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Textbook by: Denny Burzynski, Wade Ellis. E-mail the authors

# Subtraction of Whole Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to subtract whole numbers. By the end of this module, students should be able to understand the subtraction process, subtract whole numbers, and use a calculator to subtract one whole number from another whole number.

## Section Overview

• Subtraction
• Subtraction as the Opposite of Addition
• The Subtraction Process
• Subtraction Involving Borrowing
• Borrowing From Zero
• Calculators

## Subtraction

### Subtraction

Subtraction is the process of determining the remainder when part of the total is removed.

Suppose the sum of two whole numbers is 11, and from 11 we remove 4. Using the number line to help our visualization, we see that if we are located at 11 and move 4 units to the left, and thus remove 4 units, we will be located at 7. Thus, 7 units remain when we remove 4 units from 11 units.

### The Minus Symbol

The minus symbol (-) is used to indicate subtraction. For example, 114114 size 12{"11" - 4} {} indicates that 4 is to be subtracted from 11.

### Minuend

The number immediately in front of or the minus symbol is called the minuend, and it represents the original number of units.

### Subtrahend

The number immediately following or below the minus symbol is called the subtrahend, and it represents the number of units to be removed.

### Difference

The result of the subtraction is called the difference of the two numbers. For example, in 114=7114=7 size 12{"11" - 4=7} {}, 11 is the minuend, 4 is the subtrahend, and 7 is the difference.

## Subtraction as the Opposite of Addition

Subtraction can be thought of as the opposite of addition. We show this in the problems in Sample Set A.

### Sample Set A

#### Example 1

85=385=3 size 12{8 - 5=3} {} since 3+5=83+5=8 size 12{3+5=8} {}.

#### Example 2

93=693=6 size 12{9 - 3=6} {} since 6+3=96+3=9 size 12{6+3=9} {}.

### Practice Set A

Complete the following statements.

#### Exercise 1

75=75= size 12{7 - 5={}} {}


since

+5=7+5=7 size 12{+5=7} {}.

##### Solution

7-5=27-5=2 since 2+5=72+5=7

#### Exercise 2

91=91= size 12{9 - 1={}} {}


since

+1=9+1=9 size 12{+1=9} {}.

##### Solution

9-1=89-1=8 since 8+1=98+1=9

#### Exercise 3

178=178= size 12{"17" - 8={}} {}


since

+8=17+8=17 size 12{+8="17"} {}.

##### Solution

17-8=917-8=9 since 9+8=179+8=17

## The Subtraction Process

We'll study the process of the subtraction of two whole numbers by considering the difference between 48 and 35.

which we write as 13.

### Example 3: The Process of Subtracting Whole Numbers

To subtract two whole numbers,

The process

1. Write the numbers vertically, placing corresponding positions in the same column.

48 - 35 ̲ 48 - 35 ̲

2. Subtract the digits in each column. Start at the right, in the ones position, and move to the left, placing the difference at the bottom.

48 - 35 ̲ 13 48 - 35 ̲ 13

### Sample Set B

Perform the following subtractions.

#### Example 4

275 - 142 ̲ 133 275 - 142 ̲ 133

5 2 = 3 . 7 4 = 3 . 2 1 = 1 . 5 2 = 3 . 7 4 = 3 . 2 1 = 1 . alignr { stack { size 12{5 - 2=3 "." } {} # size 12{7 - 4=3 "." } {} # size 12{2 - 1=1 "." } {} } } {}

#### Example 5

46,042 -   1,031 ̲ 45,011 46,042 -   1,031 ̲ 45,011

2 1 = 1 . 4 3 = 1 . 0 0 = 0 . 6 1 = 5 . 4 0 = 4 . 2 1 = 1 . 4 3 = 1 . 0 0 = 0 . 6 1 = 5 . 4 0 = 4 . alignr { stack { size 12{2 - 1=1 "." } {} # size 12{4 - 3=1 "." } {} # size 12{0 - 0=0 "." } {} # size 12{6 - 1=5 "." } {} # size 12{4 - 0=4 "." } {} } } {}

#### Example 6

Find the difference between 977 and 235.

Write the numbers vertically, placing the larger number on top. Line up the columns properly.

977 - 235 ̲ 742 977 - 235 ̲ 742

The difference between 977 and 235 is 742.

#### Example 7

In Keys County in 1987, there were 809 cable television installations. In Flags County in 1987, there were 1,159 cable television installations. How many more cable television installations were there in Flags County than in Keys County in 1987?

