## NCERT Solutions for class 7 Maths Chapter 9: Rational Numbers## Exercise 9.1
Where, P and q are the integers and q is not equal to 0. We can use any value of the rational number with denominator greater than the numerator. If the value of the denominator is greater than the numerator, its decimal value lies between 0 and 1. So, the rational numbers between -1 and 0 are: -2/3, -4/5, -6/7, -2/4, -1/3, -2/5, -2/7, -1/5, -1/2, etc. We can specify any five rational numbers.
Where, P and q are the integers and q is not equal to 0. We can use any value of the rational number with numerator greater than the denominator. So, the rational numbers between -2 and -1 are: -5/3, -4/3, -6/4, -7/5, -3/2, -8/6, -9/7
Where, P and q are the integers and q is not equal to 0. Let's multiply both the numerator and denominator of the two rational numbers by 9 and 15. (-4 × 9)/ (5 × 9) = -36/45 (-2 × 15)/ (3 × 15) = -30/45 The five rational numbers between -30/45 and -36/45 are: -31/45, -32/45, -33/45, -34/45, and -35/45 These numbers in their lowest form can be represented as: -31/45, -32/45, -11/15, -34/45, and -7/9
Where, P and q are the integers and q is not equal to 0. The five rational numbers between -1/2 and 2/3 are: -1/3, -1/4, 0, 1/3, 1/2
Where, P and q are the integers and q is not equal to 0. In the given rational numbers, let's convert them into their lowest form. −6/10 = (−3 × 2)/ (5 × 2) = −3/5 −9/15 = (−3 × 3)/ (5 × 3) = −3/5 −12/20 = (−3 × 4)/ (5 × 4) = −3/5 Thus, the four more rational numbers will be: (−3 × 5)/ (5 × 5) = −15/25 (−3 × 6)/ (5 × 6) = −18/30 (−3 × 7)/ (5 × 7) = −21/35 (−3 × 8)/ (5 × 8) = −24/40
Where, P and q are the integers and q is not equal to 0. In the given rational numbers, let's convert them into their lowest form. −2/8 = (−1 × 2)/ (4 × 2) = −1/4 −3/12 = (−1 × 3)/ (4 × 3) = −1/4 Thus, the four more rational numbers will be: (−1 × 4)/ (4 × 4) = −4/16 (−1 × 5)/ (4 × 5) = −5/20 (−1 × 6)/ (4 × 6) = −6/24 (−1 × 7)/ (4 × 7) = −7/28
Where, P and q are the integers and q is not equal to 0. In the given rational numbers, let's convert them into their lowest form. −2/12 = (−1 × 2)/ (6 × 2) = −1/6 −3/18 = (−1 × 3)/ (6 × 3) = −1/6 −4/24 = (−1 × 4)/ (6 × 4) = −1/6 Thus, the four more rational numbers will be: (−1 × 5)/ (6 × 5) = −5/30 (−1 × 6)/ (6 × 6) = −6/36 (−1 × 7)/ (6 × 7) = −7/42 (−1 × 8)/ (6 × 8) = −8/48
Where, P and q are the integers and q is not equal to 0. The given series of rational numbers show the addition of 2 to the numerator and 3 to the denominator. 4/−6 = (2 + 2)/ − (3 + 3) 6/−9 = (4 + 2)/ − (6 + 3) Thus, the four more rational numbers will be: (6 + 2)/ − (9 + 3) = 8/−12 (8 + 2)/ − (12 + 3) = 10/−15 (10 + 2)/ − (15 + 3) = 12/−18 (12 + 2)/ − (18 + 3) =14/−21
Thus, the four equivalent rational numbers are: (−2 × 2)/ (7 × 2) = −4/14 (−2 × 3)/ (7 × 3) = −6/21 (−2 × 4)/ (7 × 4) = −8/28 (−2 × 5)/ (7 × 5) = −10/35
Thus, the four equivalent rational numbers are: (5 × 2)/ (−3 × 2) = 10/−6 (5 × 3)/ (−3 × 3) = 15/−9 (5 × 4)/ (−3 × 4) = 20/−12 (5 × 5)/ (−3 × 5) = 25/−15
Thus, the four equivalent rational numbers are: (4 × 2)/ (9 × 2) = 8/18 (4 × 3)/ (9 × 3) = 12/27 (4 × 4)/ (9 × 4) = 16/36 (4 × 5)/ (9 × 5) = 20/45
(i) 3/4
(ii) −5/8
(iii) −7/4
(iv) 7/8
The given numbers on the number line divides into three sections. P = 2 + 1/3 P = 7/3 Q = 2 + 2/3 Q = 8/3 R = -2 + 2/3 R = -4/3 S = -2 + 1/3 S = -5/3
The given pair does not represent the same rational number
(1 × 3)/ (3 × 3) = 1/3 −7/21 can be represented as: (−1 × 7)/ (3 × 7) = −1/3 1/3 is not equal to −1/3 Hence, both rational numbers does not represent the same pair.
