1.3 Subtract Whole Numbers

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 $3$ from $7$ is

$7 minus 3$

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

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 $base-10$ blocks. Remember a block represents 1 and a rod represents 10. Let’s start by modeling the subtraction expression we just considered, $7 minus 3.$

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 $7 - 3 = 4$ .

Example 2

Model the subtraction: $8 minus 2.$

$8 minus 2$ 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 $8 - 2 = 6$ .

Example 3

Model the subtraction: $13 minus 8.$

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 $13 - 8 = 5$ .

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

Subtract Whole Numbers

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

We know $7 - 3 = 4$ because $4 + 3 = 7 .$ 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.

$\begin{matrix} 7 - 3 = 4 & because & 4 + 3 = 7 \\ 13 - 8 = 5 & because & 5 + 8 = 13 \\ 43 - 26 = 17 & because & 17 + 26 = 43 \end{matrix}$

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

Subtract and then check by adding: $89 minus 61.$

Write the numbers so the ones and tens digits line up vertically.	$89; minus 61 ____$
Subtract the digits in each place value. Subtract the ones: $9 - 1 = 8$ Subtract the tens: $8 - 6 = 2$	$89; minus 61 ____; 28$
Check using addition. $28; plus 61 ____; 89$

Our answer is correct.

When we modeled subtracting $26$ from $43,$ we exchanged $1$ ten for $10$ ones. When we do this without the model, we say we borrow $1$ from the tens place and add $10$ to the ones place.

Example 7

Subtract: $43 minus 26.$

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. $13 - 6 = 7.$ We write the 7 in the ones place in the difference.
Now we subtract the tens. $3 - 2 = 1.$ We write the 1 in the tens place in the difference.
Check by adding. Our answer is correct.

Example 8

Subtract and then check by adding: $207 minus 64.$

Write the numbers so each place value lines up vertically.
Subtract the ones. $7 - 4 = 3.$ 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.
Now we subtract the tens. $10 - 6 = 4.$ 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 $1 - 0 = 1,$ we write 1 in the hundreds place in the difference.
Check by adding. Our answer is correct.

Example 9

Subtract and then check by adding: $910 minus 586.$

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 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. $10 - 6 = 4.$
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. $10 - 8 = 2$ .
Subtract the hundreds place. $8 - 5 = 3$ Write the 3 in the hundreds place in the difference.
Check by adding. Our answer is correct.

Example 10

Subtract and then check by adding: $2,162 minus 479.$

Write the numbers so each place value lines 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.	$12 - 9 = 3$
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.	$15 - 7 = 8$
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. $1 - 0 = 1.$ Write 1 in the thousands place of the difference.
Check by adding. $1 1, 6 1 8 1 3; plus 479 ______; 2, 162 ✓$

Our answer is correct.

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.

**Table 1.3**
Operation	Word Phrase	Example	Expression
Subtraction	minus	$5$ minus $1$	$5 minus 1$
	difference	the difference of $9$ and $4$	$9 minus 4$
	decreased by	$7$ decreased by $3$	$7 minus 3$
	less than	$5$ less than $8$	$8 minus 5$
	subtracted from	$1$ subtracted from $6$	$6 minus 1$

Example 11

Translate and then simplify:

ⓐ the difference of $13$ and $8$
ⓑ subtract $24$ from $43$

ⓐ
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. $13 minus 8$
Simplify. 5
ⓑ
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. $43 minus 24$
Simplify. 19

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

The temperature in Chicago one morning was $73$ degrees Fahrenheit. A cold front arrived and by noon the temperature was $27$ degrees Fahrenheit. What was the difference between the temperature in the morning and the temperature at noon?

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.	$73 minus 27$
Then we do the subtraction.
Write a sentence to answer the question.	The difference in temperatures was 46 degrees Fahrenheit.

Example 13

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

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.	$588 minus 399$
Subtract.
Write a sentence to answer the question.	The difference between the regular price and the sale price is $189.

Key Concepts

Operation	Notation	Expression	Read as	Result
Subtraction	$-$	$7 minus 3$	seven minus three	the difference of $7$ and $3$

Subtract whole numbers.
1. Write the numbers so each place value lines up vertically.
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. Continue subtracting each place value from right to left, borrowing if needed.
4. Check by adding.