We need to determine the difference between 1,159 and 809.

There were 350 more cable television installations in Flags County than in Keys County in 1987.

### Practice Set B

Perform the following subtractions.

#### Exercise 4

534 - 203 ̲ 534 - 203 ̲

331

#### Exercise 5

857 -   43 ̲ 857 -   43 ̲

814

#### Exercise 6

95,628 - 34,510 ̲ 95,628 - 34,510 ̲

61,118

#### Exercise 7

11,005 -   1,005 ̲ 11,005 -   1,005 ̲

10,000

#### Exercise 8

Find the difference between 88,526 and 26,412.

##### Solution

62,114

In each of these problems, each bottom digit is less than the corresponding top digit. This may not always be the case. We will examine the case where the bottom digit is greater than the corresponding top digit in the next section.

## Subtraction Involving Borrowing

### Minuend and Subtrahend

It often happens in the subtraction of two whole numbers that a digit in the minuend (top number) will be less than the digit in the same position in the subtrahend (bottom number). This happens when we subtract 27 from 84.

84 - 27 ̲ 84 - 27 ̲

We do not have a name for 4747 size 12{4 - 7} {}. We need to rename 84 in order to continue. We'll do so as follows:

Our new name for 84 is 7 tens + 14 ones.

= 57 = 57

Notice that we converted 8 tens to 7 tens + 1 ten, and then we converted the 1 ten to 10 ones. We then had 14 ones and were able to perform the subtraction.

### Borrowing

The process of borrowing (converting) is illustrated in the problems of Sample Set C.

### Sample Set C

#### Example 8

1. Borrow 1 ten from the 8 tens. This leaves 7 tens.
2. Convert the 1 ten to 10 ones.
3. Add 10 ones to 4 ones to get 14 ones.

#### Example 9

1. Borrow 1 hundred from the 6 hundreds. This leaves 5 hundreds.
2. Convert the 1 hundred to 10 tens.
3. Add 10 tens to 7 tens to get 17 tens.

### Practice Set C

Perform the following subtractions. Show the expanded form for the first three problems.

#### Exercise 9

53 - 35 ̲ 53 - 35 ̲

#### Exercise 10

76 - 28 ̲ 76 - 28 ̲

#### Exercise 11

872 - 565 ̲ 872 - 565 ̲

#### Exercise 12

441 - 356 ̲ 441 - 356 ̲

85

#### Exercise 13

775 -   66 ̲ 775 -   66 ̲

709

#### Exercise 14

5,663 - 2,559 ̲ 5,663 - 2,559 ̲

3,104

### Borrowing More Than Once

Sometimes it is necessary to borrow more than once. This is shown in the problems in Section 12.

### Sample Set D

Perform the Subtractions. Borrowing more than once if necessary

#### Example 10

1. Borrow 1 ten from the 4 tens. This leaves 3 tens.
2. Convert the 1 ten to 10 ones.
3. Add 10 ones to 1 one to get 11 ones. We can now perform 118118 size 12{"11" - 8} {}.
4. Borrow 1 hundred from the 6 hundreds. This leaves 5 hundreds.
5. Convert the 1 hundred to 10 tens.
6. Add 10 tens to 3 tens to get 13 tens.
7. Now 135=8135=8 size 12{"13" - 5=8} {}.
8. 53=253=2 size 12{5 - 3=2} {}.

#### Example 11

1. Borrow 1 ten from the 3 tens. This leaves 2 tens.
2. Convert the 1 ten to 10 ones.
3. Add 10 ones to 4 ones to get 14 ones. We can now perform 145145 size 12{"14" - 5} {}.
4. Borrow 1 hundred from the 5 hundreds. This leaves 4 hundreds.
5. Convert the 1 hundred to 10 tens.
6. Add 10 tens to 2 tens to get 12 tens. We can now perform 128=4128=4 size 12{"12" - 8=4} {}.
7. Finally, 40=440=4 size 12{4 - 0=4} {}.

#### Example 12

71529 - 6952 ̲ 71529 - 6952 ̲ alignr { stack { size 12{"71529"} {} # size 12{ {underline { - "6952"}} } {} } } {}

After borrowing, we have

### Practice Set D

Perform the following subtractions.

