Exercise 4.1
Question: 1
Write down the numerator of each of the following rational numbers:
(i) -7/5
(ii) 15/(-4)
(iii) (- 17)/(- 21)
(iv) 8/9
(v) 5
Solution:
Numerators are:
(i) -7
(ii) 15
(iii) -17
(iv) 8
(v) 5
Question: 2
Write down the denominator of each of the following rational numbers:
(i) – 4/5
(ii) 11/(-34)
(iii) 15/82
(iv) 15
(v) 0
Solution:
Denominators are:
(i) 5
(ii) – 34
(iii) – 82
(iv) 1
(v) 1
Question: 3
Write down the rational number whose numerator is (-3) ×4, and whose denominator is (34 – 23) ×(7 – 4).
Solution:
According to the question:
Numerator = (-3) x 4 = – 12
Denominator = (34 – 23) x (7 – 4) =11 x 3 = 33
Therefore, Rational number = -12/32
Question: 4
Write the following rational numbers as integers:
Solution:
Integers are 7, -12, 34, -73 and 95.
Question: 5
Write the following integers as rational numbers with denominator 1:
-15, 17, 85, -100
Solution:
Rational numbers of given integers with denominator 1 are:
Question: 6
Write down the rational whose numerator is the smallest three digit number and denominator is the largest four digit number.
Solution:
Smallest three digit number = 100
Largest four digit number = 9999
Therefore rational number = 100/9999
Question: 7
Seperate positive and negative rational numbers from the following rational numbers:
Solution:
Given rational numbers can be rewritten as:
Thus, positive rational numbers are:
Negative rational numbers are:
Question: 8
Which of the following rational numbers are positive:
(i) (- 8)/7
(ii) 9/8
(iii) 19/13
(iv) (- 21)/13
Solution:
The numbers can be rewritten as:
(i) (- 8)/7
(ii) 9/8
(iii) 19/13
(iv) -21/13
Positive rational numbers are (ii) and (iii), i.e., 9/8 and (-19)/(-13)
Question: 9
Which of the following rational numbers are negative?
(i) (- 3)/7
(ii) (- 5)/(- 8)
(iii) 9/(- 83)
(iv) 115/197
Solution:
The numbers can be rewritten as:
(i) (- 3)/7
(ii) (-5)/(- 8)
(iii) 9/(- 83)
(iv) 115/197
Negative rational numbers are (i) and (iii).
Exercise 4.2
Question: 1
Express each of the following as rational number with positive denominator:
(i) (-15)/(-28)
(ii) 6/(-9)
(iii) (-28)/(-11)
(iv) 19/(- 7)
Solution:
Rational number with positive denominators:
(i) Multiplying the number by -1, we get: -15 – 28 = -15 x -1 – 28 x – 1 = 1528
(ii) Multiplying the number by -1, we get: 6 – 9 = 6x – 1 – 9x – 1 = – 69
(iii) Multiplying the number by -1, we get: – 28 – 11= – 28 x – 1 – 11 x – 1 = 2811
(iv) Multiplying the number by -1, we get: 19 – 7 = 19 x – 1 – 7 x – 1= -197
Question: 2
Express 3/5 as a rational number with numerator:
(i) 6
(ii) -15
(iii) 21
(iv) – 27
Solution:
Rational number with numerator:
(i) 6 is:
(Multiplying numerator and denominator by 2)
(ii) -15 is:
(Multiplying numerator and denominator by – 5)
(iii) 21 is:
(Multiplying numerator and denominator by 7)
(iv) – 27 is:
(Multiplying numerator and denominator by – 9)
Question: 3
Express 5/7 as a rational number with denominator:
(i) -14
(ii) 70
(iii) -28
(iv) – 84
Solution:
5/7 as a rational number with denominator:
(1) -14 is:
(Multiplying numerator and denominator by -2)
(ii) 70 is:
(Multiplying numerator and denominator by 10)
(iii) – 28 is:
(Multiplying numerator and denominator by – 4)
(iv) – 84 is:
(Multiplying numerator and denominator by -12)
Question: 4
Express 3/4 as a rational number with denominator:
(i) 20
(ii) 36
(iii) 44
(iv) – 80
Solution:
3/4 as rational number with denominator:
(i) 20 is:
(Multiplying numerator and denominator by 5)
(ii) 36 is:
(Multiplying numerator and denominator by 9)
(iii) 44 is:
(Multiplying numerator and denominator by 11)
(iv) – 80 is:
(Multiplying numerator and denominator by – 20)
Question: 5
Express 2/5 as a rational number with numerator:
(i) -56
(ii) 154
(iii) -750
(iv) – 80
Solution:
2/5 as a rational number with numerator:
(i) – 56 is:
(Multiplying numerator and denominator by -28)
