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# Subtract Whole Numbers

Module by: First Last. E-mail the author

Summary: 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

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## Note:

Before you get started, take this readiness quiz.

1. 1) Model 3+43+4 using base-ten blocks.
If you missed this problem, review (Reference).
2. 2) Add: 324+586.324+586.
If you missed this problem, review (Reference).

## 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

7373

We read 7373 as seven minus three and the result is the difference of seven and three.

### Note: Subtraction Notation:

To describe subtraction, we can use symbols and words.

Operation Notation Expression Read as Result
Subtraction 7373 seven minus three the difference of 77 and 33

### Example 1

#### Problem 1

Translate from math notation to words: 8181 26142614.

##### Solution: 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 resuilt is the difference of twenty-six and fourteen.

### Note:

Translate from math notation to words:

#### Exercise 1

1. 124124
2. 29112911
##### Solution
1. twelve minus four; the difference of twelve and four
2. twenty-nine minus eleven; the difference of twenty-nine and eleven

### Note:

Translate from math notation to words:

#### Exercise 2

1. 112112
2. 29122912
##### Solution
1. eleven minus two; the difference of eleven and two
2. twenty-nine minus twelve; the difference of twenty-nine and twelve

## 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, 73.73.

 We start by modeling the first number, 7. Now take away the second number, 3. We'll circle 3 blocks to show that we are taking them away. Count the number of blocks remaining. There are 4 ones blocks left. We have shown that 7−3=47−3=4.

### Note:

Doing the Manipulative Mathematics activity Model Subtraction of Whole Numbers will help you develop a better understanding of subtracting whole numbers.

### Example 2

#### Problem 1

Model the subtraction: 82.82.

##### Solution: Solution
 8−28−2 means the difference of 8 and 2. Model the first, 8. Take away the second number, 2. Count the number of blocks remaining. There are 6 ones blocks left. We have shown that 8−2=68−2=6.

Model: 96.96.

Model: 61.61.

### Example 3

#### Problem 1

Model the subtraction: 138.138.

##### Solution: Solution
 Model the first number, 13. We use 1 ten and 3 ones. Take away the second number, 8. However, there are not 8 ones, so we will exchange the 1 ten for 10 ones. Now we can take away 8 ones. Count the blocks remaining. 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.

### Note:

#### Exercise 5

Model the subtraction: 127.127.

### Note:

#### Exercise 6

Model the subtraction: 148.148.

### Example 4

#### Problem 1

Model the subtraction: 4326.4326.

##### Solution: Solution

Because 43264326 means 4343 take away 26,26, we begin by modeling the 43.43.

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.

Now we can take away 22 tens and 66 ones.

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

4326=174326=17

### Note:

#### Exercise 7

Model the subtraction: 4227.4227.

### Note:

#### Exercise 8

Model the subtraction: 4529.4529.

## Subtract Whole Numbers

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

We know 73=473=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.

73=4because4+3=7138=5because5+8=134326=17because17+26=4373=4because4+3=7138=5because5+8=134326=17because17+26=43

### Example 5

#### Problem 1

Subtract and then check by adding:

1. 9797
2. 83.83.

##### Solution: Solution
 ⓐ 9−79−7 Subtract 7 from 9. 22 Check with addition.2+7=9✓2+7=9✓
 ⓑ 8−38−3 Subtract 3 from 8. 55 Check with addition.5+3=8✓5+3=8✓

### Note:

#### Exercise 9

Subtract and then check by adding:

7070

##### Solution

7 − 0 = 7; 7 + 0 = 7

### Note:

#### Exercise 10

Subtract and then check by adding:

6262

##### Solution

6 − 2 = 4; 2 + 4 = 6

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 6

#### Problem 1

Subtract and then check by adding: 8961.8961.

##### Solution: 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=8Subtract the tens: 8-6=28-6=2 89 −61____ 28 89 −61____ 28 Check using addition. 28 +61____ 89 28 +61____ 89

Our answer is correct.

### Note:

#### Exercise 11

Subtract and then check by adding: 8654.8654.

##### Solution

86 − 54 = 32 because 54 + 32 = 86

### Note:

#### Exercise 12

Subtract and then check by adding: 9974.9974.

##### Solution

99 − 74 = 25 because 74 + 25 = 99

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.

### Note: 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 7

#### Problem 1

Subtract: 4326.4326.

##### Solution: Solution
 Write the numbers so each place value lines up vertically. 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. 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. Check by adding. Our answer is correct.

