Why does a negative times a negative equal a positive? Think of it in terms of action and debt. If a negative number represents a debt, then multiplying by a negative number represents the removal of that debt.
(Hint: If someone takes away (−) four of your debts that are each worth ₹3 (that is, −3), you are effectively ₹12 richer! Therefore, (−3) × (−4) = +12.)
Step-by-step solution
Idea: Multiplying by a positive number means ‘add this many copies’; multiplying by a negative number means ‘take away this many copies’. Taking away a debt increases what you are worth.
- A debt of ₹3 is written −3. Being given 4 such debts changes your worth by 4 × (−3) = −12: you are ₹12 poorer.½ mark
- Multiplying by −4 means the opposite action: 4 such debts are taken away from you.½ mark
- If 4 debts of ₹3 are cancelled, you no longer owe ₹12, so your worth goes up by ₹12.½ mark
- So (−3) × (−4) = +12. A negative times a negative is positive because removing a debt is the same as gaining money.½ mark
Check: Pattern check: (−3) × 2 = −6, (−3) × 1 = −3, (−3) × 0 = 0. Each step down in the multiplier adds 3, so (−3) × (−1) = 3, …, (−3) × (−4) = 12 ✓.
Answer to write in the exam
Debt of ₹3 = −3
Multiplying by −4 = taking away 4 such debts
Removing 4 debts of ₹3 ⇒ worth increases by ₹12
∴ (−3) × (−4) = +12; negative × negative = positive.
Common mistakes that cost marks
- Thinking ‘two negatives make a bigger negative’, e.g. (−3) × (−4) = −12. That is the rule for adding debts, not multiplying.
- Mixing up the rules: (−3) + (−4) = −7 (debt plus debt is a debt), but (−3) × (−4) = +12.
- Applying the sign rule to addition: (−5) + (−2) is −7, not +7.
How this can come in the exam
(−6) × (−5) =
- −30
- 30
- −11
- 11
Show answer
(B) 30
Negative × negative = positive: 6 × 5 = 30.
Assertion (A): (−2) × (−9) = 18.
Reason (R): The sum of two negative integers is positive.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(C) A is true but R is false.
A is true. R is false: the sum of two negative integers is negative, e.g. (−2) + (−9) = −11.
Try one yourself
A shopkeeper’s records show 5 bills of ₹20 owed by him. The supplier cancels all 5 bills. Write this as a product of integers and find the change in his worth.
Show answer
(−20) × (−5) = +100: he is ₹100 better off.
More questions like this
- The temperature in the high-altitude desert of Ladakh is recorded as 4 °C at noon. By midnight, it drops by 15 °C. What is the midnight temperature?
- A spice trader takes a loan (debt) of ₹850. The next day, he makes a profit (fortune) of ₹1,200. The following week, he incurs a loss of ₹450. Write this sequence as an equation using integers and calculate his final financial standing.
- Calculate the following using Brahmagupta’s laws:
- Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., 10 − (−5) = 15).
- As society grew more complex, measuring became just as important as counting. If a farmer divides a field of wheat among his three children, how much does each get? If a recipe calls for half a cup of ghee, how do we represent that mathematically?