Practice Makes Perfect

Use Subtraction Notation

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

$15 minus 9$

fifteen minus nine; the difference of fifteen and nine

$18 minus 16$

$42 minus 35$

forty-two minus thirty-five; the difference of forty-two and thirty-five

$83 minus 64$

$675 minus 350$

six hundred seventy-five minus three hundred fifty; the difference of six hundred seventy-five and three hundred fifty

$790 minus 525$

Model Subtraction of Whole Numbers

In the following exercises, model the subtraction.

$5 minus 2$

The image uses blocks to demonstrate the subtraction problem 5 - 2.

$8 minus 4$

$6 minus 3$

The image uses blocks to demonstrate the subtraction problem 6- 3.

$7 minus 5$

$18 minus 5$

The image uses blocks to demonstrate the subtraction problem 18 - 5.

$19 minus 8$

$17 minus 8$

The image uses blocks to demonstrate the subtraction problem 17 - 8.

$17 minus 9$

$35 minus 13$

The image uses blocks to demonstrate the subtraction problem 35 - 13.

$32 minus 11$

$61 minus 47$

The image uses blocks to demonstrate the subtraction problem 61 - 47.

$55 minus 36$

Subtract Whole Numbers

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

$9 minus 4$

$9 minus 3$

$8 minus 0$

$2 minus 0$

$38 minus 16$

$45 minus 21$

$85 minus 52$

$99 minus 47$

$493 minus 370$

123

$268 minus 106$

$5,946 minus 4,625$

1,321

$7,775 minus 3,251$

$75 minus 47$

$63 minus 59$

$461 minus 239$

222

$486 minus 257$

$525 minus 179$

346

$542 minus 288$

$6,318 minus 2,799$

3,519

$8,153 minus 3,978$

$2,150 minus 964$

1,186

$4,245 minus 899$

$43,650 minus 8,982$

34,668

$35,162 minus 7,885$

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

The difference of $10$ and $3$

10 − 3; 7

The difference of $12$ and $8$

The difference of $15$ and $4$

15 − 4; 11

The difference of $18$ and $7$

Subtract $6$ from $9$

9 − 6; 3

Subtract $8$ from $9$

Subtract $28$ from $75$

75 − 28; 47

Subtract $59$ from $81$

$45$ decreased by $20$

45 − 20; 25

$37$ decreased by $24$

$92$ decreased by $67$

92 − 67; 25

$75$ decreased by $49$

$12$ less than $16$

16 − 12; 4

$15$ less than $19$

$38$ less than $61$

61 − 38; 23

$47$ less than $62$

Mixed Practice

In the following exercises, simplify.

$76 minus 47$

$91 minus 53$

$256 minus 184$

$305 minus 262$

$719 plus 341$

1,060

$647 plus 528$

$2,015 minus 1,993$

$2,020 minus 1,984$

In the following exercises, translate and simplify.

Seventy-five more than thirty-five

75 + 35; 110

Sixty more than ninety-three

$13$ less than $41$

41 − 13; 28

$28$ less than $36$

The difference of $100$ and $76$

100 − 76; 24

The difference of $1,000$ and $945$

Subtract Whole Numbers in Applications

In the following exercises, solve.

Temperature The high temperature on June $2$ in Las Vegas was $80$ degrees and the low temperature was $63$ degrees. What was the difference between the high and low temperatures?

The difference between the high and low temperature was 17 degrees

Temperature The high temperature on June $1$ in Phoenix was $97$ degrees and the low was $73$ degrees. What was the difference between the high and low temperatures?

Class size Olivia’s third grade class has $35$ children. Last year, her second grade class had $22$ children. What is the difference between the number of children in Olivia’s third grade class and her second grade class?

The difference between the third grade and second grade was 13 children.

Class size There are $82$ students in the school band and $46$ in the school orchestra. What is the difference between the number of students in the band and the orchestra?

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

The difference between the regular price and sale price is $251.

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

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

John needs to save $155 more.

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

Everyday Math

Road trip Noah was driving from Philadelphia to Cincinnati, a distance of $5 0 2$ miles. He drove $115$ miles, stopped for gas, and then drove another $23 0$ miles before lunch. How many more miles did he have to travel?

157 miles

Test Scores Sara needs $35 0$ points to pass her course. She scored $75, 50, 70, and 80$ on her first four tests. How many more points does Sara need to pass the course?

Writing Exercises

Explain how subtraction and addition are related.

Answers may vary.

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. — **Figure 1.15**

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

ⓐ
	$9 minus 7$
Subtract 7 from 9.	$2$
Check with addition. $2 + 7 = 9 ✓$

ⓑ
	$8 minus 3$
Subtract 3 from 8.	$5$
Check with addition. $5 + 3 = 8 ✓$