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🔢 Chapter 3: The World of Numbers • Exercise Set 3.3

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Q1

Exercise Set 3.3

1. Prove that the following rational numbers are equal:

(i) 2/3 and 4/6      (ii) 5/4 and 10/8      (iii) -3/5 and -6/10      (iv) 9/3 and 3

Two rational numbers a/b and c/d are equal if their cross-products satisfy ad = bc:

  • (i) 2/3 and 4/6: 2 × 6 = 12 and 3 × 4 = 12. Since 12 = 12, 2/3 = 4/6.
  • (ii) 5/4 and 10/8: 5 × 8 = 40 and 4 × 10 = 40. Since 40 = 40, 5/4 = 10/8.
  • (iii) -3/5 and -6/10: (-3) × 10 = -30 and 5 × (-6) = -30. Since -30 = -30, -3/5 = -6/10.
  • (iv) 9/3 and 3/1: 9 × 1 = 9 and 3 × 3 = 9. Since 9 = 9, 9/3 = 3.
Q2

Exercise Set 3.3

2. Find the sum:

(i) 2/5 + 3/10      (ii) 7/12 + 5/8      (iii) -4/7 + 3/14

  • (i) 2/5 + 3/10:
    LCM(5, 10) = 10 ⇒ (4 + 3) / 10 = 7/10
  • (ii) 7/12 + 5/8:
    LCM(12, 8) = 24 ⇒ (14 + 15) / 24 = 29/24 (or 1 5/24)
  • (iii) -4/7 + 3/14:
    LCM(7, 14) = 14 ⇒ (-8 + 3) / 14 = -5/14
Q3

Exercise Set 3.3

3. Find the difference:

(i) 5/6 - 1/4      (ii) 11/8 - 3/4      (iii) -7/9 - (-2/3)

  • (i) 5/6 - 1/4:
    LCM(6, 4) = 12 ⇒ (10 - 3) / 12 = 7/12
  • (ii) 11/8 - 3/4:
    LCM(8, 4) = 8 ⇒ (11 - 6) / 8 = 5/8
  • (iii) -7/9 - (-2/3):
    -7/9 + 2/3; LCM(9, 3) = 9 ⇒ (-7 + 6) / 9 = -1/9
Q4

Exercise Set 3.3

4. Find the product:

(i) (2/3) × (3/10)      (ii) (7/11) × (5/8)      (iii) (-4/7) × (5/14)

  • (i) (2/3) × (3/10): (2 × 3) / (3 × 10) = 6/30 = 1/5
  • (ii) (7/11) × (5/8): (7 × 5) / (11 × 8) = 35/88
  • (iii) (-4/7) × (5/14): (-4 × 5) / (7 × 14) = -20/98 = -10/49
Q5

Exercise Set 3.3

5. Find the quotient:

(i) (2/3) ÷ (3/10)      (ii) (7/11) ÷ (5/8)      (iii) (-4/7) ÷ (5/14)

  • (i) (2/3) ÷ (3/10): (2/3) × (10/3) = (2 × 10) / (3 × 3) = 20/9
  • (ii) (7/11) ÷ (5/8): (7/11) × (8/5) = (7 × 8) / (11 × 5) = 56/55
  • (iii) (-4/7) ÷ (5/14): (-4/7) × (14/5) = (-4 × 2) / 5 = -8/5
Q6

Exercise Set 3.3

6. Show that: (1/2 + 3/4) × 8/3 = (1/2 × 8/3) + (3/4 × 8/3)

  • Left Hand Side (LHS):
    (1/2 + 3/4) = (2/4 + 3/4) = 5/4
    LHS = (5/4) × (8/3) = 40/12 = 10/3
  • Right Hand Side (RHS):
    (1/2 × 8/3) = 8/6 = 4/3
    (3/4 × 8/3) = 24/12 = 2 = 6/3
    RHS = 4/3 + 6/3 = 10/3
  • Since LHS = RHS = 10/3, the equality is verified.
Q7

Exercise Set 3.3

7. Simplify the following using the distributive property: (7/9) × (6/7 - 3/4)

Using distributive property: a(b - c) = ab - ac

  • = (7/9 × 6/7) - (7/9 × 3/4)
  • = 6/9 - 21/36
  • = 2/3 - 7/12
  • LCM(3, 12) = 12 ⇒ (8 - 7) / 12 = 1/12
Q8

Exercise Set 3.3

8. Find the rational number x such that: (5/6)(x + 3/5) = (5/6)x + 1/2

  1. Apply the distributive law to the left-hand side:
    (5/6)x + (5/6 × 3/5) = (5/6)x + 1/2
  2. Simplify the constant term on the left:
    5/6 × 3/5 = 15/30 = 1/2
  3. The equation becomes:
    (5/6)x + 1/2 = (5/6)x + 1/2
  4. Subtract (5/6)x + 1/2 from both sides:
    0 = 0
  5. This is an identity true for all values. Therefore, x can be any rational number (x ∈ Q).