### Note:

#### Exercise 13

Subtract and then check by adding: 9358.9358.

##### Solution

93 − 58 = 35 because 58 + 35 = 93

### Note:

#### Exercise 14

Subtract and then check by adding: 8139.8139.

##### Solution

81 − 39 = 42 because 42 + 39 = 81

### Example 8

#### Problem 1

Subtract and then check by adding: 20764.20764.

##### Solution: Solution
 Write the numbers so each place value lines up vertically. Subtract the ones. 7−4=3.7−4=3. Write the 3 in the ones place in the difference. 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. Now we subtract the tens. 10−6=4.10−6=4. We write the 4 in the tens place in the difference. 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. Check by adding. Our answer is correct.

### Note:

#### Exercise 15

Subtract and then check by adding: 43952.43952.

##### Solution

439 − 52 = 387 because 387 + 52 = 439

### Note:

#### Exercise 16

Subtract and then check by adding: 31875.31875.

##### Solution

318 − 75 = 243 because 243 + 75 = 318

### Example 9

#### Problem 1

Subtract and then check by adding: 910586.910586.

##### Solution: Solution
 Write the numbers so each place value lines up vertically. Subtract the ones. We cannot subtract 6 from 0, so we borrow 1 ten and add 10 ones to the 10 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. 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. Subtract the hundreds place. 8−5=38−5=3 Write the 3 in the hundreds place in the difference. Check by adding. Our answer is correct.

### Note:

#### Exercise 17

Subtract and then check by adding: 832376.832376.

##### Solution

832 − 376 = 456 because 456 + 376 = 832

### Note:

#### Exercise 18

Subtract and then check by adding: 847578.847578.

##### Solution

847 − 578 = 269 because 269 + 578 = 847

### Example 10

#### Problem 1

Subtract and then check by adding: 2,162479.2,162479.

##### Solution: Solution
 Write the numbers so each place values line up vertically. 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. Now we can subtract the ones. 12−9=312−9=3 Write 3 in the ones place in the difference. 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. Now we can subtract the tens. 15−7=815−7=8 Write 8 in the tens place in the difference. Now we can subtract the hundreds. Write 6 in the hundreds place in the difference. 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. Check by adding.11,61813+479______2,162✓11,61813+479______2,162✓

Our answer is correct.

### Note:

#### Exercise 19

Subtract and then check by adding: 4,585697.4,585697.

##### Solution

4,585 − 697 = 3,888 because 3,888 + 697 = 4,585

### Note:

#### Exercise 20

Subtract and then check by adding: 5,637899.5,637899.

##### Solution

5,637 − 899 = 4,738 because 4,738 + 899 = 5,637

## 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 12.

Table 12
Operation Word Phrase Example Expression
Subtraction minus 55 minus 11 5151
difference the difference of 99 and 44 9494
decreased by 77 decreased by 33 7373
less than 55 less than 88 8585
subtracted from 11 subtracted from 66 6161

### Example 11

#### Problem 1

Translate and then simplify:

1. the difference of 1313 and 88
2. subtract 2424 from 4343
##### Solution: 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
• The words subtract from tells us to take the second number away from the first. We must be careful to get the order correct.

 subtract 24 from 43 Translate. 43−2443−24 Simplify. 19

### Note:

#### Exercise 21

Translate and simplify:

1. the difference of 1414 and 99
2. subtract 2121 from 3737
##### Solution
1. 14 − 9 = 5
2. 37 − 21 = 16

### Note:

#### Exercise 22

Translate and simplify:

1. 1111 decreased by 66
2. 1818 less than 6767
##### Solution
1. 11 − 6 = 5
2. 67 − 18 = 49

## 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 12

#### Problem 1

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: 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. Write a sentence to answer the question. The difference in temperatures was 46 degrees Fahrenheit.

### Note:

#### Exercise 23

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?

##### Solution

The difference is 19 degrees Fahrenheit.

### Note:

#### Exercise 24

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?

##### Solution

The difference is 17 degrees Fahrenheit.

### Example 13

#### Problem 1

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: Solution

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

### Note:

#### Exercise 26

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?

#### Exercise 104

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?

#### Exercise 105

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?

### Everyday Math

#### Exercise 107

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?

157 miles

#### Exercise 108

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

#### Exercise 109

Explain how subtraction and addition are related.

##### Solution

Answers may vary.

#### Exercise 110

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.

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

## Glossary

difference:
The difference is the result of subtracting two or more numbers.

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