Rational Numbers

rational numbers class 7

Step by Step solutions of RS Aggarwal ICSE Class-7 Maths chapter 4- Rational Numbers by Goyal Brothers Prakashan is provided.

Table Of Contents
  1. Q1: What are rational numbers? Give four examples of each of positive rationals and negative rationals. Give an example of a rational number which is neither positive nor negative.
  2. Q2: Which of the following are rational numbers?
  3. Q3: Write down the numerator and the denominator of each the following rational numbers:
  4. Q4: Which of the following are positive rational numbers?
  5. Q5: Which of the following are negative rational numbers?
  6. Q6: Find four rational numbers equivalent to each of the following:
  7. Q7: Write each of the following rational numbers with positive denominator:
  8. Q8: Express (frac{4}{9}) as a rational number with numerator:
  9. Q9: Express (frac{3}{8}) as a rational number with denominator
  10. Q10: Express (frac{-6}{11}) as a rational numerator
  11. Q11: Express (frac{2}{-7}) as a rational number with denominator
  12. Q12: Express (frac{-48}{36}) as a rational with numerator
  13. Q13: Express (frac{78}{-117}) as a rational with numerator
  14. Q14: Write each of the following rational numbers in standard form:
  15. Q15: Find the value of x such that:
  16. Q16: State whether the given statement is true or false:
  17. Q1: Which of the two rational numbers is greater in each of the following pairs?
  18. Q2: Fill in the blanks with the correct symbol out of >, = or
  19. Q3: Arrange the following rational numbers in ascending order:
  20. Q4: Arrange the following rational numbers in descending order:
  21. Q1: Add the following rational numbers:
  22. Q2: Add the following rational numbers:
  23. Q3: Evaluate:
  24. Q4: Evaluate:
  25. Q1: Find the additive inverse of:
  26. Q2: Subtract:
  27. Q3: Evaluate:
  28. Q4: The sum of two rational numbers is (frac{-5}{8}). If one of them is (frac{7}{16}), find the other.
  29. Q5: The sum of two rational numbers is -4. If one of them is (frac{-3}{5}), find the other.
  30. Q6: The sum of two rational numbers is (frac{-5}{4}). If one of them is -3, find the other.
  31. Q7: What should be added to (frac{-5}{6}) to get (frac{-2}{3})?
  32. Q8: What should be added to (frac{2}{5}) get -1?
  33. Q9: What should be subtracted from (frac{-3}{4}) to get (frac{-5}{6})?
  34. Q10: What should be subtracted from (frac{-2}{3}) to get 1?
  35. Q1: Multiply:
  36. Q2: Simplify:
  37. Q3: Simplify:
  38. Q4: Simplify:
  39. Q5: Find the cost of (3frac{1}{3}) kg of rice at ₹(40frac{1}{2}) per kg.
  40. Q6: Find the distance covered by a car in (2frac{2}{5}) hours at a speed of (46frac{2}{3}) km per hour.
  41. Q7: Write the multiplicative inverse of:
  42. Q1: Find the multiplicative inverse (or reciprocal) of each of the following rational numbers:
  43. Q2: Evaluate:
  44. Q3: The product of two rational numbers is (frac{2}{5}). If one of them is (frac{-8}{25}), find the other.
  45. Q4: The product of two rational numbers is (frac{-2}{3}). If one of them is (frac{16}{39}), find the other.
  46. Q5: By what rational number should (frac{-9}{35}) be multiplied to get (frac{3}{5})?
  47. Q6: By what rational should (frac{25}{8}) multiplied to get (frac{-20}{7})?
  48. Q7: The cost of 17 pencils is ₹(59frac{1}{2}). Find the cost of each pencil.
  49. Q8: The cost of 20 metres of ribbon is ₹335. Find the cost of each metre of it.
  50. Q9: How many pieces, each of length (2frac{3}{4}) m, can be cut from a rope of length 66 m?
  51. Q10: Fill in the blanks:
  52. Q1: Represent (frac{2}{3}) on the number line
  53. Q2: Represent (-frac{5}{7}) on the number line
  54. Q3: Represent (frac{1}{6}) on the number line
  55. Q4: Represent (-frac{3}{8}) on the number line
  56. Q5: Represent (frac{22}{7}) on the number line
  57. Q6: Represent (frac{23}{-5}) on the number line
  58. Q7: Represent (-frac{3}{4}) on the number line
  59. Q8: Represent (frac{-12}{5}) on the number line
  60. Q9: Represent (frac{13}{6}) on the number line
  61. Q1: Without actual division, show that each of the rational numbers given below is expressible as a terminating decimal:
  62. Q2: By actual division, express each of the following rational numbers as a terminating decimal:
  63. Q3: Without actual division, show that each of the rational numbers given below is expressible as a repeating decimal:
  64. Q4: By actual division, express each of the following as a repeating decimal:
  65. Q5: Fill in the blanks:
  66. Q1: The additive inverse of (frac{5}{9}) is
  67. Q2: The rational number (frac{32}{-40}) expressed in standard form is
  68. Q3: What should be added to (frac{-3}{16}) get (frac{5}{8})?
  69. Q4: The multiplicative inverse of (frac{-3}{7}) is:
  70. Q5: The sum of (-frac{1}{3}) and its multiplicative is
  71. Q6: The product of (-frac{1}{3}) and its additive is
  72. Q7: Which of the rational numbers is equivalent to (frac{-2}{7})?
  73. Q8: If (3frac{3}{4}) m of cloth is required for one suit, then how many suits be prepared from 30 m of cloth?
  74. Q1: Fill in the blanks:
  75. Q2: Write true (T) or false (F):

