DIVISION FACTS

We have four binary operations in math. They are

Addition, Subtraction, Multiplication and Division

When we use each of the above binary operations, the result may be different.   

It is because of the work done by the binary operation that we have between the two numbers. Knowing the work done by the binary operation is called the fact of that operation.

Let us consider the following example to understand the important fact of division.

Example : 

A person wants to share the money $240 equally to his four sons. How much money will each son receive ?

To get answer for the above question, we will be dividing $240 by 4.

That is, 

240 ÷ 4 or 240/4

To know the result of the above division, we are going to use multiplication.

That is,  

The result of the above division is equal to, "How many times of the denominator is equal to the numerator ?" 

More clearly, how many times 4 goes into 240 ?

60 times of the denominator 4 is equal the numerator 240.

Then, 240 ÷ 4  =  240/4  =  60

Hence, each son will receive $60.

Other Facts of Division :

(i)  Commutative property (does not hold)

(ii)  Associative property (does not hold)

(iii)  Division by zero (not defined)

(iv)  Some other properties of division

Commutative Property

Observe the following examples :

15 ÷ 5  =  15/5  =  3

5 ÷ 15  =  5/15  =  1/3

Therefore, 15 ÷ 5  ≠  5 ÷ 15

From the above example, we observe that division is not commutative. 

Hence, commutative property does not hold for division. 

Associative Property

Observe the following examples :

12 ÷ (6 ÷ 2)  =  12 ÷ 3  =  4

(12 ÷ 6) ÷ 2  =  2 ÷ 2  =  1

Therefore, 12 ÷ (6 ÷ 2)  ≠  (12 ÷ 6) ÷ 2

From the above example, we observe that division is not associative. 

Hence, associative property does not hold for division. 

Division by Zero

Division of any number (except 0) by zero  is meaningless.

Because division by zero is not defined.

That is, 

2/0  =  Undefined

8/0  =  Undefined

Some Other Properties of Division

We know that division is the inverse operation of multiplication.

Positive number / Positive number  =  Positive number

Negative number / Negative number  =  Positive number

Negative number / Positive number  =  Negative number

Positive number / Negative number  =  Negative number

For example, 

250 / 50  =  5

(-144) / (-12)  =  12

(-120) / 20  =  -6

100 / (-25)  =  -4

Problem 1 :

A store has 60 shoes arranged into 6 equal rows. How many shoes are in each row?

Solution :

Number of shoes = 60

Number of rows = 6

Number of shoes in each row = 60/6

= 10

So, there must be 10 shoes in each row.

Problem 2 :

Which equation does not belong with the other three?

8 × 3 = 24

24 ÷ 3 = 8

24 ÷ 8 = 3

4 × 6 = 24

Solution :

8 x 3 = 24

Dividing 24 by 3, we get 8.

Dividing 24 by 8, we get 3

By multiplying 4 and 6, we will get 24 only but it is not related to the remaining. So, last option is not related.

Problem 3 :

The 12 boys and 15 girls in a class are divided into teams for a game. There are 3 students on each team. How many teams are there?

Number of boys = 12

Number of girls = 15

Number of students in each group = 3

Total number of students = 12 + 15

= 27

Number of teams = 27/3

= 9 

Problem 4 :

There are 12 people at a campsite. There are 4 people in each tent. How many tents are there?

Solution :

Number of people = 12

Number of people in each tent = 4

Number of tents = 12/4

= 3

So, there are 3 tents.

Problem 5 :

Two related facts from a multiplication table are shown.

36 ÷ 6 = ____

6 × ____ = 36

What number completes the equations?

Solution :

36 ÷ 6 = 6

Using multiplication facts, 6 x 6 = 36

So, the result is 6.

Problem 6 :

Descartes has 3 tubes of tennis balls. There are 4 tennis balls in each tube. Newton gives Descartes 7 more tennis balls. How many tennis balls does Descartes have in all?

Solution :

Number of tubes of tennis balls = 3

Number of balls in each tube = 4

Total number of balls in all 3 tubes = 3 (4) 

= 12 balls

Need to add 7 more, then 12 + 7 which is 19 balls.

So, there is 19 tennis balls.

Problem 7 :

Lana's hair is 26 centimeters long. Which is another way to describe the length of Lana's hair?

a)0.026 kilometers  b)  2,600 milimeters

c) 2.6 meters          d)  260 milimeters

Solution :

Length of hair = 26 cm

1 m = 100 cm

To convert cm to m, we have to divide by 100

= 26/100

= 0.26 m

1 cm = 10 mm

26 cm = 260 mm

So, option d is correct.

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