#### Exercise 15

526 - 358 ̲ 526 - 358 ̲

168

#### Exercise 16

63,419 -   7,779 ̲ 63,419 -   7,779 ̲

55,640

#### Exercise 17

4,312 - 3,123 ̲ 4,312 - 3,123 ̲

1,189

## Borrowing from Zero

It often happens in a subtraction problem that we have to borrow from one or more zeros. This occurs in problems such as

1. 503 -   37 ̲ 503 -   37 ̲

and
2. 5000 -     37 ̲ 5000 -     37 ̲

We'll examine each case.

### Example 13: Borrowing from a single zero.

Consider the problem 503 -   37 ̲ 503 -   37 ̲

Since we do not have a name for 3737 size 12{3 - 7} {}, we must borrow from 0.

Since there are no tens to borrow, we must borrow 1 hundred. One hundred = 10 tens.

We can now borrow 1 ten from 10 tens (leaving 9 tens). One ten = 10 ones and 10 ones + 3 ones = 13 ones.

Now we can suggest the following method for borrowing from a single zero.

### Borrowing from a Single Zero

To borrow from a single zero,

1. Decrease the digit to the immediate left of zero by one.
2. Draw a line through the zero and make it a 10.
3. Proceed to subtract as usual.

### Sample Set E

#### Example 14

Perform this subtraction.

503 -   37 ̲ 503 -   37 ̲

The number 503 contains a single zero

1. The number to the immediate left of 0 is 5. Decrease 5 by 1.

5 1 = 4 5 1 = 4 size 12{5 - 1=4} {}

2. Draw a line through the zero and make it a 10.
3. Borrow from the 10 and proceed.

1 ten+10 ones1 ten+10 ones

10 ones+ 3 ones=13 ones10 ones+3 ones=13 ones

### Practice Set E

Perform each subtraction.

#### Exercise 18

906 -   18 ̲ 906 -   18 ̲

888

#### Exercise 19

5102 -   559 ̲ 5102 -   559 ̲

4,543

#### Exercise 20

9055 -   386 ̲ 9055 -   386 ̲

8,669

### Example 15: Borrowing from a group of zeros

Consider the problem 5000 -     37 ̲ 5000 -     37 ̲

In this case, we have a group of zeros.

Since we cannot borrow any tens or hundreds, we must borrow 1 thousand. One thousand = 10 hundreds.

We can now borrow 1 hundred from 10 hundreds. One hundred = 10 tens.

We can now borrow 1 ten from 10 tens. One ten = 10 ones.

From observations made in this procedure we can suggest the following method for borrowing from a group of zeros.

### Borrowing from a Group of zeros

To borrow from a group of zeros,

1. Decrease the digit to the immediate left of the group of zeros by one.
2. Draw a line through each zero in the group and make it a 9, except the rightmost zero, make it 10.
3. Proceed to subtract as usual.

### Sample Set F

Perform each subtraction.

#### Example 16

40,000 -     125 ̲ 40,000 -     125 ̲

The number 40,000 contains a group of zeros.

1. The number to the immediate left of the group is 4. Decrease 4 by 1.

4 1 = 3 4 1 = 3 size 12{4 - 1=3} {}

2. Make each 0, except the rightmost one, 9. Make the rightmost 0 a 10.

3. Subtract as usual.

#### Example 17

8,000,006 -       41,107 ̲ 8,000,006 -       41,107 ̲

The number 8,000,006 contains a group of zeros.

1. The number to the immediate left of the group is 8. Decrease 8 by 1.

8 1 = 7 8 1 = 7 size 12{8 - 1=7} {}

2. Make each zero, except the rightmost one, 9. Make the rightmost 0 a 10.

3. To perform the subtraction, we’ll need to borrow from the ten.

1 ten = 10 ones 10 ones + 6 ones = 16 ones 1 ten = 10 ones 10 ones + 6 ones = 16 ones

### Practice Set F

Perform each subtraction.

#### Exercise 21

21,007 -   4,873 ̲ 21,007 -   4,873 ̲

16,134

#### Exercise 22

10,004 -   5,165 ̲ 10,004 -   5,165 ̲

4,839

#### Exercise 23

16,000,000 -      201,060 ̲ 16,000,000 -      201,060 ̲

15,789,940

## Calculators

In practice, calculators are used to find the difference between two whole numbers.

### Sample Set G

Find the difference between 1006 and 284.

 Display Reads Type 1006 1006 Press − − size 12{ - } {} 1006 Type 284 284 Press = 722

The difference between 1006 and 284 is 722.