Exercise: 4-H

Q1: Without actual division, show that each of the rational numbers given below is expressible as a terminating decimal:

Step: A rational number \(\frac{p}{q}\) is expressible as a terminating decimal if the denominator \(q\) has no prime factor other than 2 and/or 5, i.e., the denominator must be of the form \(2^m \times 5^n\), where \(m, n\) are non-negative integers.

i. \(\frac{11}{16}\)

Step 1: Prime factorization of 16 \[ 16 = 2^4 \] Only prime factor is 2 → Terminating
Answer: \(\frac{11}{16}\) is a terminating decimal.

ii. \(\frac{17}{20}\)

Step 1: Prime factorization of 20 \[ 20 = 2^2 \times 5 \] Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{17}{20}\) is a terminating decimal.

iii. \(\frac{44}{125}\)

Step 1: Prime factorization of 125 \[ 125 = 5^3 \] Only prime factor is 5 → Terminating
Answer: \(\frac{44}{125}\) is a terminating decimal.

iv. \(\frac{9}{80}\)

Step 1: Prime factorization of 80 \[ 80 = 2^4 \times 5 \] Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{9}{80}\) is a terminating decimal.

v. \(\frac{123}{200}\)

Step 1: Prime factorization of 200 \[ 200 = 2^3 \times 5^2 \] Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{123}{200}\) is a terminating decimal.

vi. \(\frac{129}{320}\)

Step 1: Prime factorization of 320 \[ 320 = 2^6 \times 5 \] Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{129}{320}\) is a terminating decimal.

vii. \(\frac{431}{500}\)

Step 1: Prime factorization of 500 \[ 500 = 2^2 \times 5^3 \] Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{431}{500}\) is a terminating decimal.

viii. \(\frac{807}{1250}\)

Step 1: Prime factorization of 1250 \[ 1250 = 2 \times 5^4 \] Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{807}{1250}\) is a terminating decimal.

Q2: By actual division, express each of the following rational numbers as a terminating decimal:

i. \(\frac{11}{8}\)

    1.375
   _______
8 | 11.000
   - 8
    -----
     30
   - 24
    -----
      60
    - 56
    -----
       40
     - 40
      -----
        0

Answer: 1.375

ii. \(\frac{23}{16}\)

     1.4375
    ________
16 | 23.0000
    -16
     ----
      70
    - 64
     ----
       60
     - 48
     ----
       120
      -112
       ----
         80
       - 80
        ----
         0

Answer: 1.4375

iii. \(\frac{76}{125}\)

      0.608
     _______
125 | 76.000
      -0
      -----
       760
     - 750
      -----
        100
     -  000
       -----
        1000
      - 1000
        ----
         0

Answer: 0.608

iv. \(\frac{103}{40}\)

      2.575
    _______
40 | 103.000
     -80
     -----
      230
     -200
      -----
       300
     - 280
     -----
        200
      - 200
       -----
         0

Answer: 2.575

v. \(\frac{17}{80}\)

     0.2125
    _______
80 | 17.0000
    - 0
     -----
     170
    -160
     -----
      100
     - 80
      -----
       200
     - 160
     -----
        400
      - 400
       -----
         0

Answer: 0.2125

vi. \(\frac{2}{25}\)

  0.08
    ______
25 | 2.00
    - 0
    ----
     200
   - 200
    ----
      0

Answer: 0.08

vii. \(\frac{1}{125}\)

     0.008
     _______
125 | 1.000
      - 0
      ----
      100
    - 000
      ----
      1000
    - 1000
      ----
        0

Answer: 0.008

viii. \(\frac{309}{1250}\)

       0.2472
      __________
1250 | 309.0000
       -0
       -----
       3090
     - 2500
       -----
        5900
      - 5000
        -----
         9000
       - 8750
        -----
          2500
        - 2500
          -----
            0

Answer: 0.2472

Q3: Without actual division, show that each of the rational numbers given below is expressible as a repeating decimal:

i. \(\frac{23}{24}\)

Step 1: Prime factorisation of denominator: \(24 = 2^3 \times 3\)
Step 2: The denominator contains 3 (a prime other than 2 or 5).
Answer: Repeating decimal

ii. \(\frac{79}{30}\)

Step 1: Prime factorisation of 30 = \(2 \times 3 \times 5\)
Step 2: The denominator contains 3 (other than 2, 5).
Answer: Repeating decimal

iii. \(\frac{100}{9}\)

Step 1: Prime factorisation of 9 = \(3^2\)
Step 2: Only prime factor is 3 (not 2 or 5).
Answer: Repeating decimal

iv. \(\frac{205}{27}\)

