←
Subtract Whole Numbers

1.3 Subtract Whole Numbers

Learning Objectives

By the end of this section, you will be able to:

  • Use subtraction notation
  • Model subtraction of whole numbers
  • Subtract whole numbers
  • Translate word phrases to math notation
  • Subtract whole numbers in applications

Be Prepared 1.3

Before you get started, take this readiness quiz.

Model 3+43+4 using base-ten blocks.
 If you missed this problem, review Example 1.12.

Be Prepared 1.4

Add: 324+586.324+586.
 If you missed this problem, review Example 1.20.

Use Subtraction Notation

Suppose there are seven bananas in a bowl. Elana uses three of them to make a smoothie. How many bananas are left in the bowl? To answer the question, we subtract three from seven. When we subtract, we take one number away from another to find the difference. The notation we use to subtract 33 from 77 is

7−37−3

We read 7−37−3 as seven minus three and the result is the difference of seven and three.

Subtraction Notation

To describe subtraction, we can use symbols and words.

Operation Notation Expression Read as Result
Subtraction −− 7−37−3 seven minus three the difference of 77 and 33

Example 1.26

Translate from math notation to words: ⓐ 8−18−1 ⓑ 26−1426−14.

Solution

  • ⓐ We read this as eight minus one. The result is the difference of eight and one.
  • ⓑ We read this as twenty-six minus fourteen. The result is the difference of twenty-six and fourteen.

Try It 1.51

Translate from math notation to words:

  1. ⓐ 12−412−4
  2. ⓑ 29−1129−11

Try It 1.52

Translate from math notation to words:

  1. ⓐ 11−211−2
  2. ⓑ 29−1229−12

Model Subtraction of Whole Numbers

A model can help us visualize the process of subtraction much as it did with addition. Again, we will use base-10base-10 blocks. Remember a block represents 1 and a rod represents 10. Let’s start by modeling the subtraction expression we just considered, 7−3.7−3.

We start by modeling the first number, 7. CNX_BMath_Figure_01_03_018_img-02.png
Now take away the second number, 3. We'll circle 3 blocks to show that we are taking them away. CNX_BMath_Figure_01_03_018_img-03.png
Count the number of blocks remaining. Four light blue squares with dark outlines and subtle drop shadows are arranged horizontally on a clean white background.
There are 4 ones blocks left. We have shown that 7−3=47−3=4.
This table visually demonstrates the subtraction of 3 from 7 using blocks, illustrating each step from initial modeling to the final mathematical result.

Example 1.27

Model the subtraction: 8−2.8−2.

Solution

8−28−2 means the difference of 8 and 2.
Model the first, 8. A row of eight light blue squares with dark borders, with the number '8' centered directly below them on a white background.
Take away the second number, 2. CNX_BMath_Figure_01_03_019_img-03.png
Count the number of blocks remaining. CNX_BMath_Figure_01_03_019_img-04.png
There are 6 ones blocks left. We have shown that 8−2=68−2=6.

Try It 1.53

Model: 9−6.9−6.

Try It 1.54

Model: 6−1.6−1.

Example 1.28

Model the subtraction: 13−8.13−8.

Solution

Model the first number, 13. We use 1 ten and 3 ones. Ten light blue blocks aligned horizontally, followed by a gap and then three more blocks. This simple graphic might illustrate quantities, data segments, or a step-by-step process.
Take away the second number, 8. However, there are not 8 ones, so we will exchange the 1 ten for 10 ones. Two groups of light blue squares with dark outlines are displayed on a white background, one group has 10 squares and the other has 3 squares, with the numbers 10 and 3 placed below them respectively.
Now we can take away 8 ones. A row of light blue squares. Ten squares are enclosed by a magenta oval, with an arrow pointing left underneath, indicating a loop or repetition. Four additional squares are to the right.
Count the blocks remaining. CNX_BMath_Figure_01_03_020_img-05.png
There are five ones left. We have shown that 13−8=513−8=5.