(What happens if you type 284 first and then 1006? We'll study such numbers in (Reference)Chapter 10.)

### Practice Set G

#### Exercise 24

Use a calculator to find the difference between 7338 and 2809.

4,529

#### Exercise 25

Use a calculator to find the difference between 31,060,001 and 8,591,774.

22,468,227

## Exercises

For the following problems, perform the subtractions. You may check each difference with a calculator.

### Exercise 26

15 -   8 ̲ 15 -   8 ̲

7

### Exercise 27

19 -   8 ̲ 19 -   8 ̲

### Exercise 28

11 -   5 ̲ 11 -   5 ̲

6

### Exercise 29

14 -   6 ̲ 14 -   6 ̲

### Exercise 30

12 -   9 ̲ 12 -   9 ̲

3

### Exercise 31

56 - 12 ̲ 56 - 12 ̲

### Exercise 32

74 - 33 ̲ 74 - 33 ̲

41

### Exercise 33

80 - 61 ̲ 80 - 61 ̲

### Exercise 34

350 - 141 ̲ 350 - 141 ̲

209

### Exercise 35

800 - 650 ̲ 800 - 650 ̲

### Exercise 36

35,002 - 14,001 ̲ 35,002 - 14,001 ̲

21,001

### Exercise 37

5,000,566 - 2,441,326 ̲ 5,000,566 - 2,441,326 ̲

### Exercise 38

400,605 - 121,352 ̲ 400,605 - 121,352 ̲

279,253

### Exercise 39

46,400 -   2,012 ̲ 46,400 -   2,012 ̲

### Exercise 40

77,893 -      421 ̲ 77,893 -      421 ̲

77,472

### Exercise 41

42 - 18 ̲ 42 - 18 ̲

### Exercise 42

51 - 27 ̲ 51 - 27 ̲

24

### Exercise 43

622 -   88 ̲ 622 -   88 ̲

### Exercise 44

261 -   73 ̲ 261 -   73 ̲

188

### Exercise 45

242 - 158 ̲ 242 - 158 ̲

### Exercise 46

3,422 - 1,045 ̲ 3,422 - 1,045 ̲

2,377

### Exercise 47

5,565 - 3,985 ̲ 5,565 - 3,985 ̲

### Exercise 48

42,041 - 15,355 ̲ 42,041 - 15,355 ̲

26,686

### Exercise 49

304,056 -   20,008 ̲ 304,056 -   20,008 ̲

### Exercise 50

64,000,002 -     856,743 ̲ 64,000,002 -     856,743 ̲

63,143,259

### Exercise 51

4,109 -   856 ̲ 4,109 -   856 ̲

### Exercise 52

10,113 -   2,079 ̲ 10,113 -   2,079 ̲

8,034

### Exercise 53

605 -   77 ̲ 605 -   77 ̲

### Exercise 54

59 - 26 ̲ 59 - 26 ̲

33

### Exercise 55

36,107 -   8,314 ̲ 36,107 -   8,314 ̲

### Exercise 56

92,526,441,820 - 59,914,805,253 ̲ 92,526,441,820 - 59,914,805,253 ̲

32,611,636,567

### Exercise 57

1,605 -   881 ̲ 1,605 -   881 ̲

### Exercise 58

30,000 - 26,062 ̲ 30,000 - 26,062 ̲

3,938

### Exercise 59

600 - 216 ̲ 600 - 216 ̲

### Exercise 60

9,000,003 -    726,048 ̲ 9,000,003 -    726,048 ̲

#### Solution

8,273,955

For the following problems, perform each subtraction.

### Exercise 61

Subtract 63 from 92.

#### Hint:

The word "from" means "beginning at." Thus, 63 from 92 means beginning at 92, or 92 63 92 63 size 12{"92" - "63"} {} .

### Exercise 62

Subtract 35 from 86.

51

### Exercise 63

Subtract 382 from 541.

### Exercise 64

Subtract 1,841 from 5,246.

3,405

### Exercise 65

Subtract 26,082 from 35,040.

### Exercise 66

Find the difference between 47 and 21.

26

### Exercise 67

Find the difference between 1,005 and 314.

### Exercise 68

Find the difference between 72,085 and 16.

72,069

### Exercise 69

Find the difference between 7,214 and 2,049.

### Exercise 70

Find the difference between 56,108 and 52,911.

3,197

### Exercise 71

How much bigger is 92 than 47?

### Exercise 72

How much bigger is 114 than 85?

