 4]
Since, the standard forms of two rational numbers are not same. Hence, they are not
equal.
(ii) Given (-16/20) and (20/-25)
Multiplying numerator and denominator of (-16/20) by the denominator of (20/-25)
i.e. -25.
(-16/20) × (-25/-25) = (400/-500)
Now multiply the numerator and denominator of (20/-25) by the denominator of
(-16/20) i.e. 20
(20/-25) × (20/20) = (400/-500)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iii) Given (-7/21) and (3/-9)
Multiplying numerator and denominator of (-7/21) by the denominator of (3/-9)
i.e. -9.
(-7/21) × (-9/-9) = (63/-189)
Now multiply the numerator and denominator of (3/-9) by the denominator of
(-7/21) i.e. 21
(3/-9) × (21/21) = (63/-189)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
 
 
 
 
 
 
(iv) Given (-8/-14) and (13/21) 
Multiplying numerator and denominator of (-8/-14) by the denominator of (13/21) 
i.e. 21 
(-8/-14) × (21/21) = (-168/-294) 
Now multiply the numerator and denominator of (13/21) by the denominator of  
(-8/-14) i.e. -14 
(13/21) × (-14/-14) = (-182/-294) 
Clearly, the numerators of the above obtained rational numbers are not equal.  
Hence, the given rational numbers are also not equal 
 
2. In each of the following pairs represent a pair of equivalent rational numbers, find 
the values of x. 
(i) (2/3) and (5/x) 
(ii) (-3/7) and (x/4) 
(iii) (3/5) and (x/-25) 
(iv) (13/6) and (-65/x) 
 
Solution: 
(i) Given (2/3) and (5/x) 
Also given that they are equivalent rational number so (2/3) = (5/x) 
x = (5 × 3)/2 
x = (15/2) 
 
(ii) Given (-3/7) and (x/4) 
Also given that they are equivalent rational number so (-3/7) = (x/4) 
x = (-3 × 4)/7 
x = (-12/7) 
 
(iii) Given (3/5) and (x/-25) 
Also given that they are equivalent rational number so (3/5) = (x/-25) 
x = (3 × -25)/5 
x = (-75)/5 
x = -15 
 
(iv) Given (13/6) and (-65/x) 
Also given that they are equivalent rational number so (13/6) = (-65/x) 
x = 6/13 x (- 65)  
 
 
 
 
 
 
x = 6 x (-5)  
x = -30 
 
3. In each of the following, fill in the blanks so as to make the statement true: 
(i)  A number which can be expressed in the form p/q, where p and q are integers and 
q is not equal to zero, is called a ……….. 
(ii)  If the integers p and q have no common divisor other than 1 and q is positive, then 
the rational number (p/q) is said to be in the …. 
(iii) Two rational numbers are said to be equal, if they have the same …. form 
(iv) If m is a common divisor of a and b, then (a/b) = (a ÷ m)/….. 
(v)  If p and q are positive Integers, then p/q is a ….. rational number and (p/-q) is a 
…… rational number. 
(vi) The standard form of -1 is … 
(vii)  If (p/q) is a rational number, then q cannot be …. 
(viii)  Two rational numbers with different numerators are equal, if their numerators 
are in the same …. as their denominators. 
 
Solution: 
(i) Rational number 
(ii) Standard form 
(iii) Standard 
(iv) b ÷ m 
(v) Positive, negative 
(vi) (-1/1) 
(vii) Zero 
(viii) Ratio 
 
4. In each of the following state if the statement is true (T) or false (F): 
(i) The quotient of two integers is always an integer. 
(ii) Every integer is a rational number. 
(iii) Every rational number is an integer. 
(iv) Every traction is a rational number. 
(v)  Every rational number is a fraction. 
(vi) If a/b is a rational number and m any integer, then (a/b) = (a x m)/ (b x m) 
(vii) Two rational numbers with different numerators cannot be equal. 
(viii) 8 can be written as a rational number with any integer as denominator. 
(ix) 8 can be written as a rational number with any integer as numerator. 
 
 
 
 
 
 
(x)  (2/3) is equal to (4/6). 
 
Solution: 
(i) False 
 
Explanation: 
The quotient of two integers is not necessary to be an integer 
 
(ii) True 
 
Explanation: 
Every integer can be expressed in the form of p/q, where q is not zero.  
 
(iii) False 
 
Explanation: 
Every rational number is not necessary to be an integer 
 
(iv) True 
 
Explanation: 
According to definition of rational number i.e. every integer can be expressed in the 
form of p/q, where q is not zero.  
 
(v) False 
 
Explanation: 
It is not necessary that every rational number is a fraction. 
 