As we did with addition, we can describe the models as ones blocks and tens rods, or we can simply say ones and tens.

Try It 1.55

Model the subtraction: 12−7.12−7.

Try It 1.56

Model the subtraction: 14−8.14−8.

Example 1.29

Model the subtraction: 43−26.43−26.

Solution

Because 43−2643−26 means 4343 take away 26,26, we begin by modeling the 43.43.

An image containing two items. The first item is 4 horizontal rods containing 10 blocks each. The second item is 3 individual blocks.

Now, we need to take away 26,26, which is 22 tens and 66 ones. We cannot take away 66 ones from 33 ones. So, we exchange 11 ten for 1010 ones.

This figure contains two groups. The first group on the left includes 3 rows of blue base 10 blocks and 1 red row of 10 blocks. This is labeled 4 tens. Alongside the first row of ten blocks are 3 individual blocks. This is labeled 3 ones. An arrow points to the right to the second group in which there are three rows of 10 base blocks labeled 3 tens. Next to this is a row of 3 blue individual blocks and two rows each with five individual blocks in red. This is labeled 13 ones.

Now we can take away 22 tens and 66 ones.

This image includes one row of base ten blocks at the top of the image; Next to it are seven individual blocks. Below this, is a group of two rows of base ten blocks, and two rows of 3 individual blocks with a circle around all. The arrow points to the right and shows one row of ten blocks and seven individual blocks underneath.

Count the number of blocks remaining. There is 11 ten and 77 ones, which is 17.17.

43 − 26 = 17 43 − 26 = 17

Try It 1.57

Model the subtraction: 42−27.42−27.

Try It 1.58

Model the subtraction: 45−29.45−29.

Subtract Whole Numbers

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.

We know 7−3=47−3=4 because 4+3=7.4+3=7. Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding. In the examples above, our subtractions can be checked by addition.

7−3=4because4+3=713−8=5because5+8=1343−26=17because17+26=437−3=4because4+3=713−8=5because5+8=1343−26=17because17+26=43

Example 1.30

Subtract and then check by adding:

  1. ⓐ 9−79−7
  2. ⓑ 8−3.8−3.

Solution

ⓐ
9−79−7
Subtract 7 from 9. 22
Check with addition.
2+7=9✓2+7=9✓
Illustration of subtracting 7 from 9, with expression, result, and addition-based verification.
ⓑ
8−38−3
Subtract 3 from 8. 55
Check with addition.
5+3=8✓5+3=8✓
This table demonstrates the subtraction of 3 from 8, including the problem, its result, and a verification step.

Try It 1.59

Subtract and then check by adding:

7−07−0

Try It 1.60

Subtract and then check by adding:

6−26−2

To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.

Example 1.31

Subtract and then check by adding: 89−61.89−61.

Solution

Write the numbers so the ones and tens digits line up vertically. 89 −61____ 89 −61____
Subtract the digits in each place value.

Subtract the ones: 9-1=89-1=8
Subtract the tens: 8-6=28-6=2		 89 −61____ 28 89 −61____ 28
Check using addition.
 28 +61____ 89 28 +61____ 89
This table illustrates the step-by-step process of subtracting two-digit numbers, including lining up, subtracting by place value, and checking with addition.

Our answer is correct.

Try It 1.61

Subtract and then check by adding: 86−54.86−54.

Try It 1.62

Subtract and then check by adding: 99−74.99−74.

When we modeled subtracting 2626 from 43,43, we exchanged 11 ten for 1010 ones. When we do this without the model, we say we borrow 11 from the tens place and add 1010 to the ones place.

How To

Find the difference of whole numbers.

  1. Step 1. Write the numbers so each place value lines up vertically.
  2. Step 2. Subtract the digits in each place value. Work from right to left starting with the ones place. If the digit on top is less than the digit below, borrow as needed.
  3. Step 3. Continue subtracting each place value from right to left, borrowing if needed.
  4. Step 4. Check by adding.

