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-D

Q1: Find the additive inverse of:

i. 9

Additive Inverse: The number which when added to 9 gives 0 is: -9
Answer: -9

ii. -11

Additive Inverse: The number which when added to -11 gives 0 is: 11
Answer: 11

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

Additive Inverse: The number which when added to \(\frac{-8}{13}\) gives 0 is: \(\frac{8}{13}\)
Answer: \(\frac{8}{13}\)

iv. \(\frac{5}{-6}\)

Step 1: Rewrite with negative sign in numerator: \(\frac{-5}{6}\)
Step 2: Its additive inverse is: \(\frac{5}{6}\)
Answer: \(\frac{5}{6}\)

v. 0

Additive Inverse: The number which when added to 0 gives 0 is: \(0\)
Answer: 0

Q2: Subtract:

i. \(\frac{3}{5}\) from \(\frac{1}{2}\)

Step 1: Write as: \(\frac{1}{2} – \frac{3}{5}\)
Step 2: LCM of 2 and 5 is 10 \[ \frac{1}{2} = \frac{5}{10},\quad \frac{3}{5} = \frac{6}{10} \\ \frac{5}{10} – \frac{6}{10} = \frac{-1}{10} \] Answer: \(\frac{-1}{10}\)

ii. \(\frac{-4}{7}\) from \(\frac{2}{3}\)

Step 1: Write as: \(\frac{2}{3} – \left(\frac{-4}{7}\right)\) \[ = \frac{2}{3} + \frac{4}{7} \] Step 2: LCM of 3 and 7 is 21 \[ \frac{2}{3} = \frac{14}{21},\quad \frac{4}{7} = \frac{12}{21} \\ \frac{14}{21} + \frac{12}{21} = \frac{26}{21} \] Answer: \(\frac{26}{21}\)

iii. \(\frac{-5}{6}\) from \(\frac{-3}{4}\)

Step 1: Write as: \(\frac{-3}{4} – \left(\frac{-5}{6}\right)\) \[ = \frac{-3}{4} + \frac{5}{6} \] Step 2: LCM of 4 and 6 is 12 \[ \frac{-3}{4} = \frac{-9}{12},\quad \frac{5}{6} = \frac{10}{12} \\ \frac{-9}{12} + \frac{10}{12} = \frac{1}{12} \] Answer: \(\frac{1}{12}\)

iv. \(\frac{-7}{9}\) from \(0\)

Step 1: Write as: \(0 – \left(\frac{-7}{9}\right) = \frac{7}{9}\)
Answer: \(\frac{7}{9}\)

v. \(4\) from \(\frac{-6}{11}\)

Step 1: Write as: \(\frac{-6}{11} – 4\)
Step 2: Convert 4 to fraction with denominator 11: \[ 4 = \frac{44}{11} \\ \frac{-6}{11} – \frac{44}{11} = \frac{-50}{11} \] Answer: \(\frac{-50}{11}\)

vi. \(\frac{3}{8}\) from \(\frac{-5}{6}\)

Step 1: Write as: \(\frac{-5}{6} – \frac{3}{8}\)
Step 2: LCM of 6 and 8 is 24 \[ \frac{-5}{6} = \frac{-20}{24},\quad \frac{3}{8} = \frac{9}{24} \\ \frac{-20}{24} – \frac{9}{24} = \frac{-29}{24} \] Answer: \(\frac{-29}{24}\)

Q3: Evaluate:

i. \(\frac{5}{6} – \frac{7}{8}\)

Step 1: LCM of 6 and 8 is 24 \[ \frac{5}{6} = \frac{20}{24},\quad \frac{7}{8} = \frac{21}{24} \\ \frac{20}{24} – \frac{21}{24} = \frac{-1}{24} \] Answer: \(\frac{-1}{24}\)

ii. \(\frac{5}{12} – \frac{17}{18}\)

Step 1: LCM of 12 and 18 is 36 \[ \frac{5}{12} = \frac{15}{36},\quad \frac{17}{18} = \frac{34}{36} \\ \frac{15}{36} – \frac{34}{36} = \frac{-19}{36} \] Answer: \(\frac{-19}{36}\)

iii. \(\frac{11}{15} – \frac{13}{20}\)

Step 1: LCM of 15 and 20 is 60 \[ \frac{11}{15} = \frac{44}{60},\quad \frac{13}{20} = \frac{39}{60} \\ \frac{44}{60} – \frac{39}{60} = \frac{5}{60} = \frac{1}{12} \] Answer: \(\frac{1}{12}\)

iv. \(\frac{-5}{9} – \frac{-2}{3}\)

Step 1: \(\frac{-5}{9} + \frac{2}{3}\) \[ \frac{-5}{9} + \frac{2}{3} = \frac{-5}{9} + \frac{6}{9} = \frac{1}{9} \] Answer: \(\frac{1}{9}\)

v. \(\frac{6}{11} – \frac{-3}{4}\)

Step 1: \(\frac{6}{11} + \frac{3}{4}\)
LCM of 11 and 4 is 44 \[ \frac{6}{11} = \frac{24}{44},\quad \frac{3}{4} = \frac{33}{44} \] \[ \frac{24}{44} + \frac{33}{44} = \frac{57}{44} \] Answer: \(\frac{57}{44}\)

vi. \(\frac{-2}{3} – \frac{3}{4}\)

Step 1: LCM of 3 and 4 is 12 \[ \frac{-2}{3} = \frac{-8}{12},\quad \frac{3}{4} = \frac{9}{12} \\ \frac{-8}{12} – \frac{9}{12} = \frac{-17}{12} \] Answer: \(\frac{-17}{12}\)

Q4: The sum of two rational numbers is \(\frac{-5}{8}\). If one of them is \(\frac{7}{16}\), find the other.

