1.5: Subtraction of Whole Numbers

Learning Objectives

• understand the subtraction process
• be able to subtract whole numbers
• be able to use a calculator to subtract one whole number from another whole number

Subtraction

Definition: 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.

Definition: The Minus Symbol

The minus symbol (-) is used to indicate subtraction. For example, 11 − 4 indicates that 4 is to be subtracted from 11.

Definition: Minuend

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

Definition: Subtrahend

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

Definition: Difference

The result of the subtraction is called the difference of the two numbers. For example, in 11 − 4 = 7

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

8 - 5 = 3 since 3 + 5 = 8.

Sample Set A

9 - 3 = 6 since 6 + 3 = 9.

Practice Set A

Complete the following statements.

7 - 5 = since + 5 = 7

Answer

7 - 5 = 2 since 2 + 5 = 7

Practice Set A

9 - 1 = since + 1 = 9

Answer

9 - 1 = 8 since 8 + 1 = 9

Practice Set A

17 - 8 = since + 8 = 17

Answer

17 - 8 = 9 since 9 + 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.

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.
   $\begin{array} {r} {48} \\ {\underline{-35}} \end{array}$
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.
   $\begin{array} {r} {48} \\ {\underline{-35}} \\ {13} \end{array}$

Sample Set B

Perform the following subtractions.

$\begin{array} {r} {275} \\ {\underline{-142}} \\ {133} \end{array}$

$\begin{array} {l} {5 - 2 = 3.} \\ {7 - 4 = 3.} \\ {2 - 1 = 1.} \end{array}$

Sample Set B

$\begin{array} {r} {46,042} \\ {\underline{-\ \ 1,031}} \\ {45,011} \end{array}$

$\begin{array} {l} {2 - 1 = 1.} \\ {4 - 3 = 1.} \\ {0 - 0 = 0.} \\ {6 - 1 = 5.} \\ {4 - 0 = 4.} \end{array}$

Sample Set B

Find the difference between 977 and 235.

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

$\begin{array} {r} {977} \\ {\underline{-235}} \\ {742} \end{array}$

The difference between 977 and 235 is 742.

Sample Set B

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.

$\begin{array} {r} {534} \\ {\underline{-203}} \end{array}$

Answer

331

Practice Set B

$\begin{array} {r} {857} \\ {\underline{-\ \ 43}} \end{array}$

Answer

814

Practice Set B

$\begin{array} {r} {95,628} \\ {\underline{-34,510}} \end{array}$

Answer

61,118

Practice Set B

$\begin{array} {r} {11,005} \\ {\underline{-\ \ 1,005}} \end{array}$

Answer

10,000

Practice Set B

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

Answer

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

Definition: 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.

$\begin{array} {r} {84} \\ {\underline{-27}} \end{array}$

We do not have a name for 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

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.

Definition: Borrowing

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

Sample Set C

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.

Sample Set C

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.

$\begin{array} {r} {53} \\ {\underline{-35}} \end{array}$

Answer

Practice Set C

$\begin{array} {r} {76} \\ {\underline{-28}} \end{array}$

Answer

Practice Set C

$\begin{array} {r} {872} \\ {\underline{-565}} \end{array}$

Answer

Practice Set C

$\begin{array} {r} {441} \\ {\underline{-356}} \end{array}$

Answer

85

Practice Set C

$\begin{array} {r} {775} \\ {\underline{-\ \ 66}} \end{array}$

Answer

709

Practice Set C

$\begin{array} {r} {5,663} \\ {\underline{-2,559}} \end{array}$

Answer

3,104

Borrowing More Than Once

Sometimes it is necessary to borrow more than once. This is shown in the problems in Sample Set D.

Sample Set D

Perform the Subtractions. Borrowing more than once if necessary

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 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 13 − 5 = 8.
8. 5 − 3 = 2.

Sample Set D

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 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 12 − 8 = 4.
7. Finally, 4 − 0 = 4.

Sample Set D

$\begin{array} {r} {71529} \\ {\underline{-\ \ 6952}} \end{array}$

After borrowing, we have

Practice Set D

Perform the following subtractions.

