Topic Review — Integers and Rational Numbers — Solutions

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1. Adding and subtracting integers

   1. 8 + (−3): 5
   2. −5 + (−7): −12
   3. 4 − (−6): 10
   4. −12 − (−5): −7
   5. −9 + 14: 5
   6. (−3) + (−8) − (−2): −11 + 2 = −9
   7. Temperature: −4 + 11: 7°C
   8. Ascending order: −10, −7, −1, 0, 3, 5

2. Multiplying and dividing integers

   1. (−4) × 7: −28
   2. (−6) × (−5): 30
   3. 36 ÷ (−9): −4
   4. (−48) ÷ (−8): 6
   5. (−3) × (−4) × 2: 12 × 2 = 24
   6. (−72) ÷ 9: −8
   7. (−2)⁵: Negative (odd power of a negative = negative; −32)
   8. (−3)⁴: Positive (even power of a negative = positive; 81)

3. Operations with rational numbers

   1. ½ + ¼: ²⁄₄ + ¼ = ¾
   2. ¾ − ⅙: ⁹⁄₁₂ − ²⁄₁₂ = ⁷⁄₁₂
   3. 2⅓ + 1¾: ⁷⁄₃ + ⁷⁄₄ = ²⁸⁄₁₂ + ²¹⁄₁₂ = ⁴⁹⁄₁₂ = 4¹⁄₁₂
   4. ⅗ × ⅖: ⁶⁄₂₅
   5. 1½ × 2⅔: ³⁄₂ × ⁸⁄₃ = ²⁴⁄₆ = 4
   6. ⅞ ÷ ¾: ⁷⁄₈ × ⁴⁄₃ = ²⁸⁄₂₄ = ⁷⁄₆ = 1¹⁄₆
   7. −½ + ¾: −²⁄₄ + ¾ = ¼
   8. −1¼ − (−¾): −⁵⁄₄ + ¾ = −²⁄₄ = −½

4. Order of operations

   1. 3 + 4 × 2: 3 + 8 = 11
   2. (3 + 4) × 2: 7 × 2 = 14
   3. 20 − 8 ÷ 4 + 3: 20 − 2 + 3 = 21
   4. 4 × (7 − 3)²: 4 × 16 = 64
   5. 18 ÷ (2 + 4) × 3 − 5: 18 ÷ 6 × 3 − 5 = 3 × 3 − 5 = 9 − 5 = 4
   6. (3 + −2) × (4 − −1): (1) × (5) = 5
   7. 2² + 3 × (10 − 4) ÷ 2: 4 + 3 × 6 ÷ 2 = 4 + 9 = 13
   8. (−5)² − 4 × 3 + 10 ÷ 2: 25 − 12 + 5 = 18

5. Problem solving

   1. Diver: −12 + 7 − 4: −9 m (9 m below surface)
   2. Company: (−350) × 6 + 800: −2100 + 800 = −$1300 (net loss of $1300)
   3. ½ × (3 + −1)² − ¾: ½ × 4 − ¾ = 2 − ¾ = 1¼
   4. (−3) × ☐ = −48: ☐ = −48 ÷ (−3) = 16
   5. 5 × 3² − (12 − 7) + 24 ÷ 8: 5 × 9 − 5 + 3 = 45 − 5 + 3 = 43
   6. Recipe: ⅔ × 4½: ²⁄₃ × ⁹⁄₂ = ¹⁸⁄₆ = 3 cups