NCERT Solutions for class 7 Maths Chapter 9: Rational NumbersExercise 9.11. List five rational numbers between: (i) -1 and 0 Answer: -2/3, -1/2, -2/5, -1/3, -2/7 Explanation: Rational numbers are the numbers that can be represented in the form of p/q, 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. (ii) -2 and -1 Answer: -5/3, -4/3, -6/4, -7/5, -3/2 Explanation: Rational numbers are the numbers that can be represented in the form of p/q, 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 (iii) −4/5 and -2/3 Answer: -31/45, -32/45, -11/15, -34/45, and -7/9 Explanation: Rational numbers are the numbers that can be represented in the form of p/q, 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 (iv) -1/2 and 2/3 Answer: -1/3, -1/4, 0, 1/3, 1/2 Explanation: Rational numbers are the numbers that can be represented in the form of p/q, 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 2. Write four more rational numbers in each of the following patterns: (i) −3/5, −6/10, −9/15, −12/20 … Answer: −15/25, −18/30, −21/35, −24/40 Explanation: Rational numbers are the numbers that can be represented in the form of p/q, 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 (ii) −1/4, −2/8, −3/12 … Answer: −4/16, −5/20, −6/24, −7/28 Explanation: Rational numbers are the numbers that can be represented in the form of p/q, 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 (iii) −1/6, −2/12, −3/18, −4/24 … Answer: −5/30, −6/36, −7/42, −8/48 Explanation: Rational numbers are the numbers that can be represented in the form of p/q, 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 (iv) −2/3, 2/−3, 4/−6, 6/−9 … Answer: 8/−12, 10/−15, 12/−18, 14/−21 Explanation: Rational numbers are the numbers that can be represented in the form of p/q, 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 3. Give four rational numbers equivalent to: (i) −2/7 Answer: −4/14, −6/21, −8/28, −10/35 Explanation: To find the equivalent rational numbers, we will multiply the rational number will 2, 3, 4, and 5. 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 (ii) 5/−3 Answer: 10/−6, 15/−9, 20/−12, 25/−15 Explanation: To find the equivalent rational numbers, we will multiply the rational number will 2, 3, 4, and 5. 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 (iii) 4/9 Answer: 8/18, 12/27, 16/36, 20/45 Explanation: To find the equivalent rational numbers, we will multiply the rational number will 2, 3, 4, and 5. 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 4. Draw the number line and represent the following rational numbers on it: (i) 3/4 Answer: (ii) −5/8 Answer: (iii) −7/4 Answer: (iv) 7/8 Answer: 5. The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S. Answer: P = 7/3, Q = 8/3, R = -4/3, S = -5/3 Explanation: 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 6. Which of the following pairs represent the same rational number? (i) −7/21 and 3/9 Answer: No. The given pair does not represent the same rational number Explanation: 3/9 can be represented as: (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. (ii) − 16/20 and 20/− 25 Answer: Yes The given pair represents the same rational number Explanation: − 16/20 can be represented as: (−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.