Example 1.32

Subtract: 43−26.43−26.

Solution

Write the numbers so each place value lines up vertically. A vertical subtraction problem is displayed, showing 43 minus 26, set up for manual calculation. A horizontal line underneath indicates where the answer would be placed.
Subtract the ones. We cannot subtract 6 from 3, so we borrow 1 ten. This makes 3 tens and 13 ones. We write these numbers above each place and cross out the original digits. A vertical subtraction problem setup, demonstrating the regrouping (borrowing) method for 43 minus 26. The 4 is shown as 3, and the 3 as 13, preparing for the subtraction calculation.
Now we can subtract the ones. 13−6=7.13−6=7. We write the 7 in the ones place in the difference. ..
Now we subtract the tens. 3−2=1.3−2=1. We write the 1 in the tens place in the difference. A handwritten example demonstrating subtraction with borrowing, showing 43 minus 26 equals 17.
Check by adding.

A vertical addition problem showing 17 plus 26 equals 43, with a black checkmark verifying the correct answer. The numbers are neatly aligned, demonstrating a basic arithmetic operation.
 Our answer is correct.
A step-by-step visual guide demonstrating the process of subtracting numbers with borrowing, including textual instructions and corresponding illustrations.

Try It 1.63

Subtract and then check by adding: 93−58.93−58.

Try It 1.64

Subtract and then check by adding: 81−39.81−39.

Example 1.33

Subtract and then check by adding: 207−64.207−64.

Solution

Write the numbers so each place value lines up vertically. A vertical subtraction problem shows 207 minus 64, with a line beneath to indicate the answer should follow.
Subtract the ones. 7−4=3.7−4=3.
Write the 3 in the ones place in the difference.		 ..
Subtract the tens. We cannot subtract 6 from 0 so we borrow 1 hundred and add 10 tens to the 0 tens we had. This makes a total of 10 tens. We write 10 above the tens place and cross out the 0. Then we cross out the 2 in the hundreds place and write 1 above it. A vertical subtraction problem of 207 - 64, demonstrating the initial borrowing from the hundreds to the tens place, and the calculation of the units digit as 3.
Now we subtract the tens. 10−6=4.10−6=4. We write the 4 in the tens place in the difference. A vertical subtraction problem demonstrating the borrowing method. The calculation shows 107 minus 64, resulting in 43, with regrouping indicated by the crossed-out digits and new values above them.
Finally, subtract the hundreds. There is no digit in the hundreds place in the bottom number so we can imagine a 0 in that place. Since 1−0=1,1−0=1, we write 1 in the hundreds place in the difference.A vertical subtraction problem showing 207 minus 64, with borrowing steps indicated, resulting in the answer 143.
Check by adding.
..
Our answer is correct.
This table illustrates the step-by-step process of subtracting multi-digit numbers with borrowing, followed by a verification step.

Try It 1.65

Subtract and then check by adding: 439−52.439−52.

Try It 1.66

Subtract and then check by adding: 318−75.318−75.

Example 1.34

Subtract and then check by adding: 910−586.910−586.

Solution

Write the numbers so each place value lines up vertically. A vertical subtraction problem is presented, with 910 on top and 586 below it, separated by a horizontal line, and a minus sign to the left of 586, indicating the operation.
Subtract the ones. We cannot subtract 6 from 0, so we borrow 1 ten and add 10 ones to the 0 ones we had. This makes 10 ones. We write a 0 above the tens place and cross out the 1. We write the 10 above the ones place and cross out the 0. Now we can subtract the ones. 10−6=4.10−6=4.A vertical subtraction problem setup, showing 910 minus 586. The image illustrates the first step of borrowing for the ones column, where the '1' in the tens place becomes '0' and the '0' becomes '10'.
Write the 4 in the ones place of the difference. ..
Subtract the tens. We cannot subtract 8 from 0, so we borrow 1 hundred and add 10 tens to the 0 tens we had, which gives us 10 tens. Write 8 above the hundreds place and cross out the 9. Write 10 above the tens place. ..
Now we can subtract the tens. 10−8=210−8=2. An image illustrating long subtraction with borrowing: 910 minus 586. The borrowing steps for units and tens are shown, yielding 4 and 2 respectively. However, the final result of 24 is incorrect, as it should be 324.
Subtract the hundreds place. 8−5=38−5=3 Write the 3 in the hundreds place in the difference. Vertical subtraction problem 910 - 586 = 324, demonstrating the regrouping technique used to solve it.
Check by adding.

