Multiplying & Dividing Integers — Solutions

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1. Multiply Integers

   1. (−3) × 7 = −21
   2. (−5) × (−4) = 20
   3. 8 × (−6) = −48
   4. (−9) × (−3) = 27
   5. (−2) × 11 = −22
   6. (−7) × (−7) = 49
   7. 4 × (−12) = −48
   8. (−10) × 5 = −50

2. Divide Integers

   1. (−24) ÷ 6 = −4
   2. (−36) ÷ (−9) = 4
   3. 40 ÷ (−8) = −5
   4. (−55) ÷ 5 = −11
   5. (−48) ÷ (−6) = 8
   6. 72 ÷ (−9) = −8
   7. (−63) ÷ 7 = −9
   8. (−100) ÷ (−10) = 10

3. Powers of Negative Numbers

   1. (−3)² = 9
   2. −3² = −9
   3. (−2)³ = −8
   4. −4² = −16
   5. (−5)² = 25
   6. (−1)⁵ = −1
   7. (−2)⁴ = 16
   8. −2⁴ = −16

4. Determine the Sign First

   1. (−3) × (−5) × 2: Positive (30)
   2. (−4) × (−6) × (−1): Negative (−24)
   3. (−7)²: Positive (49)
   4. −7²: Negative (−49)
   5. (−2) × 8 × (−3): Positive (48)
   6. (−10) ÷ (−2) × (−1): Negative (−5)

5. Find the Missing Integer

   1. n × (−4) = −20: n = 5
   2. (−6) × n = 42: n = −7
   3. n ÷ (−3) = 8: n = −24
   4. (−36) ÷ n = −9: n = 4
   5. n × (−7) = 0: n = 0
   6. (−5) × n = −45: n = 9
   7. n ÷ 4 = −12: n = −48
   8. 54 ÷ n = −6: n = −9

6. Real-World Problem Solving

   1. (−150) × 8 = −$1200 (loss of $1200)
   2. (−3) × 6 = −18°C
   3. −52 ÷ (−4) = 13 weeks
   4. (−15) × 7 = −105 m (depth of 105 m)
   5. (−85) ÷ 5 = −$17 each
   6. 8 minutes = 16 intervals of 30 s; (−2) × 16 = −32°C

7. Mixed Multiplication and Division

   1. (−4) × 3 ÷ (−6) = −12 ÷ (−6) = 2
   2. (−48) ÷ (−4) × (−2) = 12 × (−2) = −24
   3. 30 ÷ (−5) × (−3) = −6 × (−3) = 18
   4. (−7) × (−3) ÷ 7 = 21 ÷ 7 = 3
   5. (−2) × 6 ÷ (−4) = −12 ÷ (−4) = 3
   6. (−9) × (−4) ÷ (−6) = 36 ÷ (−6) = −6
   7. (−60) ÷ 12 × (−5) = −5 × (−5) = 25
   8. 5 × (−8) ÷ (−10) = −40 ÷ (−10) = 4

8. Explain the Sign Rules

   1. Student’s error: Error: negative × negative = positive (not negative). Two negatives multiplied always give a positive. (−3) × (−4) = +12.
   2. (−1)⁶ vs (−1)⁷: An even number of negative factors produces a positive product; an odd number produces a negative. (−1)⁶ = +1 (6 pairs of negatives). (−1)⁷ = −1 (one extra negative left over).
   3. (−2)×(−3)×(−1)×4 sign: Three negatives = odd number of negatives = negative. Calculation: 6 × (−1) × 4 = −6 × 4 = −24. Sign prediction confirmed: negative.
   4. Negative ÷ negative = positive: If you owe 3 people $6 each, your debt is −$18. If we want to know how many groups of (−$6) fit into (−$18), the answer is 3 (a positive number of groups). −18 ÷ (−6) = 3.

9. Integer Patterns and Sequences

   1. 2, −6, 18, −54, …: Multiplier: ×(−3). Next three: 162, −486, 1458.
   2. −1, 3, −9, 27, …: Multiplier: ×(−3). Next three: −81, 243, −729.
   3. 64, −32, 16, −8, …: Multiplier: ×(−½). Next three: 4, −2, 1.
   4. −5, 10, −20, 40, …: Multiplier: ×(−2). Next three: −80, 160, −320.

10. Extended Real-World Problems

    1. Mining company ore removal: (−250) × 12 = −3000 tonnes. The total change is −3000 tonnes (3000 tonnes removed).
    2. Temperature changes: First 6 hours: (−4) × 6 = −24°C. Next 3 hours: (+2) × 3 = +6°C. Net change = −24 + 6 = −18°C.
    3. Share price: Change over 5 days: (−3) × 5 = −$15. Change over next 4 days: 4 × 4 = +$16. Net change = −15 + 16 = +$1. Final price = $50 + $1 = $51.