(vi) True 
 
Explanation: 
If a/b is a rational number and m any integer, then (a/b) = (a x m)/ (b x m) is one of the 
rule of rational numbers 
 
(vii) False 
 
Page 5


 
Exercise 4.5 page no: 4.20 
1. Which of the following rational numbers are equal?
(i) (-9/12) and (8/-12)
(ii) (-16/20) and (20/-25)
(iii) (-7/21) and (3/-9)
(iv) (-8/-14) and (13/21)
Solution: 
(i) Given (-9/12) and (8/-12)
The standard form of (-9/12) is (-3/4) [on diving the numerator and denominator of
given number by their HCF i.e. by 3]
The standard form of (8/-12) = (-2/3) [on diving the numerator and denominator of
given number by their HCF i.e. by 4]
Since, the standard forms of two rational numbers are not same. Hence, they are not
equal.
(ii) Given (-16/20) and (20/-25)
Multiplying numerator and denominator of (-16/20) by the denominator of (20/-25)
i.e. -25.
(-16/20) × (-25/-25) = (400/-500)
Now multiply the numerator and denominator of (20/-25) by the denominator of
(-16/20) i.e. 20
(20/-25) × (20/20) = (400/-500)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
(iii) Given (-7/21) and (3/-9)
Multiplying numerator and denominator of (-7/21) by the denominator of (3/-9)
i.e. -9.
(-7/21) × (-9/-9) = (63/-189)
Now multiply the numerator and denominator of (3/-9) by the denominator of
(-7/21) i.e. 21
(3/-9) × (21/21) = (63/-189)
Clearly, the numerators of the above obtained rational numbers are equal.
Hence, the given rational numbers are equal
 
 
 
 
 
 
(iv) Given (-8/-14) and (13/21) 
Multiplying numerator and denominator of (-8/-14) by the denominator of (13/21) 
i.e. 21 
(-8/-14) × (21/21) = (-168/-294) 
Now multiply the numerator and denominator of (13/21) by the denominator of  
(-8/-14) i.e. -14 
(13/21) × (-14/-14) = (-182/-294) 
Clearly, the numerators of the above obtained rational numbers are not equal.  
Hence, the given rational numbers are also not equal 
 
2. In each of the following pairs represent a pair of equivalent rational numbers, find 
the values of x. 
(i) (2/3) and (5/x) 
(ii) (-3/7) and (x/4) 
(iii) (3/5) and (x/-25) 
(iv) (13/6) and (-65/x) 
 
Solution: 
(i) Given (2/3) and (5/x) 
Also given that they are equivalent rational number so (2/3) = (5/x) 
x = (5 × 3)/2 
x = (15/2) 
 
(ii) Given (-3/7) and (x/4) 
Also given that they are equivalent rational number so (-3/7) = (x/4) 
x = (-3 × 4)/7 
x = (-12/7) 
 
(iii) Given (3/5) and (x/-25) 
Also given that they are equivalent rational number so (3/5) = (x/-25) 
x = (3 × -25)/5 
x = (-75)/5 
x = -15 
 
(iv) Given (13/6) and (-65/x) 
Also given that they are equivalent rational number so (13/6) = (-65/x) 
x = 6/13 x (- 65)  
 
 
 
 
 
 
x = 6 x (-5)  
x = -30 
 
3. In each of the following, fill in the blanks so as to make the statement true: 
(i)  A number which can be expressed in the form p/q, where p and q are integers and 
q is not equal to zero, is called a ……….. 
(ii)  If the integers p and q have no common divisor other than 1 and q is positive, then 
the rational number (p/q) is said to be in the …. 
(iii) Two rational numbers are said to be equal, if they have the same …. form 
(iv) If m is a common divisor of a and b, then (a/b) = (a ÷ m)/….. 
(v)  If p and q are positive Integers, then p/q is a ….. rational number and (p/-q) is a 
…… rational number. 
(vi) The standard form of -1 is … 
(vii)  If (p/q) is a rational number, then q cannot be …. 
(viii)  Two rational numbers with different numerators are equal, if their numerators 
are in the same …. as their denominators. 
 
Solution: 
(i) Rational number 
(ii) Standard form 
(iii) Standard 
(iv) b ÷ m 
(v) Positive, negative 
(vi) (-1/1) 
(vii) Zero 
(viii) Ratio 
 
4. In each of the following state if the statement is true (T) or false (F): 
(i) The quotient of two integers is always an integer. 
(ii) Every integer is a rational number. 
(iii) Every rational number is an integer. 
(iv) Every traction is a rational number. 
(v)  Every rational number is a fraction. 
(vi) If a/b is a rational number and m any integer, then (a/b) = (a x m)/ (b x m) 
(vii) Two rational numbers with different numerators cannot be equal. 
(viii) 8 can be written as a rational number with any integer as denominator. 
(ix) 8 can be written as a rational number with any integer as numerator. 
 
 
 
 
 
 
(x)  (2/3) is equal to (4/6). 
 
Solution: 
(i) False 
 
Explanation: 
The quotient of two integers is not necessary to be an integer 
 
(ii) True 
 
Explanation: 
Every integer can be expressed in the form of p/q, where q is not zero.  
 
(iii) False 
 
Explanation: 
Every rational number is not necessary to be an integer 
 
(iv) True 
 
Explanation: 
According to definition of rational number i.e. every integer can be expressed in the 
form of p/q, where q is not zero.  
 
(v) False 
 
Explanation: 
It is not necessary that every rational number is a fraction. 
 
(vi) True 
 
Explanation: 
If a/b is a rational number and m any integer, then (a/b) = (a x m)/ (b x m) is one of the 
rule of rational numbers 
 
(vii) False 
 
 
 
 
 
 
 
Explanation: 
They can be equal, when simplified further. 
 
(viii) False 
 
Explanation: 
8 can be written as a rational number but we can’t write 8 with any integer as 
denominator. 
 
(ix) False 
 
Explanation: 
8 can be written as a rational number but we can’t with any integer as numerator. 
 
(x) True 
 
Explanation: 
When convert it into standard form they are equal