Class 7 - Multiplication and Division of Integers

The multiplication and division of integers are fundamental operations in arithmetic that are used in everyday situations. These operations are governed by rules based on the signs of the integers involved. This section explores the rules, properties, and applications of multiplying and dividing integers, along with clear examples to master multiplication and division of integers.

Table of Contents

• What are Multiplication and Division of Integers?
• Rules for Multiplication of Integers
• Properties of Multiplication of Integers
• Rules for Division of Integers
• Properties of Division of Integers
• Solved Examples on Multiplication and Division of Integers

 

What are Multiplication and Division of Integers?

Multiplication of integers refers to the repeated addition of a number and follows specific rules based on the signs of the integers involved.
Division of integers is the inverse operation of multiplication and involves distributing or partitioning a number into equal parts. Similar to multiplication, division also follows sign rules.

Rules for Multiplication of Integers

Multiplication of integers means repeated addition while considering the signs (+ or –) of the numbers. Given below are the sign rules to follow while multiplying integers:

• When two integers with the same sign are multiplied, the result is positive

• When two integers with the same sign are multiplied, the result is a negative value. 

 

 

Properties of Multiplication of Integers

Multiplication of integers follows important mathematical properties:

• Closure property:
  The product of two integers is always an integer.
  If a and b are two integers, then a × b = ab is also an integer.
  For example, 6 × (–8) = −48,  −48 is also an integer.

• Commutative property:
  Two integers can be multiplied in any order, as the product remains the same.
  If a and b are two integers, then a × b = b × a.
  For example, 10 × (−7) = −7 × 10 = −70

• Associative property:
  The product of any three integers remains the same even if the order of grouping the integers is changed.
  If a, b and c are three integers, then a × (b × c) = (a × b) × c.
  For example, 6 × [(−7) × (−5)] = 6 × (35) = 210
  [6 × (−7)] × (−5) = −42 × (−5) = 210
  Therefore, 6 × [(−7) × (−5)] = [6 × (−7)] × (−5)

• Distributive property:
  If a, b and c are three integers, then a × (b + c) = a × b + a × c
  For example, 3 × [−6 + (−2)] = [3 × (−6)] + [3 × (−2)] =  −18 + (−6) = −24 

• Multiplication by zero:
  The product of any integer (positive or negative) and zero is always zero.
  If a is any integer, then a × 0 = 0 × a = 0

• Multiplicative identity:
  When we multiply one with any integer, we get the integer itself as the product.
  For any integer a, we have a × 1 = a.

 

Rules for Division of Integers

Division of integers is the process of splitting or grouping numbers. It is the inverse of multiplication. Given below are the sign rules to follow while dividing integers:

• When two integers with the same sign are divided, the result is positive.

• When two integers with different signs are divided, the result is a negative value.
  
  

 

Properties of Division of Integers

Division of integers follows important mathematical properties:

• Closure property:
  Integers are not closed under division
  If a and b are any two integers, then a ÷ b is not always an integer.
  For example, (−10) ÷ (−4) = 10 / 4 = 5 / 2, which is not an integer.

• Commutative property:
  Integers are not commutative under division. If a and b are any two integers, then a ÷ b ≠ b ÷ a
  The commutative property does not hold true for the division of integers.
  For example, (−77) ÷ (11) = −7, but (11) ÷ (−77) = 1/7, -7 ≠ 1/7

• Associative property:
  If a, b and c are any three integers, then a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c.
  Therefore, the associative property does not hold true for division of integers.
  For example, −8 ÷ (4 ÷ 2) = −8 ÷ 2 = −4
  (−8 ÷ 4) ÷ 2 = −2 ÷ 2 = −1
  −4 ≠ −1

• Identity property:
  When we divide any integer by 1, we get the same integer as the quotient.

 

Solved Examples on Multiplication and Division of Integers

Example 1: Find (−8) × 3 + 12 ÷ (−4)
Solution: Applying bodmas and rules of multiplication and division of integers,
12 ÷ (−4) = −3
(−8) × 3 = −24
−24 + (−3) = −27
(−8) × 3 + 12 ÷ (−4) = −27

Example 2: Find (−10) × (−2) ÷ 5 + (−6)
Solution: Applying bodmas and rules of multiplication and division of integers,
(−10) × (−2) = 20
20 ÷ 5 = 4
4 + (−6) = −2
(−10) × (−2) ÷ 5 + (−6) = -2

Example 3: A submarine descends 6 metres every minute. What is its position after 4 minutes?
Solution: Given, the submarine descends 6 metres every minute. Since the submarine is descending, it represents a negative direction.
Position in 4 minutes = (-6) × 4 = -24 m
The submarine is at -24 metres (24 metres below sea level).

Example 4: A total loss of ₹360 is shared equally among 6 partners. What is each partner’s share?
Solution: Total loss = -360
Number of partners = 6
Loss of each partner = -360 ÷ 6 = −60
Each partner incurred a loss of ₹60.

Example 5: An elevator descends into a coal mine at the rate of 6 m/minute. If the descent starts from 30 m above the ground level, how long will it take to reach –240 m?
Solution: Given, Starting position = +30 m
Final position = −240 m

Total change in position = Final − Initial
= (−240) − 30
= −270 m
Rate of descent = −6 m/min
Time taken = Total change in position ÷ Rate
= (−270) ÷ (−6)
= 45 minutes
Therefore, The elevator will take 45 minutes to reach −240 m.

Example 6: Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) when a = 8, b = – 2, and c = –2.
Solution: a÷(b+c) = 8÷(−2+(−2)) = 8÷(−4) = −2
(a÷b) + (a÷c) = [8÷(−2)] + [8÷(−2)] = −4 + (−4) = −8
LHS ≠ RHS
−2 ≠ −8
Hence, a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)