The given pair represents the same rational number
(−4 × 4)/ (5 × 4) = −4/5 20/− 25 can be represented as: (4 × 5)/ (−5 × 5) = 4/−5 = −4/5 Thus, both the rational numbers are equal. ## Note: A negative sign on a rational number can be present either with the numerator or the denominator. The result is the same in both the cases.
The given pair represents the same rational number
2/3 Negative sign present at both the numerator and the denominator results in a positive rational number. Thus, both the rational numbers are equal.
The given pair represents the same rational number
(−3 × 1)/ (5 × 1) = − 3/5 - 12/20 can be represented as: (−3 × 4)/ (5 × 4) = − 3/5 Thus, both the rational numbers are equal.
The given pair represents the same rational number
(8 × 1)/ (−5 × 1) = 8/−5 - 24/15 can be represented as: (8 × 3)/ (−5 × 3) = 8/−5 Thus, both the rational numbers are equal.
The given pair does not represent the same rational number
(−1 × 1)/ (3 × 3) = -1/9 Both the rational numbers are not equal. Hence, it does not represent the similar pairs.
The given pair does not represent the same rational number
(5 × −1)/ (9 × −1) = 5/9 5/9 is not equal to 5/−9 Hence, it does not represent the similar pairs.
(−4 × 2)/ (3 × 2) = −4/3
= (5 × 5)/ (5 × 9) = 5/9
(−11 × 4)/ (18 × 4) = −11/18
(−4 × 2)/ (5 × 2) = −4/5
Hence, −5/7 < 2/3
Let's first compare 4/5 and 5/7 by multiplying the numbers by 7 and 5 to make their denominators equal. (4 × 7)/ (5 × 7) __ (5 × 5)/ (7 × 5) 28/35 __ 25/35 28/35 > 25/35 Now, reversing the order with negative signs, we get: −28/35 < −25/35 −4/5 < −5/7
Two negative rational numbers can be compared by ignoring their negative signs and then reversing the order. First rational number is already present in its lowest form. Let's convert the second rational number to its lowest form. −14/16 = (−7 × 2)/ (8 × 2) = −7/8 Both the rational numbers are equal. Hence, −7/8 = 14/−16
Let's first compare 8/5 and 7/4 by multiplying the numbers by 4 and 5 to make their denominators equal. (8 × 4)/ (5 × 4) __ (7 × 5)/ (4 × 5) 32/20 __ 35/20 32/20 < 35/20 Now, reversing the order with negative signs, we get: −32/20 > −35/20 −8/5 > −7/4
Two negative rational numbers can be compared by ignoring their negative signs and then reversing the order. Let's first compare 1/3 and 1/4 by multiplying the numbers by 4 and 3 to make their denominators equal. (1 × 4)/ (3 × 4) __ (1 × 3)/ (4 × 3) 4/12 __ 3/12 4/12 > 3/12 Now, reversing the order with negative signs, we get: −4/12 < −3/12 1/−3 < −1/4
Hence, both the rational numbers are equal. 5/−11 = −5/11
Hence, 0 > −7/6
(2 × 2)/ (3 × 2), (5 × 3), (2 × 3) 4/6, 15/6 4/6 < 15/6 Thus, 15/6 or 5/2 is greater.
−5/6, (−4 × 2) /(3 × 2) −5/6, −8/6 In negative rational numbers, the comparison works in the opposite method as compared to the positive rational numbers. 5/6 < 8/6 −5/6 > −8/6 Thus, −5/6 is greater
2/− 3 can also be represented as - 2/3 - (3 × 3)/ (4 × 3), − (2 × 4)/ (3 × 4) - 9/12, -8/12 9/12 > 8/12 - 9/12 < -8/12 Thus, -8/12 or 2/− 3 is greater
Thus, 1/4 is greater
− 3 2/7 = −23/7 − 3 4/5 = − 19/5 Two rational numbers with different denominators can be compared by making them equal. Let's multiply the given rational numbers by 5 and 7. −23/7, −19/5 − (23 × 5)/ (7 × 5), − (19 × 7)/ (5 × 7) −115/35, −133/35 −115/35 > −133/35 Thus, −115/35 or − 3 2/7 is greater
So, −3/5 < −2/5 < −1/5 It means that −3/5 is the smallest and −1/5 is the greatest.
−1/3 = (−1 × 3)/ (3 × 3) = −3/9 −4/3 = (−4 × 3)/ (3 × 3) = −12/9 The numbers can now be represented as: −3/9, −2/9, −12/9 −12/9 < −3/9 < −2/9 Or −4/3 < −1/3 < −2/9 It means that −4/3 is the smallest and −2/9 is the greatest.