Step 1: Prime factorisation of 27 = \(3^3\)
Step 2: Contains 3 only, so repeating decimal.
Answer: Repeating decimal

v. \(\frac{461}{60}\)

Step 1: Prime factorisation of 60 = \(2^2 \times 3 \times 5\)
Step 2: Contains 3 → repeating decimal.
Answer: Repeating decimal

vi. \(\frac{1003}{112}\)

Step 1: Prime factorisation of 112 = \(2^4 \times 7\)
Step 2: Contains 7 → repeating decimal.
Answer: Repeating decimal

vii. \(\frac{127}{225}\)

Step 1: Prime factorisation of 225 = \(3^2 \times 5^2\)
Step 2: Contains 3 → repeating decimal.
Answer: Repeating decimal

viii. \(\frac{219}{440}\)

Step 1: Prime factorisation of 440 = \(2^3 \times 5 \times 11\)
Step 2: Contains 11 → repeating decimal.
Answer: Repeating decimal

Q4: By actual division, express each of the following as a repeating decimal:

i. \(\frac{103}{9}\)

Step 1: Divide 103 by 9.

     11.4444...
   _____________
9 | 103.000000
   - 9
   ______
     13
    - 9
   ______
      40
     -36
   ______
       40

Answer: \(\frac{103}{9} = 11.\overline{4}\)

ii. \(\frac{7}{12}\)

Step 1: Divide 7 by 12.

     0.583333...
    _______________
12 | 7.000000
    -0
    -----
     70
   - 60
   -----
     100
    - 96
    -----
       40
     - 36
      -----
        40

Answer: \(\frac{7}{12} = 0.58\overline{3}\)

iii. \(\frac{101}{15}\)

Step 1: Divide 101 by 15.

      6.7333...
    ______________
15 | 101.000
    - 90
    -----
      110
     -105
     -----
        50
      - 45
       -----
         50

Answer: \(\frac{101}{15} = 6.7\overline{3}\)

iv. \(\frac{303}{11}\)

Step 1: Divide 303 by 11.

      27.545454...
    ________________
11 | 303.000000
    -22
    -----
      83
     -77
     -----
       60
      -55
      -----
        50
       -44
       -----
         60

Answer: \(\frac{303}{11} = 27.\overline{54}\)

v. \(\frac{212}{143}\)

Step 1: Divide 212 by 143.

      1.482517482517...
     ________________
143 | 212.000000
     -143
     -----
       690
      -572
      -----
       1180
      -1144
       -----
         360
        -286
        -----
          740
         -715
         -----
           250
          -143
          -----
           1070
          -1001
           -----
             690

Answer: \(\frac{212}{143} = 1.\overline{482517}\)

vi. \(\frac{16}{7}\)

Step 1: Divide 16 by 7.

     2.285714285714...
  ______________________
7 | 16.000000000
   -14
    -----
     20
    -14
    -----
      60
     -56
     -----
       40
      -35
      -----
        50
       -49
       -----
         10
         -7
        -----
          30
         -28
         -----
           20

Answer: \(\frac{16}{7} = 2.\overline{285714}\)

vii. \(\frac{227}{30}\)

Step 1: Divide 227 by 30.

     7.56666...
   _______________
30 | 227.0000
    -210
    -----
      170
     -150
     -----
       200
      -180
      -----
        200

Answer: \(\frac{227}{30} = 7.5\overline{6}\)

viii. \(\frac{2000}{33}\)

Step 1: Divide 2000 by 33.

     60.606060...
    ___________________
33 | 2000.0000
    -198
     -----
       20
      - 0
      -----
       200
      -198
      -----
         20
        - 0
        -----
         200
        -198
        -----
           20

Answer: \(\frac{2000}{33} = 60.\overline{60}\)

Q5: Fill in the blanks:

i. \(\frac{2}{3} =\) ____

Step 1: Divide 2 by 3.

     0.6666...
   ____________
3 | 2.000000
   -0
   -----
    20
   -18
   -----
     20
    -18
    -----
      20

Answer: \(\frac{2}{3} = 0.\overline{6}\)

ii. \(\frac{11}{30} =\) ____

Step 1: Divide 11 by 30.

     0.3666...
    __________
30 | 11.0000
    - 0
    -----
     110
     -90
    -----
      200
     -180
     -----
       200

Answer: \(\frac{11}{30} = 0.3\overline{6}\)

iii. \(\frac{13}{11} =\) ____

Step 1: Divide 13 by 11.

     1.181818...
    ___________
11 | 13.00000
    -11
    -----
      20
    - 11
     -----
       90
     - 88
      -----
        20

Answer: \(\frac{13}{11} = 1.\overline{18}\)

iv. \(\frac{23}{55} =\) ____

Step 1: Divide 23 by 55.

     0.418181...
    _______________
55 | 23.00000
    - 0
    -----
     230
    -220
    -----
      100
     - 55
     -----
       450
      -440
      -----
        100

Answer: \(\frac{23}{55} = 0.4\overline{18}\)

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