(iii) −2/− 3 and 2/3 Answer: Yes The given pair represents the same rational number Explanation:−2/− 3 can be represented as: 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. (iv) −3/5 and - 12/20 Answer: Yes The given pair represents the same rational number Explanation: − 3/5 can be represented as: (−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. (v) 8/− 5 and - 24/15 Answer: Yes The given pair represents the same rational number Explanation: 8/−5 can be represented as: (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. (vi) 1/3 and -1/9 Answer: No. The given pair does not represent the same rational number Explanation: -1/9 can be represented as: (−1 × 1)/ (3 × 3) = -1/9 Both the rational numbers are not equal. Hence, it does not represent the similar pairs. (vii) −5/−9 and 5/−9 Answer: No. The given pair does not represent the same rational number Explanation: −5/−9 can be represented as: (5 × −1)/ (9 × −1) = 5/9 5/9 is not equal to 5/−9 Hence, it does not represent the similar pairs. 7. Rewrite the following rational numbers in the simplest form: (i) −8/6 Answer: −4/3 Explanation: −8/6 can be represented as: (−4 × 2)/ (3 × 2) = −4/3 (ii) 25/45 Answer: 5/9 Explanation: 25/45 can be represented as: = (5 × 5)/ (5 × 9) = 5/9 (iii) - 44/72 Answer: −11/18 Explanation: −44/72 can be represented as: (−11 × 4)/ (18 × 4) = −11/18 (iv) −8/10 Answer: −4/5 Explanation: −8/10 can be represented as: (−4 × 2)/ (5 × 2) = −4/5 8. Fill in the boxes with the correct symbol out of >, <, or =. (i) −5/7 __ 2/3 Answer: < Explanation: A negative rational number is always less than the positive rational number. Hence, −5/7 < 2/3 (ii) −4/5 __ −5/7 Answer: < Explanation: Two negative rational numbers can be compared by ignoring their negative signs and then reversing the order. 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 (iii) −7/8 __ 14/−16 Answer: = Explanation: 14/−16 can also be represented as −14/16 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 (iv) −8/5 __ −7/4 Answer: > Explanation: Two negative rational numbers can be compared by ignoring their negative signs and then reversing the order. 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 (v) 1/−3 __ −1/4 Answer: < Explanation: 1/−3 can also be represented as −1/3. 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 (vi) 5/−11 __ −5/11 Answer: = Explanation: 5/−11 can also be represented as −5/11. Hence, both the rational numbers are equal. 5/−11 = −5/11 (vii) 0 __ −7/6 Answer: > Explanation: A negative number is always less than 0. Hence, 0 > −7/6 9. Which is greater in each of the following? (i) 2/3, 5/2 Answer: 5/2 Explanation: Two rational numbers with different denominators can be compared by making them equal. Let's multiply the given rational numbers by 2 and 3. (2 × 2)/ (3 × 2), (5 × 3), (2 × 3) 4/6, 15/6 4/6 < 15/6 Thus, 15/6 or 5/2 is greater. (ii) −5/6, −4/3 Answer: −5/6 Explanation: Two rational numbers with different denominators can be compared by making them equal. Let's multiply the second rational number by 2. −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 (iii) - 3/4, 2/− 3 Answer: 2/− 3 Explanation: Two rational numbers with different denominators can be compared by making them equal. Let's multiply the given rational numbers by 3 and 4. 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 (iv) −1/4, 1/4 Answer: 1/4 Explanation: A positive rational number is always greater than the negative rational number. Thus, 1/4 is greater (v) − 3 2/7, − 3 4/5 Answer: − 3 2/7 Explanation: To compare the mixed rational numbers, let's convert them into improper fractions. − 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 10. Write the following rational numbers in ascending order: (i) −3/5, −2/5, −1/5 Answer: −3/5 < −2/5 < −1/5 Explanation: The denominators of the given rational numbers are already the same. The comparing in the case of negative numbers works in an opposite way as compared to the positive numbers. So, −3/5 < −2/5 < −1/5 It means that −3/5 is the smallest and −1/5 is the greatest. (ii) −1/3, −2/9, −4/3 Answer: −4/3 < −1/3 < −2/9 Explanation: Let's convert the denominators of the first and the third rational to 9 for easy