$\begin{array} {r} {526} \\ {\underline{-358}} \end{array}$

Answer

168

Practice Set D

$\begin{array} {r} {63,419} \\ {\underline{-\ \ 7,779}} \end{array}$

Answer

55,640

Practice Set D

$\begin{array} {r} {4,312} \\ {\underline{-3,123}} \end{array}$

Answer

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

$\begin{array} {r} {503} \\ {\underline{-\ \ 37}} \\ {\text{and}\ \ \ \ } \\ {5000} \\ {\underline{-\ \ \ \ 37}} \end{array}$

We'll examine each case.

Borrowing from a single zero.

Consider the problem $\begin{array} {r} {503} \\ {\underline{-\ \ 37}} \end{array}$

Since we do not have a name for 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

Perform this subtraction.

$\begin{array} {r} {503} \\ {\underline{-\ \ 37}} \end{array}$

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
   
2. Draw a line through the zero and make it a 10.
3. Borrow from the 10 and proceed.
   $\begin{array} {l} {\text{1 ten + 10 ones}} \\ {\text{10 ones + 3 ones = 13 ones}} \end{array}$

Practice Set E

Perform each subtraction.

$\begin{array} {r} {906} \\ {\underline{-\ \ 18}} \end{array}$

Answer

888

Practice Set E

$\begin{array} {r} {5102} \\ {\underline{-\ \ 559}} \end{array}$

Answer

4,543

Practice Set E

$\begin{array} {r} {9055} \\ {\underline{-\ \ 386}} \end{array}$

Answer

8,669

Borrowing from a group of zeros

Consider the problem $\begin{array} {r} {5000} \\ {\underline{-\ \ \ \ 37}} \end{array}$

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.

$\begin{array} {r} {40,000} \\ {\underline{-\ \ \ \ \ 125}} \end{array}$

Solution

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
2. Make each 0, except the rightmost one, 9. Make the rightmost 0 a 10.
3. Subtract as usual.

Example $\PageIndex{1}$

$\begin{array} {r} {8,000,006} \\ {\underline{-\ \ \ \ \ 41,107}} \end{array}$

Solution

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
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.
   $\begin{array} {l} {\text{1 ten = 10 ones}} \\ {\text{10 ones + 6 ones = 16 ones}} \end{array}$

Practice Set F

Perform each subtraction.

$\begin{array} {r} {21,007} \\ {\underline{-\ \ 4,873}} \end{array}$

Answer

16,134

Practice Set F

$\begin{array} {r} {10,004} \\ {\underline{-\ \ 5,165}} \end{array}$

Answer

4,839

Practice Set F

$\begin{array} {r} {16,000,000} \\ {\underline{-\ \ \ \ \ 201,060}} \end{array}$

Answer

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	−−	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 Chapter 10.)

