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-I

Multiple Choice Type

Q1: The additive inverse of \(\frac{5}{9}\) is

Step 1: Understand the definition of additive inverse.
The **additive inverse** of a number is the number that when **added** to the original gives **zero**.

Step 2: Find the number which when added to \(\frac{5}{9}\) gives 0.
Let the additive inverse be \(x\). So,
\[ \frac{5}{9} + x = 0 \] Solving for \(x\), \[ x = -\frac{5}{9} \]Answer: b. \(-\frac{5}{9}\)

Q2: The rational number \(\frac{32}{-40}\) expressed in standard form is

Step 1: Recall that the standard form of a rational number has the negative sign (if any) only in the numerator.

Step 2: Simplify \(\frac{32}{-40}\) by dividing numerator and denominator by their greatest common divisor (GCD).
GCD of 32 and 40 is 8.
\[ \frac{32}{-40} = \frac{32 \div 8}{-40 \div 8} = \frac{4}{-5} \]Step 3: Rewrite \(\frac{4}{-5}\) as \(\frac{-4}{5}\) to put the negative sign in numerator (standard form).
\[ \frac{4}{-5} = \frac{-4}{5} \]Answer: d. \(\frac{-4}{5}\)

Q3: What should be added to \(\frac{-3}{16}\) get \(\frac{5}{8}\)?

Step 1: Let the required number be \(x\). Then,
\[ \frac{-3}{16} + x = \frac{5}{8} \]Step 2: Solve for \(x\):
\[ x = \frac{5}{8} – \frac{-3}{16} = \frac{5}{8} + \frac{3}{16} \]Step 3: Find common denominator (LCM of 8 and 16 is 16) and add:
\[ \frac{5}{8} = \frac{5 \times 2}{8 \times 2} = \frac{10}{16} \] \[ x = \frac{10}{16} + \frac{3}{16} = \frac{13}{16} \]Answer: d. \(\frac{13}{16}\)

Q4: The multiplicative inverse of \(\frac{-3}{7}\) is:

Step 1: Recall that the multiplicative inverse (or reciprocal) of a rational number \(\frac{a}{b}\) is \(\frac{b}{a}\), such that
\[ \frac{a}{b} \times \frac{b}{a} = 1 \]Step 2: For \(\frac{-3}{7}\), the reciprocal is
\[ \frac{7}{-3} = \frac{-7}{3} \]Step 3: Check the product:
\[ \frac{-3}{7} \times \frac{-7}{3} = \frac{21}{21} = 1 \]Answer: a. \(\frac{-7}{3}\)

Q5: The sum of \(-\frac{1}{3}\) and its multiplicative is

Step 1: Find the multiplicative inverse (reciprocal) of \(-\frac{1}{3}\).
\[ \text{Multiplicative inverse of } -\frac{1}{3} = -3 \]Step 2: Now, add the number and its reciprocal:
\[ -\frac{1}{3} + (-3) = -\frac{1}{3} – 3 \]Step 3: Convert \(-3\) to fraction with denominator 3:
\[ -3 = -\frac{9}{3} \]Step 4: Add the fractions:
\[ -\frac{1}{3} – \frac{9}{3} = -\frac{10}{3} \]Step 5: Convert \(-\frac{10}{3}\) into a mixed fraction:
\[ -\frac{10}{3} = -3\frac{1}{3} \]Answer: d. \(-3\frac{1}{3}\)

Q6: The product of \(-\frac{1}{3}\) and its additive is

Step 1: Find the additive inverse of \(-\frac{1}{3}\).
\[ \text{Additive inverse of } -\frac{1}{3} = \frac{1}{3} \]Step 2: Now, find the product of \(-\frac{1}{3}\) and \(\frac{1}{3}\):
\[ -\frac{1}{3} \times \frac{1}{3} = -\frac{1 \times 1}{3 \times 3} = -\frac{1}{9} \]Answer: c. \(-\frac{1}{9}\)

Q7: Which of the rational numbers is equivalent to \(\frac{-2}{7}\)?

Step 1: Understand that two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if:
\[ a \times d = b \times c \]Step 2: Check each option one by one:a. \(\frac{-14}{21}\): Check if \(-2 \times 21 = 7 \times -14\) \[ -2 \times 21 = -42, \quad 7 \times (-14) = -98 \quad \Rightarrow \text{Not equal} \]b. \(\frac{-8}{14}\): Check if \(-2 \times 14 = 7 \times -8\) \[ -2 \times 14 = -28, \quad 7 \times (-8) = -56 \quad \Rightarrow \text{Not equal} \]c. \(\frac{-14}{49}\): Check if \(-2 \times 49 = 7 \times -14\) \[ -2 \times 49 = -98, \quad 7 \times (-14) = -98 \quad \Rightarrow \text{Equal!} \]d. \(\frac{-6}{28}\): Check if \(-2 \times 28 = 7 \times -6\) \[ -2 \times 28 = -56, \quad 7 \times (-6) = -42 \quad \Rightarrow \text{Not equal} \]Answer: c. \(\frac{-14}{49}\)

Q8: If \(3\frac{3}{4}\) m of cloth is required for one suit, then how many suits be prepared from 30 m of cloth?

Step 1: Convert the mixed fraction \(3\frac{3}{4}\) to an improper fraction:
\[ 3\frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4} \]Step 2: Let the number of suits be \(x\). Then,
\[ x \times \frac{15}{4} = 30 \]Step 3: Solve for \(x\):
\[ x = \frac{30}{\frac{15}{4}} = 30 \times \frac{4}{15} = \frac{30 \times 4}{15} = \frac{120}{15} = 8 \]Answer: c. 8 suits

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