(ii) 154 is:
(Multiplying numerator and denominator by 77)
(iii) – 750 is:
(Multiplying numerator and denominator by -375)
(iv) 500 is:
(Multiplying numerator and denominator by 250)
Question: 6
Express (-192)/108 as a rational number with numerator:
(i) 64
(ii) -16
(iii) 32
(iv) – 48
Solution:
Rational number with numerator:
(i) 64 as numerator:
(Dividing the numerator and denomintor by -3)
(ii) -16 as numerator:
(Dividing the numerator and denomintor by 12)
(iii) 32 as numerator:
(Dividing the numerator and denomintor by – 6)
(iv) – 48 as numerator:
(Dividing the numerator and denomintor by 4)
Question: 7
Express 168/(-294) as a rational number with denominator:
(i) 14
(ii) -7
(iii) – 49
(iv) 1470
Solution:
Rational number with denominator:
(i) 14 as denominator:
(Dividing the numerator and denomintor by -21)
(ii) -7 as denominator:
(Dividing the numerator and denomintor by 42)
(iii) – 49 as denominator:
(Dividing the numerator and denomintor by 6)
(iv) 1470 as denominator:
(Multiplying the numerator and denomintor by – 5)
Question: 8
Write (-14)/42 form so that numerator is equal to:
(i) -2
(ii) 7
(iii) 42
(iv) -70
Solution:
Rational number with numerator:
(i) -2 is:
(Dividing numerator and denominator by 7)
(ii) 7 is:
(Dividing numerator and denominator by -2)
(iii) 42 is:
(Multiplying numerator and denominator by -3)
(iv) -70 is:
(Multiplying numerator and denominator by 5)
Question: 9
Select those rational numbers which can be written as a rational number with numerator 6:
Solution:
Given rational numbers that can be written as a rational number with numerator 6 are:
1/22 (On multiplying by 6) = 6/132, 2/3 (On multiplying by 3)
= 6/9, 3/4 (On multiplying by 2)
= 6/8, (-6)/7 (On multiplying by -1)
= 6/(-7)
Question: 10
Select those rational numbers which can be written as a rational number with denominator 4:
Solution:
Given rational numbers that can be written as a rational number with denominator 4 are:
7/8 (On dividing by 2) = 3.5/4,
64/16 (On dividing by 4) = 16/4,
36/-12(On dividing by 3) = 12/-4 = -12/4,
16/17 can’t be expressed with a denominator 4.
5/- 4(On multiplying by -1) = – 5/4
140/28(On dividing by 7) = 20/4
Question: 11
In each of the following, find an equivalent form of the rational number having
common denominator:
(i) 3/4 and 5/12
(ii) 2/3, 7/6 and 11/12
(iii) 5/7, 3/8, 9/14 and 20/21<
Solution:
Equivalent forms of the rational number having common denominator are:
(i) 3/4 = (3 × 3)/(4 × 3) = 9/12 and 512.
(ii) 2/3 = (2 × 4)/(3 × 4) = 8/12 and 7/6 = (7×2)/(6×2) = 14/12 and 11/12
Forms are 8/12, 14/12 and 11/12
(iii) 5/7 = (5 × 24)/(7 × 24) = 120/168, 3/8 = (3 × 21)/(8 × 21) = 63/168, 9/14
= (9 × 12)/(14 × 12)
= 108/168 and 20/21
= (20 × 8)/(21×8)
=160/168
Forms are 120/168, 63/168, 108/168 and 160/168.
Exercise 4.3
Question: 1
Determine whether the following rational numbers are in the lowest form or not:
(i) 65/84
(ii) (-15)/32
(iii) 24/128
(iv) (-56)/(-32)
Solution:
(i) We observe that 65 and 84 have no common factor their HCF is 1.
Thus, 65/84 is in its lowest form.
(ii) We observe that -15 and 32 have no common factor i.e., their HCF is 1.
Thus, -15/32 is in its lowest form.
(iii) HCF of 24 and 128 is not 1.
Thus, given rational number is not in its simplest form.
(iv) HCF of 56 and 32 is 8.
Thus, given rational number is not in its simplest form.
Question: 2
Express each of the following numbers to the lowest form:
(i) 4/22
(ii) (- 36)/180
(iii) 132/428
(iv) 32/56
Solution:
Lowest form of:
(i) 4/22 is:
4 = 2 × 2, 22 = 2 × 11
HCF of 4 and 22 is 2.
Dividing the fraction by 2, we get 2/11.
(ii) -36/180 is:
36 = 3 × 3 × 2 × 2, 180 = 5 × 3 × 3 × 2 × 2
HCF of 36 and 180 is 36.
Dividing the fraction by 36, we get -1/5.
(iii) 132/- 428 is:
132 = 2 × 3 × 2 × 11, 428 = 2 × 2 × 107
HCF of 132 and 428 is 4.
Dividing the fraction by 4, we get 33/-107.
