Apply the sign rules for multiplying positive and negative numbers
Apply the same sign rules for dividing positive and negative numbers
Combine sign rules with BODMAS in multi-step calculations
Interpret negative numbers in real-life contexts (temperature, bank balances)
Real-World Applications
Finance: A debt of −£50 doubled is −£100 — multiplying negatives appears in bank statements, profit/loss accounts, and credit calculations.
Science: Dividing a temperature drop equally between days involves dividing a negative number — e.g. −12°C over 4 days = −3°C per day.
Business: A company loses £200 per week for 5 weeks — the total loss is \(5 \times (-200) = -£1000\). Sharing losses equally between partners uses division of negatives.
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When multiplying, the signs of the two numbers determine whether the answer is positive or negative. There are four combinations — learn these as a rule:
Same signs → Positive | Different signs → Negative
(+) × (+) = + | (−) × (−) = + | (+) × (−) = − | (−) × (+) = − Three or more factors: count the negatives — odd number of negatives → negative; even number of negatives → positive
e.g. (−2) × (−3) × (−1) has 3 negatives (odd) → answer is negative (= −6)
Average of −6, −9, −12: sum = −27. Average = \((-27) \div 3 = \mathbf{-9}\)
Quick test: Check your answer by multiplying back. If \((-12) \div 4 = -3\), then \(-3 \times 4\) should equal \(-12\). ✓
✓ Quick Check 2 — Sign rules for division
Question 1 of 10
Combined Operations with Negatives
When a calculation involves negative numbers and multiple operations, always multiply and divide before you add or subtract (the full order-of-operations rule, BODMAS, is covered in Lesson 2.6) — apply the sign rules at each step.
Watch out: Always multiply and divide before you add or subtract. In \((-3) \times 5 + 20\), do the multiplication first to get \(-15 + 20 = 5\). If you add first you get the wrong answer.
✓ Quick Check 3 — Combined operations with BODMAS
Question 1 of 10
Practice: Mixed Negative Number Calculations
Work out each answer. Include the sign (e.g. -12 or 6).
#
Question
Answer
Result
1
\((-4) \times 5\)
2
\((-6) \times (-3)\)
3
\((-24) \div (-6)\)
4
\(30 \div (-5)\)
5
\((-3) \times 4 + 10\)
6
\((-8) \times (-2)\)
7
\(45 \div (-9)\)
8
\((-7) \times 3 - 5\)
9
\((-2) \times (-2) \times (-2)\)
10
\((-40) \div 8 + 1\)
⚠ Things to Watch Out For
Forgetting the sign rule: Negative × Negative = POSITIVE. This is the rule most students get wrong. −3 × −4 = +12, not −12.
Two negatives in division: (−20) ÷ (−4) = +5. Same rule as multiplication — same signs give positive, different signs give negative.
Powers of negative numbers: (−3)² = (−3) × (−3) = +9. But −3² = −(3²) = −9. The bracket makes a crucial difference.
Mixing operations: In −2 × 3 + −4 × −2: apply BODMAS — multiply first: −6 + 8 = 2. Don't mix the signs from different operations.
Negative ÷ positive = negative: −12 ÷ 4 = −3. Different signs always give a negative result in multiplication and division.