A vertical addition problem showing 324 plus 586 equals 910. The carries are indicated above the numbers, and a checkmark confirms the correctness of the sum.

Our answer is correct.
Step-by-step guide demonstrating multi-digit subtraction with borrowing, showing instructional text alongside visual examples and a final check.

Try It 1.67

Subtract and then check by adding: 832−376.832−376.

Try It 1.68

Subtract and then check by adding: 847−578.847−578.

Example 1.35

Subtract and then check by adding: 2,162−479.2,162−479.

Solution

Write the numbers so each place value lines up vertically. CNX_BMath_Figure_01_03_028_img-02.png
Subtract the ones. Since we cannot subtract 9 from 2, borrow 1 ten and add 10 ones to the 2 ones to make 12 ones. Write 5 above the tens place and cross out the 6. Write 12 above the ones place and cross out the 2. A vertical subtraction problem of 2162 minus 479, illustrating the borrowing process. The '2' in the ones place becomes '12' by borrowing from the '6' in the tens place, which then becomes '5'.
Now we can subtract the ones. 12−9=312−9=3
Write 3 in the ones place in the difference. A vertical subtraction problem where 2162 is being subtracted by 479. The image shows the first step of borrowing, where the '2' in the ones place becomes '12' and the '6' in the tens place becomes '5', resulting in '3' for the ones column.
Subtract the tens. Since we cannot subtract 7 from 5, borrow 1 hundred and add 10 tens to the 5 tens to make 15 tens. Write 0 above the hundreds place and cross out the 1. Write 15 above the tens place.CNX_BMath_Figure_01_03_028_img-06.png
Now we can subtract the tens.15−7=815−7=8
Write 8 in the tens place in the difference.CNX_BMath_Figure_01_03_028_img-05.png
Now we can subtract the hundreds.A step-by-step vertical subtraction problem demonstrating the borrowing or regrouping method, with 2062 minus 479 resulting in 1583.
Write 6 in the hundreds place in the difference.A long subtraction problem showing 2162 minus 479, with borrowing indicated, resulting in 1683. The final answer, 1683, is partially highlighted in red (the '6').
Subtract the thousands. There is no digit in the thousands place of the bottom number, so we imagine a 0. 1−0=1.1−0=1. Write 1 in the thousands place of the difference.CNX_BMath_Figure_01_03_028_img-09.png
Check by adding.

11,61813+479______2,162✓11,61813+479______2,162✓

Our answer is correct.

Try It 1.69

Subtract and then check by adding: 4,585−697.4,585−697.

Try It 1.70

Subtract and then check by adding: 5,637−899.5,637−899.

Translate Word Phrases to Math Notation

As with addition, word phrases can tell us to operate on two numbers using subtraction. To translate from a word phrase to math notation, we look for key words that indicate subtraction. Some of the words that indicate subtraction are listed in Table 1.3.

Operation Word Phrase Example Expression
Subtraction minus 55 minus 11 5−15−1
difference the difference of 99 and 44 9−49−4
decreased by 77 decreased by 33 7−37−3
less than 55 less than 88 8−58−5
subtracted from 11 subtracted from 66 6−16−1
Table 1.3

Example 1.36

Translate and then simplify:

  1. ⓐ the difference of 1313 and 88
  2. ⓑ subtract 2424 from 4343

Solution

  • ⓐ

    The word difference tells us to subtract the two numbers. The numbers stay in the same order as in the phrase.

    the difference of 13 and 8
    Translate. 13−813−8
    Simplify. 5
    Steps to translate a verbal mathematical phrase into an expression and simplify it to its numerical value.
  • ⓑ

    The words subtract from tells us to take the first number away from the second. We must be careful to get the order correct.

    subtract 24 from 43
    Translate. 43−2443−24
    Simplify. 19
    This table illustrates the steps to translate a word problem into a mathematical expression and then simplify it to a numerical result.