29

### Exercise 73

How much bigger is 3,006 than 1,918?

### Exercise 74

How much bigger is 11,201 than 816?

10,385

### Exercise 75

How much bigger is 3,080,020 than 1,814,161?

### Exercise 76

In Wichita, Kansas, the sun shines about 74% of the time in July and about 59% of the time in November. How much more of the time (in per­cent) does the sun shine in July than in No­vember?

15%

### Exercise 77

The lowest temperature on record in Concord, New Hampshire in May is 21°F, and in July it is 35°F. What is the difference in these lowest tem­peratures?

### Exercise 78

In 1980, there were 83,000 people arrested for prostitution and commercialized vice and 11,330,000 people arrested for driving while in­toxicated. How many more people were arrested for drunk driving than for prostitution?

11,247,000

### Exercise 79

In 1980, a person with a bachelor's degree in ac­counting received a monthly salary offer of $1,293, and a person with a marketing degree a monthly salary offer of$1,145. How much more was offered to the person with an accounting de­gree than the person with a marketing degree?

### Exercise 80

In 1970, there were about 793 people per square mile living in Puerto Rico, and 357 people per square mile living in Guam. How many more people per square mile were there in Puerto Rico than Guam?

436

### Exercise 81

The 1980 population of Singapore was 2,414,000 and the 1980 population of Sri Lanka was 14,850,000. How many more people lived in Sri Lanka than in Singapore in 1980?

### Exercise 82

In 1977, there were 7,234,000 hospitals in the United States and 64,421,000 in Mainland China. How many more hospitals were there in Mainland China than in the United States in 1977?

57,187,000

### Exercise 83

In 1978, there were 3,095,000 telephones in use in Poland and 4,292,000 in Switzerland. How many more telephones were in use in Switzerland than in Poland in 1978?

For the following problems, use the corresponding graphs to solve the problems.

### Exercise 84

How many more life scientists were there in 1974 than mathematicians? (Media 29)

165,000

### Exercise 85

How many more social, psychological, mathe­matical, and environmental scientists were there than life, physical, and computer scientists? (Media 29)

### Exercise 86

How many more prosecutions were there in 1978 than in 1974? (Media 30)

74

### Exercise 87

How many more prosecutions were there in 1976-1980 than in 1970-1975? (Media 30)

### Exercise 88

How many more dry holes were drilled in 1960 than in 1975? (Media 31)

4,547

### Exercise 89

How many more dry holes were drilled in 1960, 1965, and 1970 than in 1975, 1978 and 1979? (Media 31)

For the following problems, replace the ☐ with the whole number that will make the subtraction true.

### Exercise 90

14 - ̲ 3 14 - ̲ 3

11

### Exercise 91

21 - ̲ 14 21 - ̲ 14

### Exercise 92

35 - ̲ 25 35 - ̲ 25

10

### Exercise 93

16 - ̲ 9 16 - ̲ 9

### Exercise 94

28 - ̲ 16 28 - ̲ 16

#### Solution

12

For the following problems, find the solutions.

### Exercise 95

Subtract 42 from the sum of 16 and 56.

### Exercise 96

Subtract 105 from the sum of 92 and 89.

76

### Exercise 97

Subtract 1,127 from the sum of 2,161 and 387.

### Exercise 98

Subtract 37 from the difference between 263 and 175.

51

### Exercise 99

Subtract 1,109 from the difference between 3,046 and 920.

### Exercise 100

Add the difference between 63 and 47 to the dif­ference between 55 and 11.

60

### Exercise 101

Add the difference between 815 and 298 to the difference between 2,204 and 1,016.

### Exercise 102

Subtract the difference between 78 and 43 from the sum of 111 and 89.

165

### Exercise 103

Subtract the difference between 18 and 7 from the sum of the differences between 42 and 13, and 81 and 16.

### Exercise 104

Find the difference between the differences of 343 and 96, and 521 and 488.

214

### Exercises for Review

#### Exercise 105

((Reference)) In the number 21,206, how many hundreds are there?

#### Exercise 106

((Reference)) Write a three-digit number that has a zero in the ones position.

#### Exercise 107

((Reference)) How many three-digit whole numbers are there?

#### Exercise 108

((Reference)) Round 26,524,016 to the nearest million.

27,000,000

#### Exercise 109

((Reference)) Find the sum of 846+221+116846+221+116 size 12{"846"+"221"+"116"} {}.

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