Operations with Integers — Question 4
Back to all questions(a) Let -3 × x = 27 ⇒ x = \(\frac{27}{-3}=-\frac{(27)}{3}\) = -9 (b) 5 × x = -35 ⇒ x = \(\frac{-35}{5}=-\frac{(35)}{5}\) = -7 (c) x × -8 = -56 ⇒ x = \(\frac {-56}{-8}\) = 7 (d) x × (-12) = 132 ⇒ x = \(\frac{132}{-12}=-\frac{-(132)}{12}\) = -11 (e) x ÷ (-8) = 7 ⇒ x × \(\frac {1}{-8}\) = 7 ⇒ x = 7 × (-8) = -56 (f) x ÷ 12 = -11 ⇒ x × \(\frac {1}{12}\) = -11 ⇒ x = -11 × 12 = -132 Figure It Out (Pages 42-44)
Chapter Overview: Integers
This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
Exam Weightage: ~6 marks | Difficulty: Medium
Key Formulas
| Formula | When to Use |
|---|---|
| (+) × (+) = (+), (+) × (-) = (-) | Sign rule for multiplication |
| (-) × (+) = (-), (-) × (-) = (+) | Sign rule for multiplication |
| a × (b + c) = a×b + a×c | Distributive property |
| a × 1 = a; a × 0 = 0 | Identity and zero property |
| Division sign rules same as multiplication | Division of integers |
Must-Know Concepts
- Sign rules: same signs → positive, different signs → negative
- Closure property holds for multiplication but NOT for division:
- Commutative and associative properties hold for multiplication but NOT for division:
- Division by zero is undefined:
- Distributive property: a × (b + c) = a×b + a×c
Common Mistakes to Avoid
- Writing (-2) × (-3) = -6 instead of +6
- Claiming division is commutative
- Dividing by zero and writing 0 as the result
Scoring Tips
- Master sign rules first — they apply to both multiplication and division
- Use the pattern approach to understand why negative × negative = positive
- Practice with number line for visual understanding
- Memorize: even number of negatives = positive result
Frequently Asked Questions
Why is negative × negative = positive?
Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.
Why can't we divide by zero?
Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.
Does closure hold for division of integers?
No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.