Try It 1.71

Translate and simplify:

  1. ⓐ the difference of 1414 and 99
  2. ⓑ subtract 2121 from 3737

Try It 1.72

Translate and simplify:

  1. ⓐ 1111 decreased by 66
  2. ⓑ 1818 less than 6767

Subtract Whole Numbers in Applications

To solve applications with subtraction, we will use the same plan that we used with addition. First, we need to determine what we are asked to find. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question, using the appropriate units.

Example 1.37

The temperature in Chicago one morning was 7373 degrees Fahrenheit. A cold front arrived and by noon the temperature was 2727 degrees Fahrenheit. What was the difference between the temperature in the morning and the temperature at noon?

Solution

We are asked to find the difference between the morning temperature and the noon temperature.

Write a phrase. the difference of 73 and 27
Translate to math notation. Difference tells us to subtract. 73−2773−27
Then we do the subtraction. The vertical subtraction of 73 by 27, resulting in 46, with the borrowing process from the tens column (7 to 6, 3 to 13) clearly shown.
Write a sentence to answer the question. The difference in temperatures was 46 degrees Fahrenheit.
This table illustrates the step-by-step process of translating a word phrase into a subtraction problem and calculating its numerical difference.

Try It 1.73

The high temperature on June1stJune1st in Boston was 7777 degrees Fahrenheit, and the low temperature was 5858 degrees Fahrenheit. What was the difference between the high and low temperatures?

Try It 1.74

The weather forecast for June 22 in St Louis predicts a high temperature of 9090 degrees Fahrenheit and a low of 7373 degrees Fahrenheit. What is the difference between the predicted high and low temperatures?

Example 1.38

A washing machine is on sale for $399.$399. Its regular price is $588.$588. What is the difference between the regular price and the sale price?

Solution

We are asked to find the difference between the regular price and the sale price.

Write a phrase. the difference between 588 and 399
Translate to math notation. 588−399588−399
Subtract. ..
Write a sentence to answer the question. The difference between the regular price and the sale price is $189.
This table illustrates the step-by-step process of solving a subtraction word problem, from phrasing to the final answer.

Try It 1.75

A television set is on sale for $499.$499. Its regular price is $648.$648. What is the difference between the regular price and the sale price?

Try It 1.76

A patio set is on sale for $149.$149. Its regular price is $285.$285. What is the difference between the regular price and the sale price?

Section 1.3 Exercises

Practice Makes Perfect

Use Subtraction Notation

In the following exercises, translate from math notation to words.

141.

15 − 9 15 − 9

142.

18 − 16 18 − 16

143.

42 − 35 42 − 35

144.

83 − 64 83 − 64

145.

675 − 350 675 − 350

146.

790 − 525 790 − 525

Model Subtraction of Whole Numbers

In the following exercises, model the subtraction.

147.

5 − 2 5 − 2

148.

8 − 4 8 − 4

149.

6 − 3 6 − 3

150.

7 − 5 7 − 5

151.

18 − 5 18 − 5

152.

19 − 8 19 − 8

153.

17 − 8 17 − 8

154.

17 − 9 17 − 9

155.

35 − 13 35 − 13

156.

32 − 11 32 − 11

157.

61 − 47 61 − 47

158.

55 − 36 55 − 36

Subtract Whole Numbers

In the following exercises, subtract and then check by adding.

159.

9 − 4 9 − 4

160.

9 − 3 9 − 3

161.

8 − 0 8 − 0

162.

2 − 0 2 − 0

163.

38 − 16 38 − 16

164.