comparison. −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. (iii) −3/7, −3/2, −3/4 Answer: −3/2 < −3/4 < −3/7 Explanation: Let's convert the denominators of the given rational numbers to 28 for easy comparison. −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 9.21. Find the sum: (i) 5/4 + (−11/4) Answer: -3/2 Explanation: 5/4 + (−11/4) = (5 + (−11))/4 = (5 - 11)/4 = -6/4 = -3/2 (ii) 5/3 + 3/5 Answer: 34/15 Explanation: 5/3 + 3/5 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 (iii) - 9/10 + 22/15 Answer: 17/30 Explanation: To add the two rational numbers, the denominator should be the same. 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 (iv) - 3/-11 + 5/9 Answer: 82/99 Explanation: - 3/-11 can also be represented as 3/11. It is because the negative sign is present both in the numerator and the denominator. 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 (v) - 8/19 + (- 2)/57 Answer: -26/57 Explanation: To add the two rational numbers, the denominator should be the same. 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 (vi) −2/3 + 0 Answer: −2/3 Explanation: 0 added to any number results in the same number. Hence, −2/3 + 0 = −2/3 (vii) −2 1/3 + 4 3/5 Answer: 34/15 Explanation: To compare the mixed rational numbers, let's convert them into improper fractions. −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 2. Find: (i) 7/24 - 17/36 Answer: -13/72 Explanation: To subtract the two rational numbers, the denominator should be the same. 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 (ii) 5/63 - (−6)/21 Answer: 23/63 Explanation: To subtract the two rational numbers, the denominator should be the same. 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 (iii) −6/13 - (−7/15) Answer: 1/195 Explanation: To subtract the two rational numbers, the denominator should be the same. 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 (iv) −3/8 - 7/11 Answer: −89/88 Explanation: To subtract the two rational numbers, the denominator should be the same. 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 (v) −2 1/9 - 6 Answer: - 73/9 Explanation: Let's convert the mixed rational number to improper rational number. −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 3. Find the product: (i) 9/2 × (−7/4) Answer: −63/8 Explanation: 9/2 × (−7/4) = (9 × −7)/ (2 × 4) = −63/8 (ii) 3/10 × (−9) Answer: −27/10 Explanation: 3/10 × (−9) = (3 × (−9))/10 = −27/10 (iii) - 6/5 × 9/11 Answer: - 54/55 Explanation: - 6/5 × 9/11 = (- 6 × 9)/ (5 × 11) = - 54/55 (iv) 3/7 × (-2/5) Answer: -6/35 Explanation: 3/7 × (-2/5) = (3 × (-2))/ (7 × 5) = -6/35 (v) 3/11 × 2/5 Answer: 6/55 Explanation: 3/11 × 2/5 = (3 × 2)/ (11 × 5) = 6/55 (vi) 3/−5 × −5 /3 Answer: 1 Explanation: 3/−5 × −5 /3 = (3 × −5)/ (−5 × 3) = (−15)/ (−15) = 15/15 = 1 4. Find the value of: (i) (−4) ÷ 2/3 Answer: −6 Explanation: (−4) ÷ 2/3 = (−4) × (Reciprocal of 2/3) = (−4) × 3/2 = ((−4) × 3)/2 = −6 (ii) - 3/5 ÷ 2 Answer: - 3/10 Explanation: - 3/5 ÷ 2 = - 3/5 × (Reciprocal of 2) = - 3/5 × 1/2 = (- 3 × 1)/ (5 × 2) = (- 3)/10 = - 3/10 (iii) −4/5 ÷ − (3) Answer: 4/15 Explanation: −4/5 ÷ − (3) = −4/5 × (Reciprocal of − (3)) = −4/5 × −1/3 = (−4 × −1)/ (5 × 3) = 4/15 (iv) - 1/8 ÷ 3/4 Answer: -1/6 Explanation: - 1/8 ÷ 3/4 = - 1/8 × (Reciprocal of 3/4) = - 1/8 × 4/3 = (-1 × 4)/ (8 × 3) = -1/6 (v) −2/13 ÷ 1/7 Answer: −14/13 Explanation: −2/13 ÷ 1/7 = −2/13 × (Reciprocal of 1/7) = −2/13 × 7/1 = (−2 × 7)/ (13 × 1) = −14/13 (vi) −7/12 ÷ (−2/13) Answer: 91/24 Explanation: −7/12 ÷ (−2/13) = −7/12 × (Reciprocal of (−2/13)) = −7/12 × −13/2 = (−7 × −13)/ (12 × 2) = 91/24 (vii) 3/13 ÷ (−4/65) Answer: −15/4 Explanation: 3/13 ÷ (−4/65) = 3/13 × (Reciprocal of ((−4/65)) = 3/13 × −65/4 = (3 × −65)/ (13 × 4) (13 × 5 = 65) = (3 × −5)/ (1 × 4) = −15/4 |