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-C

Q1: Add the following rational numbers:

i. \(\frac{5}{11}\) and \(\frac{4}{11}\)

Step 1: Since denominators are same, add numerators: \[ \frac{5}{11} + \frac{4}{11} = \frac{5 + 4}{11} = \frac{9}{11} \] Answer: \(\frac{9}{11}\)

ii. \(\frac{-3}{8}\) and \(\frac{5}{8}\)

Step 1: Same denominators, add numerators: \[ \frac{-3}{8} + \frac{5}{8} = \frac{-3 + 5}{8} = \frac{2}{8} = \frac{1}{4} \] Answer: \(\frac{1}{4}\)

iii. \(\frac{-6}{13}\) and \(\frac{8}{13}\)

Step 1: Add numerators: \[ \frac{-6}{13} + \frac{8}{13} = \frac{-6 + 8}{13} = \frac{2}{13} \] Answer: \(\frac{2}{13}\)

iv. \(\frac{-8}{15}\) and \(\frac{-7}{15}\)

Step 1: Add numerators: \[ \frac{-8}{15} + \frac{-7}{15} = \frac{-8 – 7}{15} = \frac{-15}{15} = -1 \] Answer: \(-1\)

v. \(\frac{-13}{20}\) and \(\frac{17}{20}\)

Step 1: Add numerators: \[ \frac{-13}{20} + \frac{17}{20} = \frac{-13 + 17}{20} = \frac{4}{20} = \frac{1}{5} \] Answer: \(\frac{1}{5}\)

vi. \(\frac{-3}{8}\) and \(\frac{5}{-8}\)

Step 1: Note \(\frac{5}{-8} = \frac{-5}{8}\). Add numerators: \[ \frac{-3}{8} + \frac{-5}{8} = \frac{-3 – 5}{8} = \frac{-8}{8} = -1 \] Answer: \(-1\)

Q2: Add the following rational numbers:

i. \(\frac{-2}{3}\) and \(\frac{3}{4}\)

Step 1: Find LCM of denominators 3 and 4 = 12
Step 2: Express both fractions with denominator 12: \[ \frac{-2}{3} = \frac{-2 \times 4}{3 \times 4} = \frac{-8}{12}, \quad \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \] Step 3: Add numerators: \[ \frac{-8}{12} + \frac{9}{12} = \frac{-8 + 9}{12} = \frac{1}{12} \] Answer: \(\frac{1}{12}\)

ii. \(\frac{-4}{9}\) and \(\frac{5}{6}\)

Step 1: LCM of 9 and 6 = 18
Step 2: Convert fractions: \[ \frac{-4}{9} = \frac{-4 \times 2}{9 \times 2} = \frac{-8}{18}, \quad \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \] Step 3: Add numerators: \[ \frac{-8}{18} + \frac{15}{18} = \frac{-8 + 15}{18} = \frac{7}{18} \] Answer: \(\frac{7}{18}\)

iii. \(\frac{-5}{18}\) and \(\frac{11}{27}\)

Step 1: LCM of 18 and 27 = 54
Step 2: Convert fractions: \[ \frac{-5}{18} = \frac{-5 \times 3}{18 \times 3} = \frac{-15}{54}, \quad \frac{11}{27} = \frac{11 \times 2}{27 \times 2} = \frac{22}{54} \] Step 3: Add numerators: \[ \frac{-15}{54} + \frac{22}{54} = \frac{-15 + 22}{54} = \frac{7}{54} \] Answer: \(\frac{7}{54}\)

iv. \(\frac{-7}{12}\) and \(\frac{-5}{24}\)

Step 1: LCM of 12 and 24 = 24
Step 2: Convert fractions: \[ \frac{-7}{12} = \frac{-7 \times 2}{12 \times 2} = \frac{-14}{24}, \quad \frac{-5}{24} = \frac{-5}{24} \] Step 3: Add numerators: \[ \frac{-14}{24} + \frac{-5}{24} = \frac{-14 – 5}{24} = \frac{-19}{24} \] Answer: \(\frac{-19}{24}\)

v. \(\frac{-1}{18}\) and \(\frac{-7}{27}\)

Step 1: LCM of 18 and 27 = 54
Step 2: Convert fractions: \[ \frac{-1}{18} = \frac{-1 \times 3}{18 \times 3} = \frac{-3}{54}, \quad \frac{-7}{27} = \frac{-7 \times 2}{27 \times 2} = \frac{-14}{54} \] Step 3: Add numerators: \[ \frac{-3}{54} + \frac{-14}{54} = \frac{-3 – 14}{54} = \frac{-17}{54} \] Answer: \(\frac{-17}{54}\)

vi. \(\frac{21}{-4}\) and \(\frac{-11}{8}\)

Step 1: Write \(\frac{21}{-4} = \frac{-21}{4}\). LCM of 4 and 8 = 8
Step 2: Convert fractions: \[ \frac{-21}{4} = \frac{-21 \times 2}{4 \times 2} = \frac{-42}{8}, \quad \frac{-11}{8} = \frac{-11}{8} \] Step 3: Add numerators: \[ \frac{-42}{8} + \frac{-11}{8} = \frac{-42 – 11}{8} = \frac{-53}{8} \] Answer: \(\frac{-53}{8}\)

Q3: Evaluate:

i. \(\frac{2}{-3} + \frac{-4}{9}\)