45 − 21 45 − 21

165.

85 − 52 85 − 52

166.

99 − 47 99 − 47

167.

493 − 370 493 − 370

168.

268 − 106 268 − 106

169.

5,946 − 4,625 5,946 − 4,625

170.

7,775 − 3,251 7,775 − 3,251

171.

75 − 47 75 − 47

172.

63 − 59 63 − 59

173.

461 − 239 461 − 239

174.

486 − 257 486 − 257

175.

525 − 179 525 − 179

176.

542 − 288 542 − 288

177.

6,318 − 2,799 6,318 − 2,799

178.

8,153 − 3,978 8,153 − 3,978

179.

2,150 − 964 2,150 − 964

180.

4,245 − 899 4,245 − 899

181.

43,650 − 8,982 43,650 − 8,982

182.

35,162 − 7,885 35,162 − 7,885

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

183.

The difference of 1010 and 33

184.

The difference of 1212 and 88

185.

The difference of 1515 and 44

186.

The difference of 1818 and 77

187.

Subtract 66 from 99

188.

Subtract 88 from 99

189.

Subtract 2828 from 7575

190.

Subtract 5959 from 8181

191.

4545 decreased by 2020

192.

3737 decreased by 2424

193.

9292 decreased by 6767

194.

7575 decreased by 4949

195.

1212 less than 1616

196.

1515 less than 1919

197.

3838 less than 6161

198.

4747 less than 6262

Mixed Practice

In the following exercises, simplify.

199.

76 − 47 76 − 47

200.

91 − 53 91 − 53

201.

256 − 184 256 − 184

202.

305 − 262 305 − 262

203.

719 + 341 719 + 341

204.

647 + 528 647 + 528

205.

2,015 − 1,993 2,015 − 1,993

206.

2,020 − 1,984 2,020 − 1,984

In the following exercises, translate and simplify.

207.

Seventy-five more than thirty-five

208.

Sixty more than ninety-three

209.

1313 less than 4141

210.

2828 less than 3636

211.

The difference of 100100 and 7676

212.

The difference of 1,0001,000 and 945945

Subtract Whole Numbers in Applications

In the following exercises, solve.

213.

Temperature The high temperature on June 22 in Las Vegas was 8080 degrees and the low temperature was 6363 degrees. What was the difference between the high and low temperatures?

214.

Temperature The high temperature on June 11 in Phoenix was 9797 degrees and the low was 7373 degrees. What was the difference between the high and low temperatures?

215.

Class size Olivia’s third grade class has 3535 children. Last year, her second grade class had 2222 children. What is the difference between the number of children in Olivia’s third grade class and her second grade class?

216.

Class size There are 8282 students in the school band and 4646 in the school orchestra. What is the difference between the number of students in the band and the orchestra?

217.

Shopping A mountain bike is on sale for $399.$399. Its regular price is $650.$650. What is the difference between the regular price and the sale price?

218.

Shopping A mattress set is on sale for $755.$755. Its regular price is $1,600.$1,600. What is the difference between the regular price and the sale price?

219.

Savings John wants to buy a laptop that costs $840.$840. He has $685$685 in his savings account. How much more does he need to save in order to buy the laptop?

220.

Banking Mason had $1,125$1,125 in his checking account. He spent $892.$892. How much money does he have left?

Everyday Math

221.

Road trip Noah was driving from Philadelphia to Cincinnati, a distance of 502502 miles. He drove 115115 miles, stopped for gas, and then drove another 230230 miles before lunch. How many more miles did he have to travel?

222.

Test Scores Sara needs 350350 points to pass her course. She scored 75,50,70,and8075,50,70,and80 on her first four tests. How many more points does Sara need to pass the course?

Writing Exercises

223.

Explain how subtraction and addition are related.

224.

How does knowing addition facts help you to subtract numbers?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment table for subtraction skills, allowing students to rate their proficiency as 'Confidently', 'With some help', or 'No-I don't get it!' across five different learning objectives.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?