Division of Whole Numbers

1. Definition of Division

Example:

Suppose that we have twelve students in the class and we want to divide the class into three equal groups.  How many should be in each group?

Solution:

We can ask the alternative question,  "Three times what number equals twelve?"
The answer to this question is four.

Division is defined as this reverse of multiplication.

We write

4
3 | 12
or        12 ÷ 3  =  4

We call the number 12 her the dividend, the number 3 the divisor, and the number 4 the quotient.

quotient
divisor | dividend
or        dividend ÷ divisor  =  quotient
2. Properties of Division

1. Division by Oneself

Example

Suppose that you had \$100 and had to distribute all the money to 100 people so that each person received the same amount of money.  How much would each person get?

Solution

If you gave each person \$1 you would achieve your goal.  This comes directly from the identity property of one.  Since the the questions asks what number times 100 equals 100.

In general we conclude,

 Any number divided by itself equals 1

Examples

100 ÷ 100  =  1        38 ÷ 38  =  1        15 ÷ 15  =  1

2. Division by 1

Example

Now lets suppose that you have twelve pieces of paper and need to give them to exactly one person.  How many pieces of paper does that person receive?

Solution

Since the only person to collect the paper is the receiver, that person gets all twelve pieces.  This also comes directly from the identity property of one, since one times twelve equals twelve.

In general we conclude,

 Any number divided by 1 equals itself

Examples

12 ÷ 1  =  12        42 ÷ 1  =  42        33 ÷ 1  =  33

3. When Zero is the Dividend

Example

Now lets suppose that you have zero pieces of pizza and need to distribute your pizza to four friends so that each person receives the same number of pieces.  How many pieces of pizza does that person receive?

Solution

Since you have no pizza to give, you give zero slices of pizza to each person.  This comes directly from the multiplicative property of zero, since zero times four equals zero.

In general we conclude,

 Zero divided by any nonzero number equals zero

Examples

0 ÷ 4  =  0        0 ÷ 1  =  0        0 ÷ 24  =  0

4. The Problem With Dividing by Zero

Example

Finally lets suppose that you have five bags of garbage and you have to get rid of all the garbage, but have no places to put the garbage.  How can you distribute your garbage to no places and still get rid of it all?

Solution

You can't!  This is an impossible problem.  There is no way to divide by zero.

In general we conclude,

 Dividing by zero is impossible

Examples

5 ÷ 0  =  undefined        0 ÷ 0  =  undefined        1 ÷ 0  =  undefined

3. Division With Remainder

Often when we work out a division problem, the answer is not a whole number.  We can then write the answer as a whole number plus a remainder that is less than the divisor.

Example

34 ÷ 5

Solution

Since there is no whole number when multiplied by five produces 34, we find the nearest number without going over.  Notice that

5 x 6  =  30         and         5 x 7  =  35

hence 6 is the nearest number without going over.  Now notice that 30 is 4 short of 34.  We write

34 ÷ 5  =  6 R 4    "6 with a remainder of 4"

Example

4321 ÷ 6

Solution

720
6 | 4321
42
6 x 7  = 42
12
43 - 42  =  1 and drop down the 2
12
6 x 2  =  12
01
12 - 12  =  0 and drop down the 1
0
6 x 0  =  0
1
1 - 0  =  1

We can conclude that

4321 ÷ 6  =  720 R1

In general we write

(divisor x quotient) + remainder  =  dividend

Example

511
37 | 18932
185
37 x 5  = 185
43
189 - 185  =  4 and drop down the 3
37
37 x 1  =  37
62
43 - 37  =  6 and drop down the 2
37
37 x 1  =  37
25
62 - 37  =  25

We can conclude that

18932 ÷ 37  =  511 R25

Divide

A.  6275 ÷ 8

B.  3828 ÷ 7

C.  324337 ÷ 43

D.  6749 ÷ 103

1. Applications

Example

You are the manager of a ski resort and noticed that during the month of January you sold a total of 111,359 day ski tickets.  What was the average number of tickets that were sold that month?

Solution

Since there are 31 days in January, we need to divide the total number of tickets by 31

3589
31 | 111259
93
31 x 3  = 93
182
111 - 93  =  18 and drop down the 2
155
31 x 5  =  155
275
182 - 155  =  27 and drop down the 5
248
31 x 8  =  248
279
275 - 248  =  27

279    31 x 9  =  279

0

The ski resort averaged 3,589 ticket sales per day in the month of January.

Exercise

You are buying a custom refrigerator with a rectangular front.  If you only have enough space for the width to be 48 inches and you need the face to have an area of 2,976 square inches, how high must the refrigerator be?

Solution

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