Practice Set G

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

Answer

4,529

Practice Set G

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

Answer

22,468,227

Exercises

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

Exercise $\PageIndex{1}$

$\begin{array} {r} {15} \\ {\underline{-\ \ 8}} \end{array}$

Answer

7

Exercise $\PageIndex{2}$

$\begin{array} {r} {19} \\ {\underline{-\ \ 8}} \end{array}$

Exercise $\PageIndex{3}$

$\begin{array} {r} {11} \\ {\underline{-\ \ 5}} \end{array}$

Answer

6

Exercise $\PageIndex{4}$

$\begin{array} {r} {14} \\ {\underline{-\ \ 6}} \end{array}$

Exercise $\PageIndex{5}$

$\begin{array} {r} {12} \\ {\underline{-\ \ 9}} \end{array}$

Answer

3

Exercise $\PageIndex{6}$

$\begin{array} {r} {56} \\ {\underline{-12}} \end{array}$

Exercise $\PageIndex{7}$

$\begin{array} {r} {74} \\ {\underline{-33}} \end{array}$

Answer

41

Exercise $\PageIndex{8}$

$\begin{array} {r} {80} \\ {\underline{-61}} \end{array}$

Exercise $\PageIndex{9}$

$\begin{array} {r} {350} \\ {\underline{-141}} \end{array}$

Answer

209

Exercise $\PageIndex{10}$

$\begin{array} {r} {800} \\ {\underline{-650}} \end{array}$

Exercise $\PageIndex{11}$

$\begin{array} {r} {35,002} \\ {\underline{-14,001}} \end{array}$

Answer

21,001

Exercise $\PageIndex{12}$

$\begin{array} {r} {5,000,566} \\ {\underline{-2,441,326}} \end{array}$

Exercise $\PageIndex{13}$

$\begin{array} {r} {400,605} \\ {\underline{-121,352}} \end{array}$

Answer

279,253

Exercise $\PageIndex{14}$

$\begin{array} {r} {46,400} \\ {\underline{-\ \ 2,012}} \end{array}$

Exercise $\PageIndex{15}$

$\begin{array} {r} {77,893} \\ {\underline{-\ \ \ \ \ 421}} \end{array}$

Answer

77,472

Exercise $\PageIndex{16}$

$\begin{array} {r} {42} \\ {\underline{-18}} \end{array}$

Exercise $\PageIndex{17}$

$\begin{array} {r} {51} \\ {\underline{-27}} \end{array}$

Answer

24

Exercise $\PageIndex{18}$

$\begin{array} {r} {622} \\ {\underline{-\ \ 88}} \end{array}$

Exercise $\PageIndex{19}$

$\begin{array} {r} {261} \\ {\underline{-\ \ 73}} \end{array}$

Answer

188

Exercise $\PageIndex{20}$

$\begin{array} {r} {242} \\ {\underline{-158}} \end{array}$

Exercise $\PageIndex{21}$

$\begin{array} {r} {3,422} \\ {\underline{-1,045}} \end{array}$

Answer

2,377

Exercise $\PageIndex{22}$

$\begin{array} {r} {5,565} \\ {\underline{-3,985}} \end{array}$

Exercise $\PageIndex{23}$

$\begin{array} {r} {42,041} \\ {\underline{-15,355}} \end{array}$

Answer

26,686

Exercise $\PageIndex{24}$

$\begin{array} {r} {304,056} \\ {\underline{-\ \ 20,008}} \end{array}$

Exercise $\PageIndex{25}$

$\begin{array} {r} {64,000,002} \\ {\underline{-\ \ \ \ \ 856,743}} \end{array}$

Answer

63,143,259

Exercise $\PageIndex{26}$

$\begin{array} {r} {4,109} \\ {\underline{-\ \ \ 856}} \end{array}$

Exercise $\PageIndex{27}$

$\begin{array} {r} {10,113} \\ {\underline{-\ \ 2,079}} \end{array}$

Answer

8,034

Exercise $\PageIndex{28}$

$\begin{array} {r} {605} \\ {\underline{-\ \ 77}} \end{array}$

Exercise $\PageIndex{27}$

$\begin{array} {r} {59} \\ {\underline{-26}} \end{array}$

Answer

33

Exercise $\PageIndex{28}$

$\begin{array} {r} {36,107} \\ {\underline{-\ \ 8,314}} \end{array}$

Exercise $\PageIndex{29}$

$\begin{array} {r} {92,526,441,820} \\ {\underline{-59,914,805,253}} \end{array}$

Answer

32,611,636,567

Exercise $\PageIndex{30}$

$\begin{array} {r} {1,605} \\ {\underline{-\ \ 881}} \end{array}$

Exercise $\PageIndex{31}$

$\begin{array} {r} {30,000} \\ {\underline{-26,062}} \end{array}$

Answer

3,938

Exercise $\PageIndex{32}$

$\begin{array} {r} {600} \\ {\underline{-216}} \end{array}$

Exercise $\PageIndex{33}$

$\begin{array} {r} {90,000,003} \\ {\underline{-\ \ \ 726,048}} \end{array}$

Answer

8,273,955

For the following problems, perform each subtraction.

Exercise $\PageIndex{34}$

Subtract 63 from 92.

