Chapter 4: Rational Numbers Exercise – 4.5
Question: 1
Which of the following 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) The standard form of -9/12 is -9/3, 12/3 = -34
The standard form of 8/-12 is 8/-4, 12/-4 = -2/3
Since, the standard forms of two rational numbers are not same. Hence, they are not equal.
(ii) Since, LCM of 20 and 25 is 100.
Therefore making the denominators equal,
Therefore, -16/20 = 20/-25 .
(iii) Since, LCM of 21 and 9 is 63.
Therefore making the denominators equal,
(iv) Since, LCM of 14 and 21 is 42.
Therefore making the denominators equal,
Question: 2
If each of the following pairs represents 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) 2/3 = 5/x, then x = (5 × 3)/2 = 15/2
(ii) -3/7 = x/4, then x = (-3/7) × 4 = -12/7
(iii) 3/5 = x/-25, then x = 3/5 x (-25) = -75/5 = -15
(iv) 13/6 = – 65/x , then x = 6/13 x (- 65) = 6 x (-5) = -30
Question: 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,
(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 rational number
(iii) standard form
(iv) a/b = (a ÷ m)/(b ÷ m )
(v) positive rational number, negative rational number
(vi) -1/1
(vii) Zero
(viii) ratio
Question: 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,
(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; not necessary
(ii) True; every integer can be expressed in the form of p/q, where q is not zero.
(iii) False; not necessary
(iv) True; every fraction can be expressed in the form of p/q, where q is not zero.
(v) False; not necessary
(vi) True
(vii) False; they can be equal, when simplified further.
(viii) False
(ix) False
(x) True; in the standard form, they are equal.