Practice Maths

L34 — Multiplying & Dividing Integers

Key Terms

sign rule
When multiplying or dividing integers: same signs → positive result; different signs → negative result.
product
The result of multiplying two numbers. E.g. the product of −3 and 4 is −12.
quotient
The result of dividing one number by another. E.g. the quotient of −12 and −4 is 3.

Sign Rules for Multiplication and Division

  • Same signs → Positive: (+) × (+) = +, and (−) × (−) = +
  • Different signs → Negative: (+) × (−) = −, and (−) × (+) = −
  • The same rules apply to division.

Worked Examples

(−3) × 4: different signs → negative → −12

(−6) × (−5): same signs → positive → 30

(−15) ÷ 3: different signs → negative → −5

(−20) ÷ (−4): same signs → positive → 5

(−2)³: (−2) × (−2) × (−2) = 4 × (−2) = −8  (odd power → negative)

(−3)²: (−3) × (−3) = 9  (even power → positive)

Why Does Negative Times Negative Equal Positive?

This surprises many students, but it follows logically from two perspectives:

  • The Debt Story: Owing $5 every day for 3 days: (−5) × 3 = −$15. Now reverse it: if someone cancels 3 days of $5 debts, you gain $15. That’s (−5) × (−3) = +15. Cancelling debts means gaining money — positive!
  • The Pattern Method: 3 × 3 = 9, 2 × 3 = 6, 1 × 3 = 3, 0 × 3 = 0, −1 × 3 = −3, −2 × 3 = −6. Each step decreases by 3. Now reverse the second factor: 3 × (−3) = −9, 2 × (−3) = −6, … 0 × (−3) = 0, −1 × (−3) = +3. The pattern forces the answers to become positive.

The Two Sign Rules

  • Same signs → Positive: (+) × (+) = + and (−) × (−) = +
  • Different signs → Negative: (+) × (−) = − and (−) × (+) = −

These rules apply to division too. (−12) ÷ 4 = −3 (different signs). (−12) ÷ (−4) = +3 (same signs).

Memory trick: think of negative as “bad” and positive as “good.” Good × Good = Good. Bad × Bad = Good. Good × Bad = Bad. Two “bads” always make a “good.”

Powers of Negative Numbers

When a negative number is raised to a power, count how many negatives are multiplied together:

  • (−2)² = (−2) × (−2) = +4 (two negatives → positive)
  • (−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8 (three negatives → negative)
  • (−2)⁴ = +16 (four negatives → positive)

Rule: even power → positive, odd power → negative (for any negative base).

Real-Life Applications

Temperature dropping at a constant rate: −3°C per hour for 5 hours = (−3) × 5 = −15°C total change.

Splitting a debt: −$24 over 6 days = (−24) ÷ 6 = −$4 per day.

Common mistake: (−3) × (−4) ≠ −12. Two negatives multiply to give a POSITIVE. (−3) × (−4) = +12. Say the rule aloud: “same signs, positive result.”

  1. Sign Rule Identification

    Before calculating, predict: will the answer be positive or negative?

    1. (−4) × 3
    2. (−5) × (−2)
    3. 7 × (−3)
    4. (−8) ÷ (−4)
    5. 12 ÷ (−3)
    6. (−10) ÷ 5
    7. (−6) × (−6)
    8. (−1) × 100

  2. Multiplying Integers

    1. (−3) × 5
    2. 4 × (−7)
    3. (−6) × (−4)
    4. (−9) × 3
    5. 8 × (−5)
    6. (−7) × (−7)
    7. (−2) × 11
    8. (−10) × (−6)

  3. Dividing Integers

    1. (−12) ÷ 4
    2. 20 ÷ (−5)
    3. (−18) ÷ (−3)
    4. (−24) ÷ 6
    5. 36 ÷ (−9)
    6. (−45) ÷ (−9)
    7. (−32) ÷ 8
    8. (−56) ÷ (−7)

  4. Powers of Negative Numbers

    1. (−2)2
    2. (−2)3
    3. (−3)2
    4. (−1)5
    5. (−4)2
    6. (−2)4
    7. (−10)3
    8. (−5)2

  5. Mixed Operations

    Calculate each expression:

    1. (−3) × 4 + 5
    2. 10 ÷ (−2) + 8
    3. (−6) × (−2) − 15
    4. (−20) ÷ 4 × (−3)
    5. 3 × (−5) + 2 × 7
    6. (−4)2 − 20

  6. True or False

    1. (−3) × (−4) = −12
    2. A negative number multiplied by a positive number is always negative.
    3. (−1) × (−1) × (−1) = 1
    4. Dividing a negative by a negative always gives a positive.
    5. (−6)2 = −36
    6. The product of three negative numbers is always negative.

  7. Find the Missing Number

    1. (−4) × __ = 28
    2. __ ÷ (−3) = 7
    3. __ × (−5) = −40
    4. (−36) ÷ __ = −9
    5. __ × (−6) = 54
    6. 24 ÷ __ = −8

  8. Temperature Changes

    1. The temperature drops 3°C every hour for 5 hours. Write a multiplication expression and find the total temperature change.
    2. A city records −24°C over 6 days as equal daily drops. What was the drop each day?
    3. The temperature is −4°C. It drops by the same amount 3 times and reaches −19°C. How much did it drop each time?
    4. If the temperature fell 2°C per hour for 8 hours, what is the total change?

  9. Money and Debt

    1. A business loses $150 each week for 4 weeks. Write an expression using integers and find the total loss.
    2. A debt of $360 is paid off in equal instalments over 6 months. What is each instalment as a negative integer?
    3. A share price falls $3 each day. After how many days will it be $21 lower?
    4. A submarine descends 8 metres per minute. How far has it descended after 7 minutes? Write as an integer.

  10. Exam Challenge

    1. Without calculating, explain why the product of an even number of negative integers is always positive.
    2. Is it possible for (−a) × (−b) to be negative? Explain.
    3. Find all integer pairs (x, y) where x × y = −12 and both values are between −6 and 6.
    4. (−2)1 = −2, (−2)2 = 4, (−2)3 = −8. Without calculating, what is (−2)6? Explain.
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