(iv) – 32/ – 56 is:
32 =2 × 2 × 2 × 2 × 2, 56 = 2 × 2 × 2 × 7
HCF of 32 and 56 is 8.
Dividing the fraction by 8, we get 4/7.
Question: 3
Fill in the blanks:
Solution:
Exercise 4.4
Question: 1
Write each of the following rational numbers in the standard form:
(i) 2/10
(ii) – 8/36
(iii) 4/-16
(iv) -15/-35
(v) 299/-161
(vi) – 63/- 210
(vii) 68/- 119
(viii) – 195/275
Solution:
(i) The denominator is positive and HCF of 2 and 10 is 2.
Therefore, Dividing the numerator and denominator by 2, we get:
2/10 = 2/2, 10/2 = 1/5
(ii) The denominator is positive and HCF of 8 and 36 is 4.
Therefore, Dividing the numerator and denominator by 4, we get: (- 8)/36
= -8/4, 36/4 = -2/9
(iii) The denominator is negative.
(4x-1)/(-16x-1) = -4/16
HCF of 4 and 16 is 4.
Therefore, Dividing the numerator and denominator by 4, we get: (-4)/4, 16/4
= -1/4
(iv) -15/-35
The H.C.F. of -15 and -35 is 5.
Dividing the Nr and Dr of -15/-35 by 5, we get:
(v) 299/-161
The H.C.F. of 299 and -161 is 23.
Dividing the Nr and Dr of 299/-161 by 23, we get:
(vi) – 63/-210
The H.C.F of 63 and 210 is 21.
Dividing the Nr and Dr of -63/-210 by 21, we get:
(vii) 68/-119
The H.C.F. of 68 and 119 is 17.
Dividing the Nr and Dr of 68/-119 by 17, we get
(viii) -195/275
The H. C. F. of 195 and 275 is 5, we get:
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.
Exercise 4.6
Question: 1
Draw the number line and represent the following rational numbers on it:
(i) 2/3
(ii) 3/4
(iii) 3/8
(iv) -5/8
(v) -3/16
(vi) -7/3
(vii) 22/-7
(viii) -31/3
Solution:
Question: 2
Which of the two rational numbers in each of the following pairs of rational numbers is greater?
(i) -3/8, 0
(ii) 5/2, 0
(iii) – 4/11, 3/11
(iv) – 7/12, 5/- 8
(v) 4/9, – 3/- 7
(vi) – 5/8, 3/- 4
(vii) 5/9, -3/- 8
(viii) 5/- 8, -7/12
Solution:
(i) We know that every positive rational number is greater than zero and every negative
rational number is smaller than zero. Thus, – 3/8 > 0
(ii) 5/2 > 0.Because every positive rational number is greater than zero and every negative rational number is smaller than zero.
(iii) – 4/11 < 3/11. Because every positive rational number is greater than zero and every negative rational number is smaller than zero.
Question: 3
Fill in the blanks by the correct symbol out >, =, or < :
Solution:
Question: 4
Fill in the blanks by the correct symbol out of >, =, or < :
Solution:
(i) Because every positive number is greater than a negative number, (- 6)/7 < 7/13.
(iv) Because every positive number is greater than a negative number, 0 > (-2)/5.
Question: 5
Arrange the following rational numbers in ascending order:
Solution:
(i) Ascending order:
Since, LCM of 5, -30, -15, 10 is 30.
Multiplying the numerators and denominators to get the denominator equal to the LCM 3/5
(ii) Since, LCM of 9, -12, -18, 3 is 36.
Multiplying the numerators and denominators to get the denominator to get the denominator equal to the LCM ,
Question: 6
Arrange the following rational numbers in descending order:
Solution:
We have to arrange them in descending order.
(i) Since, LCM of 8, 16, -12, -4, 28 is 336.
Multiplying the numerators and denominators, to get the denominator equal to the LCM, 7/8
(ii) Since, LCM of 10, -30, -15, 20 is 60.
Multiplying the numerators and denominators, to get the denominator equal to LCM,
Question: 7
Which of the following statements are true:
(i) The rational number 29/23 lies to the left of zero on the number line.
(ii) The rational number-12/-17 lies to the left of zero on the number line.
(iii) The rational number 3/4 lies to the right of zero on the number line.
(iv) The rational number (-12)/(-5) and (- 7)/(-17) are on the opposite side of zero on the number line.
(v) The rational number -2/15 and 7/(-31) are on the opposite side of zero on the number line .
(vi) The rational number (- 3)/(-5) is on the right of (- 4)/7 on the number line.
Solution:
(i) False; it lies to the right of zero because it is a positive number.
(ii) False; it lies to the right of zero because it is a positive number.
(iii) True
(iv) True; they are of opposite signs.
(v) False; they both are of same signs.
(vi) True; they both are of opposite signs and positive number is greater than the negative number.
Thus, it is on the right of the negative number.
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