Hint:

The word "from" means "beginning at." Thus, 63 from 92 means beginning at 92, or 92 - 63.

Exercise $\PageIndex{35}$

Subtract 35 from 86.

Answer

51

Exercise $\PageIndex{34}$

Subtract 382 from 541.

Exercise $\PageIndex{35}$

Subtract 1,841 from 5,246.

Answer

3,405

Exercise $\PageIndex{36}$

Subtract 26,082 from 35,040.

Exercise $\PageIndex{37}$

Find the difference between 47 and 21.

Answer

26

Exercise $\PageIndex{38}$

Find the difference between 1,005 and 314.

Exercise $\PageIndex{39}$

Find the difference between 72,085 and 16.

Answer

72,069

Exercise $\PageIndex{40}$

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

Exercise $\PageIndex{41}$

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

Answer

3,197

Exercise $\PageIndex{42}$

How much bigger is 92 than 47?

Exercise $\PageIndex{43}$

How much bigger is 114 than 85?

Answer

29

Exercise $\PageIndex{44}$

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

Exercise $\PageIndex{45}$

How much bigger is 11,201 than 816?

Answer

10,385

Exercise $\PageIndex{46}$

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

Exercise $\PageIndex{47}$

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?

Answer

15%

Exercise $\PageIndex{48}$

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 $\PageIndex{49}$

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?

Answer

11,247,000

Exercise $\PageIndex{50}$

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 $\PageIndex{51}$

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?

Answer

436

Exercise $\PageIndex{52}$

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 $\PageIndex{53}$

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?

Answer

57,187,000

Exercise $\PageIndex{54}$

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 $\PageIndex{55}$

How many more life scientists were there in 1974 than mathematicians?

Answer

165,000

Exercise $\PageIndex{56}$

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

Exercise $\PageIndex{57}$

How many more prosecutions were there in 1978 than in 1974?

Answer

74

Exercise $\PageIndex{58}$

How many more prosecutions were there in 1976-1980 than in 1970-1975?

Exercise $\PageIndex{59}$

How many more dry holes were drilled in 1960 than in 1975?

Answer

4,547

Exercise $\PageIndex{60}$

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

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

Exercise $\PageIndex{61}$

$\begin{array} {r} {14} \\ {\underline{-☐}} \\ {3} \end{array}$

Answer

11

Exercise $\PageIndex{62}$

$\begin{array} {r} {21} \\ {\underline{-☐}} \\ {14} \end{array}$

Exercise $\PageIndex{63}$

$\begin{array} {r} {35} \\ {\underline{-☐}} \\ {25} \end{array}$

Answer

10

Exercise $\PageIndex{64}$

$\begin{array} {r} {16} \\ {\underline{-☐}} \\ {9} \end{array}$

Exercise $\PageIndex{65}$

$\begin{array} {r} {28} \\ {\underline{-☐}} \\ {16} \end{array}$

Answer

12

For the following problems, find the solutions.

Exercise $\PageIndex{66}$

Subtract 42 from the sum of 16 and 56.

Exercise $\PageIndex{67}$

Subtract 105 from the sum of 92 and 89.

Answer

76

Exercise $\PageIndex{68}$

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

Exercise $\PageIndex{69}$

Subtract 37 from the difference between 263 and 175.

Answer

51

Exercise $\PageIndex{70}$

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

Exercise $\PageIndex{71}$

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

Answer

60

Exercise $\PageIndex{72}$

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

Exercise $\PageIndex{73}$

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

Answer

165

Exercise $\PageIndex{74}$

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

Exercise $\PageIndex{75}$

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

Answer

214

Exercises for Review

Exercise $\PageIndex{76}$

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

Exercise $\PageIndex{77}$

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

Answer

330 (answers may vary)

Exercise $\PageIndex{78}$

How many three-digit whole numbers are there?

Exercise $\PageIndex{79}$

Round 26,524,016 to the nearest million.

Answer

27,000,000

Exercise $\PageIndex{80}$

Find the sum of 846 + 221 + 116.