Step 1: Rewrite \(\frac{2}{-3} = \frac{-2}{3}\)
Step 2: Find LCM of denominators 3 and 9 = 9
Step 3: Convert fractions: \[ \frac{-2}{3} = \frac{-2 \times 3}{3 \times 3} = \frac{-6}{9}, \quad \frac{-4}{9} = \frac{-4}{9} \] Step 4: Add numerators: \[ \frac{-6}{9} + \frac{-4}{9} = \frac{-6 – 4}{9} = \frac{-10}{9} \] Answer: \(\frac{-10}{9}\)

ii. \(\frac{-1}{2} + \frac{-3}{4}\)

Step 1: LCM of 2 and 4 = 4
Step 2: Convert fractions: \[ \frac{-1}{2} = \frac{-1 \times 2}{2 \times 2} = \frac{-2}{4}, \quad \frac{-3}{4} = \frac{-3}{4} \] Step 3: Add numerators: \[ \frac{-2}{4} + \frac{-3}{4} = \frac{-2 – 3}{4} = \frac{-5}{4} \] Answer: \(\frac{-5}{4}\)

iii. \(\frac{7}{-9} + \frac{-5}{6}\)

Step 1: Rewrite \(\frac{7}{-9} = \frac{-7}{9}\)
Step 2: LCM of 9 and 6 = 18
Step 3: Convert fractions: \[ \frac{-7}{9} = \frac{-7 \times 2}{9 \times 2} = \frac{-14}{18}, \quad \frac{-5}{6} = \frac{-5 \times 3}{6 \times 3} = \frac{-15}{18} \] Step 4: Add numerators: \[ \frac{-14}{18} + \frac{-15}{18} = \frac{-14 – 15}{18} = \frac{-29}{18} \] Answer: \(\frac{-29}{18}\)

iv. \(2 + \frac{-3}{4}\)

Step 1: Write 2 as \(\frac{8}{4}\) (since denominator is 4)
Step 2: Add fractions: \[ \frac{8}{4} + \frac{-3}{4} = \frac{8 – 3}{4} = \frac{5}{4} \] Answer: \(\frac{5}{4}\)

v. \(3 + \frac{-5}{6}\)

Step 1: Write 3 as \(\frac{18}{6}\)
Step 2: Add fractions: \[ \frac{18}{6} + \frac{-5}{6} = \frac{18 – 5}{6} = \frac{13}{6} \] Answer: \(\frac{13}{6}\)

vi. \(-4 + \frac{2}{3}\)

Step 1: Write -4 as \(\frac{-12}{3}\)
Step 2: Add fractions: \[ \frac{-12}{3} + \frac{2}{3} = \frac{-12 + 2}{3} = \frac{-10}{3} \] Answer: \(\frac{-10}{3}\)

Q4: Evaluate:

i. \(\frac{-3}{8} + \frac{5}{8} + \frac{7}{8}\)

Step 1: All denominators are same (8), so add numerators: \[ \frac{-3 + 5 + 7}{8} = \frac{9}{8} \] Answer: \(\frac{9}{8}\)

ii. \(\frac{11}{3} + \frac{-5}{3} + \frac{-2}{3}\)

Step 1: All denominators are same (3), so add numerators: \[ \frac{11 – 5 – 2}{3} = \frac{4}{3} \] Answer: \(\frac{4}{3}\)

iii. \(-1 + \frac{2}{-3} + \frac{5}{6}\)

Step 1: Rewrite \(\frac{2}{-3} = \frac{-2}{3}\)
Step 2: Express -1 as fraction with denominator 6: \[ -1 = \frac{-6}{6} \] Step 3: Find LCM of denominators 3 and 6 = 6
Step 4: Convert all to denominator 6: \[ \frac{-2}{3} = \frac{-4}{6}, \quad \frac{5}{6} = \frac{5}{6}, \quad -1 = \frac{-6}{6} \] Step 5: Add numerators: \[ \frac{-6}{6} + \frac{-4}{6} + \frac{5}{6} = \frac{-6 – 4 + 5}{6} = \frac{-5}{6} \] Answer: \(\frac{-5}{6}\)

iv. \(\frac{7}{26} + \frac{-11}{13} + 2\)

Step 1: Express 2 as fraction with denominator 26: \[ 2 = \frac{52}{26} \] Step 2: Convert \(\frac{-11}{13}\) to denominator 26: \[ \frac{-11}{13} = \frac{-22}{26} \] Step 3: Add all fractions: \[ \frac{7}{26} + \frac{-22}{26} + \frac{52}{26} = \frac{7 – 22 + 52}{26} = \frac{37}{26} \] Answer: \(\frac{37}{26}\)

v. \(3 + \frac{-7}{8} + \frac{-3}{4}\)

Step 1: Express 3 as fraction with denominator 8: \[ 3 = \frac{24}{8} \] Step 2: Convert \(\frac{-3}{4}\) to denominator 8: \[ \frac{-3}{4} = \frac{-6}{8} \] Step 3: Add all fractions: \[ \frac{24}{8} + \frac{-7}{8} + \frac{-6}{8} = \frac{24 – 7 – 6}{8} = \frac{11}{8} \] Answer: \(\frac{11}{8}\)

vi. \(\frac{-13}{8} + \frac{7}{16} + \frac{-3}{4}\)

Step 1: Find LCM of denominators 8, 16, and 4 = 16
Step 2: Convert all fractions to denominator 16: \[ \frac{-13}{8} = \frac{-26}{16}, \quad \frac{7}{16} = \frac{7}{16}, \quad \frac{-3}{4} = \frac{-12}{16} \] Step 3: Add numerators: \[ \frac{-26}{16} + \frac{7}{16} + \frac{-12}{16} = \frac{-26 + 7 – 12}{16} = \frac{-31}{16} \] Answer: \(\frac{-31}{16}\)

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