−3/7 = − (3 × 4)/ (7 × 4) = −12/28 −3/2 = − (3 × 14)/ (2 × 14) = −42/28 −3/4 = − (3 × 7)/ (4 × 7) = −21/28 The numbers can now be represented as: −12/28, −42/28, −21/28 −42/28 < −21/28 < −12/28 Or −3/2 < −3/4 < −3/7 It means that −3/2 is the smallest and −3/7 is the greatest. ## Exercise 9.2
= (5 + (−11))/4 = (5 - 11)/4 = -6/4 = -3/2
To add the two rational numbers, the denominator should be the same. Let's multiply the given rational numbers by 5 and 3. (5 × 5)/ (3 × 5) + (3 × 3)/ (5 × 3) = 25/15 + 9/15 = (25 + 9)/15 = 34/15
Let's multiply the given rational numbers by 3 and 2. - 9/10 + 22/15 = - (9 × 3)/ (10 × 3) + (22 × 2)/ (15 × 2) = - 27/30 + 44/30 = (-27 + 44)/30 = (44 - 27)/30 = 17/30
To add the two rational numbers, the denominator should be the same. Let's multiply the given rational numbers by 9 and 11. 3/11 + 5/9 = (3 × 9)/ (11 × 9) + (5 × 11)/ (9 × 11) = 27/99 + 55/99 = (27 + 55)/99 = 82/99
19 × 3 = 57 Let's multiply the first rational number by 3 to make the denominators equal. = - (8 × 3)/(19 × 3) + (- 2)/57 = -24/57 + (- 2)/57 = (-24 - 2)/ 57 = -26/57
Hence, −2/3 + 0 = −2/3
−2 1/3 = −7/3 4 3/5 = 23/5 To add the two rational numbers, the denominator should be the same. Let's multiply the given rational numbers by 5 and 3. = −7/3 + 23/5 = − (7 × 5)/ (3 × 5) + (23 × 3)/ (5 × 3) = − 35/15 + 69/15 = (−35 + 69)/15 = (69 - 35)/15 = 34/15
Let's multiply the given rational numbers by 3 and 2. 7/24 - 17/36 = (7 × 3)/ (24 × 3) - (17 × 2)/ (36 × 2) = 21/72 - 34/72 = (21 - 34)/72 = -13/72
21 × 3 = 63 Let's multiply the second rational number by 3. = 5/63 - (−6)/21 = 5/63 - (−6 × 3)/ (21 × 3) = 5/63 - (−18)/63 = (5 - (−18)/63 = (5 + 18)/ 63 = 23/63
Let's multiply the given rational numbers by 15 and 13. = −6/13 - (−7/15) = (−6 × 15)/ (13 × 15) - (−7 × 13)/ (15 × 13) = −90/195 - (−91)/195 = (−90 - (−91))/195 = (−90 + 91)/195 = 1/195
Let's multiply the given rational numbers by 11 and 8. −3/8 - 7/11 = (−3 × 11)/ (8 × 11) - (7 × 8)/ (11 × 8) = (−33)/88 - (56)/88 = (−33 - 56)/88 = −89/88
−2 1/9 = - 19/9 To subtract the two rational numbers, the denominator should be the same. Let's multiply the second rational number by 9. - 19/9 - (6 × 9)/ (1 × 9) = - 19/9 - 54/9 = (- 19 - 54)/9 = - 73/9
= (9 × −7)/ (2 × 4) = −63/8
= (3 × (−9))/10 = −27/10
= (- 6 × 9)/ (5 × 11) = - 54/55
= (3 × (-2))/ (7 × 5) = -6/35
= (3 × 2)/ (11 × 5) = 6/55
= (3 × −5)/ (−5 × 3) = (−15)/ (−15) = 15/15 = 1
= (−4) × (Reciprocal of 2/3) = (−4) × 3/2 = ((−4) × 3)/2 = −6
= - 3/5 × (Reciprocal of 2) = - 3/5 × 1/2 = (- 3 × 1)/ (5 × 2) = (- 3)/10 = - 3/10
= −4/5 × (Reciprocal of − (3)) = −4/5 × −1/3 = (−4 × −1)/ (5 × 3) = 4/15
= - 1/8 × (Reciprocal of 3/4) = - 1/8 × 4/3 = (-1 × 4)/ (8 × 3) = -1/6
= −2/13 × (Reciprocal of 1/7) = −2/13 × 7/1 = (−2 × 7)/ (13 × 1) = −14/13
= −7/12 × (Reciprocal of (−2/13)) = −7/12 × −13/2 = (−7 × −13)/ (12 × 2) = 91/24
= 3/13 × (Reciprocal of ((−4/65)) = 3/13 × −65/4 = (3 × −65)/ (13 × 4) (13 × 5 = 65) = (3 × −5)/ (1 × 4) = −15/4 |