Let the other rational number be \(x\).
According to the question: \[ x + \frac{7}{16} = \frac{-5}{8} \]Step 1: Transpose \(\frac{7}{16}\) to the RHS \[ x = \frac{-5}{8} – \frac{7}{16} \]Step 2: Take LCM of 8 and 16
LCM of 8 and 16 is 16 \[ \frac{-5}{8} = \frac{-10}{16} \] So, \[ x = \frac{-10}{16} – \frac{7}{16} = \frac{-17}{16} \]Answer: The other number is \(\frac{-17}{16}\)

Q5: The sum of two rational numbers is -4. If one of them is \(\frac{-3}{5}\), find the other.

Let the other rational number be \(x\).
According to the question: \[ x + \frac{-3}{5} = -4 \]Step 1: Transpose \(\frac{-3}{5}\) to the RHS \[ x = -4 – \left(-\frac{3}{5}\right) \]Step 2: Simplify the signs \[ x = -4 + \frac{3}{5} \]Step 3: Express \(-4\) with denominator 5 \[ -4 = \frac{-20}{5} \\ x = \frac{-20}{5} + \frac{3}{5} = \frac{-17}{5} \]Answer: The other number is \(\frac{-17}{5}\)

Q6: The sum of two rational numbers is \(\frac{-5}{4}\). If one of them is -3, find the other.

Let the other rational number be \(x\).
According to the question: \[ x + (-3) = \frac{-5}{4} \]Step 1: Transpose -3 to the RHS \[ x = \frac{-5}{4} + 3 \]Step 2: Convert 3 to a rational number with denominator 4 \[ 3 = \frac{12}{4} \\ x = \frac{-5}{4} + \frac{12}{4} \]Step 3: Add the rational numbers \[ x = \frac{-5 + 12}{4} = \frac{7}{4} \]Answer: The other number is \(\frac{7}{4}\)

Q7: What should be added to \(\frac{-5}{6}\) to get \(\frac{-2}{3}\)?

Let the required number be \(x\).
According to the question: \[ x + \frac{-5}{6} = \frac{-2}{3} \]Step 1: Transpose \(\frac{-5}{6}\) to the RHS \[ x = \frac{-2}{3} – \frac{-5}{6} \]Step 2: Replace minus-minus with plus \[ x = \frac{-2}{3} + \frac{5}{6} \]Step 3: Take LCM of denominators LCM of 3 and 6 is 6. \[ \frac{-2}{3} = \frac{-4}{6} \]Step 4: Add the rational numbers \[ x = \frac{-4}{6} + \frac{5}{6} = \frac{1}{6} \]Answer: \(\frac{1}{6}\) should be added to \(\frac{-5}{6}\) to get \(\frac{-2}{3}\)

Q8: What should be added to \(\frac{2}{5}\) get -1?

Let the required number be \(x\).
According to the question: \[ x + \frac{2}{5} = -1 \]Step 1: Transpose \(\frac{2}{5}\) to the RHS \[ x = -1 – \frac{2}{5} \]Step 2: Express -1 with denominator 5 \[ -1 = \frac{-5}{5} \] So, \[ x = \frac{-5}{5} – \frac{2}{5} \]Step 3: Subtract the numerators \[ x = \frac{-5 – 2}{5} = \frac{-7}{5} \]Answer: \(\frac{-7}{5}\) should be added to \(\frac{2}{5}\) to get -1

Q9: What should be subtracted from \(\frac{-3}{4}\) to get \(\frac{-5}{6}\)?

Let the required number be \(x\).
According to the question: \[ \frac{-3}{4} – x = \frac{-5}{6} \]Step 1: Transpose \(x\) to RHS and \(\frac{-5}{6}\) to LHS \[ x = \frac{-3}{4} – \left(\frac{-5}{6}\right) \\ x = \frac{-3}{4} + \frac{5}{6} \]Step 2: Find LCM of denominators 4 and 6 \[ \text{LCM of 4 and 6} = 12 \]Step 3: Convert to like denominators \[ \frac{-3}{4} = \frac{-9}{12}, \quad \frac{5}{6} = \frac{10}{12} \]Step 4: Add the fractions \[ x = \frac{-9}{12} + \frac{10}{12} = \frac{1}{12} \]Answer: \(\frac{1}{12}\) should be subtracted from \(\frac{-3}{4}\) to get \(\frac{-5}{6}\)

Q10: What should be subtracted from \(\frac{-2}{3}\) to get 1?

Let the required number be \(x\).
According to the question: \[ \frac{-2}{3} – x = 1 \]Step 1: Transpose \(x\) to RHS and 1 to LHS \[ x = \frac{-2}{3} – 1 \]Step 2: Express 1 as a fraction with denominator 3 \[ 1 = \frac{3}{3} \]Step 3: Subtract the fractions \[ x = \frac{-2}{3} – \frac{3}{3} = \frac{-5}{3} \]Answer: \(\frac{-5}{3}\) should be subtracted from \(\frac{-2}{3}\) to get 1

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