Learn Divide Integers Methods

Key Concepts

• Divide integers with different signs
• Divide integers with the same sign
• Write equivalent quotients of integers

1.6 Division of Integers 

Introduction: 

Division is the inverse operation of multiplication. 

Let us see an example for whole numbers. 

Dividing 20 by 5 means finding an integer which when multiplied with 5 gives us 20, such an integer is 4.  

Since 5 × 4 = 20 

So, 20 ÷ 5 = 4 and 20 ÷ 4 = 5 

Therefore, for each multiplication statement of whole numbers, there are two division statements. 

Properties of division of integers:  

• Closure under division 

Division of integers does not follow the closure property.  

Let’s consider the following pairs of integers.  

(−12) ÷ (−6) = 2 (Result is an integer) 

(−5) ÷ (−10) =

1/2 (Result is not an integer) 

We observe that integers are not closed under division. 

•  Commutative property of division 

Division of integers is not commutative for integer.  

Let’s consider the following pairs of integers.  

(–14) ÷ (–7) = 2 

(–7) ÷ (–14) = 1/2

(–14) ÷ (–7) ≠ (–7) ÷ (–14) 

We observe that division is not commutative for integers. 

• Division of an integer by zero 

Any integer divided by zero is meaningless. 

Example: 5 ÷ 0 = not defined  

Zero divided by an integer other than zero is equal to zero. 

Example: 0 ÷ 6 = 0  

• Division of an integer by 1 

When we divide an integer by 1 it gives the same integer.  

Example: (– 7) ÷ 1 = (– 7) 

This shows that negative integer divided by 1 gives the same negative integer. So, any integer divided by 1 gives the same integer. 

In general, for any integer a, a ÷ 1 = a 

Rules for the division of integers: 

Rule 1: The quotient value of two positive integers will always be a positive integer. 

Rule 2:  For two negative integers the quotient value will always be a positive integer. 

Rule 3: The quotient value of one positive integer and one negative integer will always be a negative integer. 

The following table will help you remember rules for dividing integers: 

Types of Integers	Result	Example
Both Integers Positive	Positive	16 ÷ 8 = 2
Both Integers Negative	Positive	–16 ÷ –8 = 2
1 Positive and 1 Negative	Negative	–16 ÷ 8 = –2

1.6.1 Division of Integers with Different Signs 

Steps: 

1. First divide them as whole numbers. 

2. Then put a minus sign (–) before the quotient. We, thus, get a negative integer. 

Example: 

    (–10) ÷ 2 = (– 5) 

              (–32) ÷ (8) = (– 4) 

In general, for any two positive integers a and b, a ÷ (– b) = (– a) ÷ b where b ≠ 0. 

1.6.2 Division of Integers with the Same Sign 

Steps: 

1. Divide them as whole numbers. 

2. Then put a positive sign (+). That is, we get a positive integer. 

Example:  

(–15) ÷ (–3) = 5 

(–21) ÷ (–7) = 3 

In general, for any two positive integers a and b, (– a) ÷ (– b) = a ÷ b where b ≠ 0. 

1.6.3 Write Equivalent Quotients of Integers  

What is Quotient in division? 

The number obtained by dividing one number by another number. 

 If p and q are integers, then

−(p / q)

=-p/q   =p/−q. 

Example: Show that the quotients of

−(18 / 4) , −18 / 4, and 18/−4 are equivalent. 

−(18/4) = −(18÷4)=−(4.5)= −4.5

−18 / 4 = −18÷4 =  −4.5

18 / −4 = 18÷−4 = −4.5

∴The quotients of

−(18 / 4) , −18 / 4, and 18/−4 are equivalent to −4.5 

Example: Evaluate [(– 8) + 4)] ÷ [(–5) + 1] 

Solution: [(– 8) + 4)] ÷ [(–5) + 1] 

           = (−4) ÷ (−4) 

           = 1 

Example: Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) when a = 8, b = – 2, c = 4. 

Solution: L.H.S = a ÷ (b + c)  

                             = 8 ÷ (−2 + 4) 

                             = 8 ÷ 2 = 4 

                  R.H.S = (a ÷ b) + (a ÷ c) 

                             = [8 ÷ (−2)] + (8 ÷ 4) 

                             = (−4) + 2 

                             = −2 

Here, L.H.S ≠ R.H.S 

Hence verified. 

Exercise

1. $4000 is distributed among 25 women for the work completed by them at a construction site. Calculate the amount given to each woman.
2. 66 people are invited to a birthday party. The suppliers have to arrange tables for the invitees. 7 people can sit around a table. How many tables should the suppliers arrange for the invitees?
3. Calculate the number of hours in 2100 minutes.
4. When the teacher of class 6 asked a question, some students raised their hands. But instead of raising one hand, each student raised both their hands. If there are 56 hands in total, how many students raised their hands?
5. Emma needs 3 apples to make a big glass of apple juice. If she has 51 apples, how many glasses of juice can she make?
6. Simplify –40 ÷ (–5)
7. Find the quotient of .
8. Why is the quotient of two negative integers positive?
9. Classify the quotient –50 ÷ 5 as positive, negative, zero or undefined.
10. Simplify –28 ÷ (–7).

What we have learned

• Restate that the quotient of two integers with unlike signs is a negative integer.
• Restate that the quotient of two integers with like signs is a positive integer.
• Perform division of two integers with like signs.
• Perform division of two integers with unlike signs.
• Describe the procedure for dividing integers with like signs.
• Describe the procedure for dividing integers with unlike signs.
• Apply the procedures for integer division to complete the exercises.

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